# PRIMES: Research Papers

2016 Research Papers

121) Piotr Suwara (MIT) and Albert Yue (PRIMES), An Index-Type Invariant of Knot Diagrams Giving Bounds for Unknotting Framed Unknots (arXiv.org, 7 Jul 2017)

We introduce a new knot diagram invariant called the *Self-Crossing Index* (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves.

We also investigate the relation between SCI and Arnold's curve invariant St, as well as the relation with Hass and Nowik's invariant, which generalizes cowrithe. In particular, the change of SCI under $\Omega$3 moves depends only on the forward/backward character of the move, similar to how the change of St or cowrithe depends only on the positive/negative quality of the move.

120) Alec Leng, Independence of the Miller-Rabin and Lucas Probable Prime Tests (30 Mar 2017)

In the modern age, public-key cryptography has become a vital component for secure online communication. To implement these cryptosystems, rapid primality testing is necessary in order to generate keys. In particular, probabilistic tests are used for their speed, despite the potential for pseudoprimes. So, we examine the commonly used Miller-Rabin and Lucas tests, showing that numbers with many nonwitnesses are usually Carmichael or Lucas-Carmichael numbers in a specific form. We then use these categorizations, through a generalization of Korselt’s criterion, to prove that there are no numbers with many nonwitnesses for both tests, affirming the two tests’ relative independence. As Carmichael and Lucas-Carmichael numbers are in general more difficult for the two tests to deal with, we next search for numbers which are both Carmichael and Lucas-Carmichael numbers, experimentally finding none less than $10^{16}$. We thus conjecture that there are no such composites and, using multivariate calculus with symmetric polynomials, begin developing techniques to prove this.

119) Andrew Gritsevskiy and Adithya Vellal, Development and Biological Analysis of a Neural Network Based Genomic Compression System (3 Mar 2017)

The advent of Next Generation Sequencing (NGS) technologies has resulted in a barrage of genomic data that is now available to the scientific community. This data contains information that is driving fields such as precision medicine and pharmacogenomics, where clinicians use a patient’s genetics in order to develop custom treatments. However, genomic data is immense in size, which makes it extremely costly to store, transport and process. A genomic compression system which takes advantage of intrinsic biological patterns can help reduce the costs associated with this data while also identifying important biological patterns. In this project, we aim to create a compression system which uses unsupervised neural networks to compress genomic data. The complete compression suite, GenComp, is compared to existing genomic data compression methods. The results are then analyzed to discover new biological features of genomic data. Testing showed that GenComp achieves at least 40 times more compression than existing variant compression solutions, while providing comparable decoding times in most applications. GenComp also provides some insight into genetic patterns, which has significant potential to aid in the fields of pharmacogenomics and precision medicine. Our results demonstrate that neural networks can be used to significantly compress genomic data while also assisting in better understanding genetic biology.

118) Vivek Bhupatiraju, John Kuszmaul, and Vinjai Vale, On the Viability of Distributed Consensus by Proof of Space (3 Mar 2017)

In this paper, we present our implementation of Proof of Space (PoS) and our study of its viability in distributed consensus. PoS is a new alternative to the commonly used Proof of Work, which is a protocol at the heart of distributed consensus systems such as Bitcoin. PoS resolves the two major drawbacks of Proof of Work: high energy cost and bias towards individuals with specialized hardware. In PoS, users must store large “hard-to-pebble” PTC graphs, which are recursively generated using subgraphs called superconcentrators. We implemented two types of superconcentrators to examine their differences in performance. Linear superconcentrators are about 1:8 times slower than butterfly superconcentrators, but provide a better lower bound on space consumption. Finally, we discuss our simulation of using PoS to reach consensus in a peer-to-peer network. We conclude that Proof of Space is indeed viable for distributed consensus. To the best of our knowledge, we are the first to implement linear superconcentrators and to simulate the use of PoS to reach consensus on a decentralized network.

117) Albert Yue, An Index-Type Invariant of Knot Diagrams and Bounds for Unknotting Framed Knots (3 Mar 2017)

We introduce a new knot diagram invariant called self-crossing index, or $\mathrm{SCI}$. We found that $\mathrm{SCI}$ changes by at most $\pm 1$ under framed Reidemeister moves, and specifically provides a lower bound for the number of 3 moves. We also found that $\mathrm{SCI}$ is additive under connected sums, and is a Vassiliev invariant of order 1. We also conduct similar calculations with Hass and Nowik's diagram invariant and cowrithe, and present a relationship between forward/backward, ascending/descending, and positive/negative 3 moves.

116) Valerie Zhang, Computer-Based Visualizations and Manipulations of Matching Paths (2 Mar 2017)

Given n points in the 2-D plane, a matching path is a path that starts at one of these n points and ends at a different one without going through any of the other n - 2 points. Matching paths, as well as an important operation called the Hurwitz move, come up naturally in the study of complex algebraic varieties. At the heart of the Hurwitz move is the twist operation, which “twists” one matching path along another to produce a new (third) matching path. Performing the twist operation by hand, however, is not only tedious but also prone to errors and unnecessary complications. Therefore, using computer-based methods to represent matching paths and perform the twist operation makes sense. In this project, which was coded in Java, computer-based methods are developed to perform the twist operation efficiently and accurately, providing a framework for visualizing and manipulating matching paths with computers. The computer program performs fast computations and represents matching paths as simply as possible in a simple visual interface. This program could be utilized when solving open problems in symplectic geometry: potential applications include characterizing the overtwistedness of contact manifolds, as well as better understanding braid group actions.

115) Harshal Sheth, Nihar Sheth, and Aashish Welling, Read-Copy Update in a Garbage Collected Environment (1 Mar 2017)

Read-copy update (RCU) is a synchronization mechanism that allows efficient parallelism when there are a high number of readers compared to writers. The primary use of RCU is in Linux, a highly popular operating system kernel. The Linux kernel is written in C, a language that is not garbage collected, and yet the functionality that RCU provides is effectively that of a “poor man’s garbage collector” (P. E. McKenney). RCU in C is also complicated to use, and this can lead to bugs. The purpose of this paper is to investigate whether RCU implemented in a garbage collected language (Go) is easier to use while delivering comparable performance to RCU in C. This is tested through the implementation and benchmarking of 4 linked lists, 2 using RCU and 2 using mutexes. One RCU linked list and one mutex linked list are implemented in each language. This paper finds that RCU in a garbage collected language is indeed significantly easier to use, has similar overall performance to, and on very high read loads, outperforms, RCU in C.

114) Xiangyao Yu (MIT), Siye Zhu (PRIMES), Justin Kaashoek (PRIMES), Andrew Pavlo (Carnegie Mellon University), and Srinivas Devadas (MIT), Taurus: A Parallel Transaction Recovery Method Based on Fine-Granularity Dependency Tracking (28 Feb 2017)

Logging is crucial to performance in modern multicore main-memory database management systems (DBMSs). Traditional data logging (ARIES) and command logging algorithms enforce a sequential order among log records using a global log sequence number (LSN). Log flushing and recovery after a crash are both performed in the LSN order. This serialization of transaction logging and recovery can limit the system performance at high core count. In this paper, we propose Taurus to break the LSN abstraction and enable parallel logging and recovery by tracking fine-grained dependencies among transactions. The dependency tracking lends Taurus three salient features. (1) Taurus decouples the transaction logging order with commit order and allows transactions to be flushed to persistent storage in parallel independently. Transactions that are persistent before commit can be discovered and ignored by the recovery algorithm using the logged dependency information. (2) Taurus can leverage multiple persistent devices for logging. (3) Taurus can leverage multiple devices and multiple worker threads for parallel recovery. Taurus improves logging and recovery parallelism for both data and command logging. .

113) Louis Golowich (PRIMES), Chiheon Kim (MIT), and Richard Zhou (PRIMES), Maximum Size of a Family of Pairwise Graph-Different Permutations (arXiv.org, 27 Feb 2017)

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.

112) Sathwik Karnik, On the Classification and Algorithmic Analysis of Carmichael Numbers (arXiv.org, 26 Feb 2017)

In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers and prime numbers.

111) Felix Wang, Functional equations in Complex Analysis and Number Theory (26 Feb 2017)

We study the following questions:

(1) What are all solutions to $f\circ \hat{f} = g\circ \hat{g}$ with $f,g,\hat{f},\hat{g}\in\mathbb{C}(X)$ being complex rational functions?

(2) For which rational functions $f(X)$ and $g(X)$ with rational coefficients does the equation $f(a)=g(b)$ have infinitely many solutions with $a,b\in$ $Q$?

We utilize various algebraic, geometric and analytic results in order to resolve both (1) and a variant of (2) in case the numerator of $f(X)-g(Y)$ is an irreducible polynomial in $\mathbb{C}[X,Y]$. Our results have applications in various mathematical fields, such as complex analysis, number theory, and dynamical systems. Our work resolves a 1973 question of Fried, and makes significant progress on a 1924 question of Ritt and a 1997 question of Lyubich and Minsky. In addition, we prove a quantitative refinement of a 2015 conjecture of Cahn, Jones and Spear.

110) Albert Gerovitch, Automatically Improving 3D Neuron Segmentations for Expansion Microscopy Connectomics (25 Feb 2017)

Understanding the geometry of neurons and their connections is key to comprehending brain function. This is the goal of a new optical approach to brain mapping using expansion microscopy (ExM), developed in the Boyden Lab at MIT to replace the traditional approach of electron microscopy. A challenge here is to perform image segmentation to delineate the boundaries of individual neurons. Currently, however, there is no method implemented for assessing a segmentation algorithm’s accuracy in ExM. The aim of this project is to create automated assessment of neuronal segmentation algorithms, enabling their iterative improvement. By automating the process, I aim to devise powerful segmentation algorithms that reveal the “connectome” of a neural circuit. I created software, called SEV-3D, which uses the pixel error and warping error metrics to assess 3D segmentations of single neurons. To allow better assessment beyond a simple numerical score, I visualized the results as a multilayered image. My program runs in a closed loop with a segmentation algorithm, modifying its parameters until the algorithm yields an optimal segmentation. I am further developing my application to enable evaluation of multi-cell segmentations. In the future, I aim to further implement the principles of machine learning to automatically improve the algorithms, yielding even better accuracy.

109) Laura Pierson, Signatures of Stable Multiplicity Spaces in Restrictions of Representations of Symmetric Groups (25 Feb 2017)

Representation theory is a way of studying complex mathematical structures such as groups and algebras by mapping them to linear actions on vector spaces. Recently, Deligne proposed a new way to study the representation theory of finite groups by generalizing the collection of representations of a sequence of groups indexed by positive integer rank to an arbitrary complex rank, creating an abelian tensor category. In this project, we focused on the case of the symmetric groups $S_n,$ the groups of permutations of $n$ objects. Elements of the Deligne category Rep $S_t$ can be constructed by taking a stable sequence of $S_n$ representations for increasing $n$ and interpolating the associated formulas to an arbitrary complex number $t.$ In this project, we studied the case of restriction multiplicity spaces $V_{\lambda,\rho}$, counting the number of copies of an irreducible representation $V_{\rho}$ of $S_{n-k}$ in the restriction $\text{Res}_{S_{n-k}}^{S_n} V_{\lambda}$ of an irreducible representation of $S_n.$ We found formulas for norms of orthogonal basis vectors in these spaces, and ultimately for signatures (the number of basis vectors with positive norm minus the number with negative norm), an invariant that multiplies over tensor products and has important combinatorial connections.

108) Kevin Chang, Upper Bounds for Ordered Ramsey Numbers of Small 1-Orderings (arXiv.org, 7 Feb 2017)

A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_{<} (H)$ is the minimum $n$ such that every edge-2-coloring of the complete graph on the vertex set $\{1, \ldots, n\}$ contains a copy of $H$, the $i$th smallest vertex of which either has order-label $i$ in $H$ or no order-label in $H$. <br />This paper conducts the first systematic study of ordered Ramsey numbers for $1$-orderings of small graphs. We provide upper bounds for $R_{<} (H)$ for each connected $1$-ordering $H$ on $4$ vertices. Additionally, for every $1$-ordering $H$ of the $n$-vertex path $P_n$, we prove that $R_{<} (H) \in O(n)$. Finally, we provide an upper bound for the generalized ordered Ramsey number $R_{<} (K_n, H)$ which can be applied to any $k$-ordering $H$ containing some vertex with order-label $1$.

107) Nikhil Marda, On Equal Point Separation by Planar Cell Decompositions (arXiv.org, 17 Jan 2017)

In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a property of curves called the stabbing number, defined to be the maximum countable number of intersections possible between the curve and a line in the plane. We show that large subsets of $X$ lying on Jordan curves of low stabbing number are an obstacle to equal separation. We further discuss Jordan curves of minimal stabbing number containing $X$. Our results generalize recent bounds on the Erd\H{o}s-Szekeres Conjecture, showing that for fixed $d$ and sufficiently large $n$, if $|X| \ge 2^{c_dn/d + o(n)}$ with $c_d = 1 + O(\frac{1}{\sqrt{d}})$, then there exists a subset of $n$ points lying on a Jordan curve with stabbing number at most $d$.

106) Samuel Cohen and Peter Rowley, Results of Triangles Under Discrete Curve Shortening Flow (7 Jan 2017)

In this paper, we analyze the results of triangles under discrete curve shortening flow, specifically isosceles triangles with top angles greater than $\frac{\pi}{3}$, and scalene triangles. By considering the location of the three vertices of the triangle after some small time $\epsilon$, we use the definition of the derivative to calculate a system of differential equations involving parameters that can describe the triangle. Constructing phase plane diagrams and then analyzing them, we find that the singular behavior of discrete curve shorting flow on isosceles triangles with top angles greater than $\frac{\pi}{3}$ is a point, and for scalene triangles is a line segment.

105) Matthew Hase-Liu (PRIMES) and Nicholas Triantafillou (MIT), Efficient Point-Counting Algorithms for Superelliptic Curves (7 Jan 2017; arXiv.org, 7 Sep 2017)

In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries we express in terms of multinomial coefficients. We present a fast algorithm for counting points on specific trinomial superelliptic curves and a slower, more general method for all superelliptic curves. For the first case, we reduce the problem of simplifying the entries of the Hasse-Witt matrix modulo *p* to a problem of solving quadratic Diophantine equations. For the second case, we extend Bostan et al.'s method for hyperelliptic curves to general superelliptic curves. We believe the methods we describe are asymptotically the most efficient known point-counting algorithms for certain families of trinomial superelliptic curves.

104) Seth Shelley-Abrahamson (MIT) and Alec Sun (PRIMES), Towards a Classification of Finite-Dimensional Representations of Rational Cherednik Algebras of Type D (arXiv.org, 15 Dec 2016)

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c(\lambda^\pm)$ of the rational Cherednik algebra $H_c(D_n, \mathbb{C}^n)$ of type $D$ for symmetric bipartitions $\lambda$ are infinite dimensional for all parameters $c$. In particular, all finite-dimensional irreducible representations of rational Cherednik algebras of type $D$ arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type $B$.

103) Nicholas Guo (PRIMES) and Guangyi Yue (MIT), An Application of Rational Hyperplane Arrangements in Counting Independent Sets of Graphs (arXiv.org, 13 Dec 2016)

We develop a method in counting independent sets of disjoint union of certain types of graphs. This method is based upon the $n$ to 1 covering from the points in the finite field $\mathbb{F}_q^n$ which the characteristic polynomial of a rational hyperplane arrangement in $\mathbb{R}^n$ are counting to the independent sets of the corresponding graph. We show that for graphs $G(k)$ with $k$ vertices corresponding to a class of rational hyperplane arrangements, the number of $n$-element independent sets of any disjoint union $G = G(k_1)+ G(k_2)+\cdots + G(k_s)$ depends only on the total number of vertices in the entire disjoint union, $\sum_i{k_i}$. We also give some results with broader conditions. This new technique has importance in simplifing the complexity of counting independent sets in a disjoint union of graphs, and provides closed-form solutions for certain cases.

102) Yatharth Agarwal (PRIMES), Vishnu Murale (PRIMES), Jason Hennessey (Boston University), Kyle Hogan (Boston University), and Mayank Varia (Boston University), Moving in Next Door: Network Flooding as a Side Channel in Cloud Environments (14-16 Nov 2016), published in Sara Foresti and Giuseppe Persiano, eds., *Cryptology and Network Security: 15th International Conference Proceedings, CANS 2016, Milan, Italy, November 14–16, 2016*, pp. 755-760.

Co-locating multiple tenants’ virtual machines (VMs) on the same host underpins public clouds’ affordability, but sharing physical hardware also exposes consumer VMs to side channel attacks from adversarial co-residents. We demonstrate passive bandwidth measurement to perform traffic analysis attacks on co-located VMs. Our attacks do not assume a privileged position in the network or require any communication between adversarial and victim VMs. Using a single feature in the observed bandwidth data, our algorithm can identify which of 3 potential YouTube videos a co-resident VM streamed with 66 % accuracy. We discuss defense from both a cloud provider’s and a consumer’s perspective, showing that effective defense is difficult to achieve without costly under-utilization on the part of the cloud provider or over-utilization on the part of the consumer.

101) Dhruv Rohatgi, A Connection Between Vector Bundles over Smooth Projective Curves and Representations of Quivers (31 Oct 2016)

We create a partition bijection that yields a partial result on a recent conjecture by Schiffmann relating the problems of counting over a finite field (1) vector bundles over smooth projective curves, and (2) representations of quivers.

100) Aaron Yeiser, A Next Generation Partial Differential Equation Solver (30 Oct 2016)

When solving differential equations in multiple dimensions, mesh generation is required to discretize the geometry of the domain. Current numerical techniques, such as finite element methods, are numerically unstable on meshes containing skinny triangles. Here, we develop a novel numerical method that is numerically stable even on meshes with skinny elements. Our method is spectrally accurate on each element, and the discretization size can be adapted to suit a wide variety of applications. Our algorithm alleviates the current burden on mesh generation algorithms of avoiding skinny triangles, allowing meshes to instead be optimized for the efficient solution of time-dependent partial differential equations. We can also simulate the Navier-Stokes equations at moderate Reynolds numbers with our method.

99) Tanya Khovanova (MIT) and Rafael Saavedra (PRIMES), Discreet Coin Weighings and the Sorting Strategy (arXiv.org, 23 Sep 2016)

In 2007, Alexander Shapovalov posed an old twist on the classical coin weighing problem by asking for strategies that manage to conceal the identities of specific coins while providing general information on the number of fake coins. In 2015, Diaco and Khovanova studied various cases of these "discreet strategies" and introduced the revealing factor, a measure of the information that is revealed.

In this paper we discuss a natural coin weighing strategy which we call the sorting strategy: divide the coins into equal piles and sort them by weight. We study the instances when the strategy is discreet, and given an outcome of the sorting strategy, the possible number of fake coins. We prove that in many cases, the number of fake coins can be any value in an arithmetic progression whose length depends linearly on the number of coins in each pile. We also show the strategy can be discreet when the number of fake coins is any value within an arithmetic subsequence whose length also depends linearly on the number of coins in each pile. We arrive at these results by connecting our work to the classic Frobenius coin problem. In addition, we calculate the revealing factor for the sorting strategy.

98) Kai-Siang Ang (PRIMES) and Laura P. Schaposnik (University of Illinois at Chicago), On the geometry of regular icosahedral capsids containing disymmetrons (arXiv.org, 29 Aug 2016), published in Journal of Structural Biology (19 Jan 2017)

Icosahedral virus capsids are composed of symmetrons, organized arrangements of capsomers. There are three types of symmetrons: disymmetrons, trisymmetrons, and pentasymmetrons, which have different shapes and are centered on the icosahedral 2-fold, 3-fold and 5-fold axes of symmetry, respectively. In 2010 [Sinkovits & Baker] gave a classification of all possible ways of building an icosahedral structure solely from trisymmetrons and pentasymmetrons, which requires the triangulation number T to be odd. In the present paper we incorporate disymmetrons to obtain a geometric classification of icosahedral viruses formed by regular penta-, tri-, and disymmetrons. For every class of solutions, we further provide formulas for symmetron sizes and parity restrictions on h, k, and T numbers. We also present several methods in which invariants may be used to classify a given configuration.

97) Tanya Khovanova (MIT) and Shuheng Niu (PRIMES), *m*-Modular Wythoff (arXiv.org, 2 Aug 2016)

We discuss a variant of Wythoff's Game, $m$-Modular Wythoff's Game, and identify the winning and losing positions for this game.

2015 Research Papers

96) Caleb Ji, Robin Park, and Angela Song, Combinatorial Games of No Strategy (20 Aug 2016)

In this paper, we study a particular class of combinatorial game motivated by previous research conducted by Professor James Propp, called *Games of No Strategy*, or games whose winners are predetermined. Finding the number of ways to play such games often leads to new combinatorial sequences and involves methods from analysis, number theory, and other fields. For the game *Planted Brussel Sprouts*, a variation on the well-known game Sprouts, we find a new proof that the number of ways to play is equal to the number of spanning trees on n vertices, and for *Mozes’ Game of Numbers*, a game studied for its interesting connections with other fields, we use prior work by Alon to calculate the number of ways to play the game for a certain case. Finally, in the game *Binary Fusion*, we show through both algebraic and combinatorial proofs that the number of ways to play generates Catalan’s triangle.

95) Meena Jagadeesan, The Exchange Graphs of Weakly Separated Collections (arXiv.org, 19 Aug 2016)

Weakly separated collections arise in the cluster algebra derived from the Pl\"ucker coordinates on the nonnegative Grassmannian. Oh, Postnikov, and Speyer studied weakly separated collections over a general Grassmann necklace $\mathcal{I}$ and proved the connectivity of every exchange graph. Oh and Speyer later introduced a generalization of exchange graphs that we call $\mathcal{C}$-constant graphs. They characterized these graphs in the smallest two cases. We prove an isomorphism between exchange graphs and a certain class of $\mathcal{C}$-constant graphs. We use this to extend Oh and Speyer's characterization of these graphs to the smallest four cases, and we present a conjecture on a bound on the maximal order of these graphs. In addition, we fully characterize certain classes of these graphs in the special cases of cycles and trees.

94) Nicholas Diaco, Counting Counterfeit Coins: A New Coin Weighing Problem (arXiv.org, 13 Jun 2016)

In 2007, a new variety of the well-known problem of identifying a counterfeit coin using a balance scale was introduced in the sixth International Kolmogorov Math Tournament. This paper offers a comprehensive overview of this new problem by presenting it in the context of the traditional coin weighing puzzle and then explaining what makes the new problem mathematically unique. Two weighing strategies described previously are used to derive lower bounds for the optimal number of admissible situations for given parameters. Additionally, a new weighing procedure is described that can be adapted to provide a solution for a broad spectrum of initial parameters by representing the number of counterfeit coins as a linear combination of positive integers. In closing, we offer a new form of the traditional counterfeit coin problem and provide a lower bound for the number of weighings necessary to solve it.

93) Varun Jain, Novel Relationships Between Circular Planar Graphs and Electrical Networks (20 Feb 2016)

Circular planar graphs are used to model electrical networks, which arise in classical physics. Associated with such a network is a network response matrix, which carries information about how the network behaves in response to certain potential differences. Circular planar graphs can be organized into equivalence classes based upon these response matrices. In each equivalence class, certain fundamental elements are called critical. Additionally, it is known that equivalent graphs are related by certain local transformations. Using wiring diagrams, we first investigate the number of Y-∆ transformations required to transform one critical graph in an equivalence class into another, proving a quartic bound in the order of the graph. Next, we consider positivity phenomena, studying how testing the signs of certain circular minors can be used to determine if a given network response matrix is associated with a particular equivalence class. In particular, we prove a conjecture by Kenyon and Wilson for some cases.

92) Arthur Azvolinsky, Explicit Computations of the Frozen Boundaries of Rhombus Tilings of Polygonal Domains (12 Feb 2016)

Consider a polygonal domain $\Omega$ drawn on a regular triangular lattice. A \textit{rhombus tiling} of $\Omega$ is defined as a complete covering of the domain with $60^{\textrm{o}}$-rhombi, where each one is obtained by gluing two neighboring triangles together. We consider a uniform measure on the set of all tilings of $\Omega$. As the mesh size of the lattice approaches zero while the polygon remains fixed, a random tiling approaches a deterministic limit shape. An important phenomenon that occurs with the convergence towards a limit shape is the formation of \textit{frozen facets}; that is, areas where there are asymptotically tiles of only one particular type. The sharp boundary between these ordered facet formations and the disordered region is a curve inscribed in $\Omega$. This inscribed curve is defined as the \textit{frozen boundary}. The goal of this project was to understand the purely algebraic approach, elaborated on in a paper by Kenyon and Okounkov, to the problem of explicitly computing the frozen boundary. We will present our results for a number of special cases we considered.

91) David Amirault, Better Bounds on the Rate of Non-Witnesses of Lucas Pseudoprimes (3 Feb 2016)

Efficient primality testing is fundamental to modern cryptography for the purpose of key generation. Different primality tests may be compared using their runtimes and rates of non-witnesses. With the Lucas primality test, we analyze the frequency of Lucas pseudoprimes using MATLAB. We prove that a composite integer* n* can be a strong Lucas pseudoprime to at most ^{1}⁄_{6} of parameters *P*, *Q* unless *n* belongs to a short list of exception cases, thus improving the bound from the previous result of ^{4}⁄_{15}: We also explore the properties obeyed by such exceptions and how these cases may be handled by an extended version of the Lucas primality test.

90) Daniel Guo, An Infection Spreading Model on Binary Trees (26 Jan 2016)

An important and ongoing topic of research is the study of infectious diseases and the speed at which these diseases spread. Modeling the spread and growth of such diseases leads to a more precise understanding of the phenomenon and accurate predictions of spread in real life. We consider a long-range infection model on an infinite regular binary tree. Given a spreading coefficient $\alpha>1$, the time it takes for the infection to travel from one node to another node below it is exponentially distributed with specific rate functions such as $2^{-k}k^{-\alpha}$ or $\frac{1}{\alpha^k}$, where $k$ is the difference in layer number between the two nodes. We simulate and analyze the time needed for the infection to reach layer $m$ or below starting from the root node. The resulting time is recorded and graphed for different values of $\alpha$ and $m$. Finally, we prove rigorous lower and upper bounds for the infection time, both of which are approximately logarithmic with respect to $m$. The same techniques and results are valid for other regular $d$-ary trees, in which each node has exactly $d$ children where $d>2$.

89) Jacob Klegar, Bounded Tiling-Harmonic Functions on the Integer Lattice (25 Jan 2016)

Tiling-harmonic functions are a class of functions on square tilings that minimize a specific energy. These functions may provide a useful tool in studying square Sierpinski carpets. In this paper we show two new Maximum Modulus Principles for these functions, prove Harnack's Inequality, and give a proof that the set of tiling-harmonic functions is closed. One of these Maximum Modulus Principles is used to show that bounded infinite tiling-harmonic functions must have arbitrarily long constant lines. Additionally, we give three sufficient conditions for tiling-harmonic functions to be constant. Finally, we explore comparisons between tiling and graph-harmonic functions, especially in regards to oscillating boundary values.

88) Richard Yi, A Probability-Based Model of Traffic Flow (22 Jan 2016)

Describing the behavior of traffic via mathematical modeling and computer simulation has been a challenge confronted by mathematicians in various ways throughout the last century. In this project, we introduce various existing traffic flow models and present a new, probability-based model that is a hybrid of the microscopic and macroscopic views, drawing upon current ideas in traffic flow theory. We examine the correlations found in the data of our computer simulation. We hope that our results could help civil engineers implement efficient road systems that fit their needs, as well as contribute toward the design of safely operating unmanned vehicles.

87) Kenz Kallal, Matthew Lipman, and Felix Wang, Equal Compositions of Rational Functions (21 Jan 2016)

We study the following questions:

(1) What are all solutions to $f\circ \hat{f} = g\circ \hat{g}$ in complex rational functions $f,g\in\mathbb{C}(X)$ and meromorphic functions $\hat{f}, \hat{g}$ on the complex plane?

(2) For which rational functions $f(X)$ and $g(X)$ with coefficients in an algebraic number field $K$ does the equation $f(a)=g(b)$ have infinitely many solutions with $a,b\in K$?

We utilize various algebraic, geometric and analytic results in order to resolve both questions in the case that the numerator of $f(X)-g(Y)$ is an irreducible polynomial in $\mathbb{C}[X,Y]$ of sufficiently large degree. Our work answers a 1973 question of Fried in all but finitely many cases, and makes significant progress towards answering a 1924 question of Ritt and a 1997 question of Lyubich and Minsky.

86) Dhruv Medarametla, Bounding Norms of Locally Random Matrices (21 Jan 2016)

Recently, several papers proving lower bounds for the performance of the Sum Of Squares Hierarchy on the planted clique problem have been published. A crucial part of all four papers is probabilistically bounding the norms of certain \locally random" matrices. In these matrices, the entries are not completely independent of each other, but rather depend upon a few edges of the input graph. In this paper, we study the norms of these locally random matrices. We start by bounding the norms of simple locally random matrices, whose entries depend on a bipartite graph *H* and a random graph *G*; we then generalize this result by bounding the norms of complex locally random matrices, matrices based o of a much more general graph *H* and a random graph *G*. For both cases, we prove almost-tight probabilistic bounds on the asymptotic behavior of the norms of these matrices.

85) Rachel Zhang, Statistics of Intersections of Curves on Surfaces (19 Jan 2016)

Each orientable surface with nonempty boundary can be associated with a planar model,
whose edges can then be labeled with letters that read out a surface word. Then, the curve
word of a free homotopy class of closed curves on a surface is the minimal sequence of edges of
the planar model through which a curve in the class passes. The length of a class of curves is
defined to be the number of letters in its curve word.
We fix a surface and its corresponding planar model.

Fix a free homotopy class of curves ω on the surface. For another class of curves *c*, let *i*(ω; *c*) be the minimal number of intersections
of curves in ω and *c*. In this paper, we show that the mean of the distribution of *i*(ω; *c*), for
random curve *c* of length *n*, grows proportionally with *n* and approaches μ(ω) ⋅ *n* for a constant
μ(ω). We also give an algorithm to compute μ(ω) and have written a program that calculates
μ(ω) for any curve ω on any surface. In addition, we prove that *i*(ω; *c*) approahces a Gaussian
distribution as *n* → ∞ by viewing the generation of a random curve as a Markov Chain.

84) Cristian Gutu and Fengyao Ding, SecretRoom: An Anonymous Chat Client (16 Jan 2016)

While many people would like to be able to communicate anonymously, the few existing anonymous communication systems sacrifice anonymity for performance, or viceversa. The most popular such app is Tor, which relies on a series of relays to protect anonymity. Though proven to be efficient, Tor does not guarantee anonymity in the presence of strong adversaries like ISPs and government agencies who can conduct indepth traffic analysis. In contrast, our messaging application, SecretRoom, implements an improved version of a secure messaging protocol called Dining Cryptographers Networks (DCNets) to guarantee true anonymity in moderately sized groups. However, unlike traditional DCNets, SecretRoom does not require direct communication between all participants and does not depend on the presence of honest clients for anonymity. By introducing an untrusted server that performs the DCNet protocol on behalf of the clients, SecretRoom manages to reduce the O(*n*^{2}) communication associated with traditional DCNets to O(*n*) for *n* clients. Moreover, by introducing artificially intelligent clients, SecretRoom makes the anonymity set size independent of the number of “real” clients. Ultimately SecretRoom reduces the communication to O(*n*) and allows the DCNet protocol to scale to hundreds of clients compared to a few tens of clients in traditional DCNets.

83) Girishvar Venkat, Signatures of the Contravariant Form on Representations of the Hecke Algebra and Rational Cherednik Algebra associated to *G *(*r*,1,*n*) (15 Jan 2016)

The Hecke algebra and rational Cherednik algebra of the group *G *(*r*,1,*n*) are non-commutative algebras that are deformations of certain classical algebras associated to the group. These algebras have numerous applications in representation theory, number theory, algebraic geometry and integrable systems in quantum physics. Consequently, understanding their irreducible representations is important. If the deformation parameters are generic, then these irreducible representations, called Specht modules in the case of the Hecke algebra and Verma modules in the case of the Cherednik algebra, are in bijection with the irreducible representations of *G *(*r*,1,*n*). However, while every irreducible representation of *G *(*r*,1,*n*) is unitary, the Hermitian contravariant form on the Specht modules and Verma modules may only be non-degenerate. Thus, the signature of this form provides a great deal of information about the representations of the algebras that cannot be seen by looking at the group representations. In this paper, we compute the signature of arbitrary Specht modules of the Hecke algebra and use them to give explicit formulas of the parameter values for which these modules are unitary. We also compute asymptotic limits of existing formulas for the signature character of the polynomial representations of the Cherednik algebra which are vastly simpler than the full signature characters and show that these limits are rational functions in *t*. In addition, we show that for half of the parameter values, for each *k*, the degree *k* portion of the polynomial representation is unitary for large enough *n*.

82) Mehtaab Sawhney (PRIMES) and Jonathan Weed (MIT), Further results on arc and bar k-visibility graphs (arXiv.org, 6 Jan 2016)

We consider visibility graphs involving bars and arcs in which lines of sight can pass through up to k objects. We prove a new edge bound for arc k-visibility graphs, provide maximal constructions for arc and semi-arc k-visibility graphs, and give a complete characterization of semi-arc visibility graphs. We show that the family of arc i-visibility graphs is never contained in the family of bar j-visibility graphs for any i and j, and that the family of bar i-visibility graphs is not contained in the family of bar j-visibility graphs for $i \neq j$. We also give the first thickness bounds for arc and semi-arc k-visibility graphs. Finally, we introduce a model for random semi-bar and semi-arc k-visibility graphs and analyze its properties.

81) Harshal Sheth and Aashish Welling, An Implementation and Analysis of a Kernel Network Stack in Go with the CSP Style (30 Dec 2015)

Modern operating system kernels are written in lower level languages such as C. Although the low level functionalities of C are often useful within kernels, they also give rise to several classes of bugs. Kernels written in higher level languages avoid many of these potential problems, at the possible cost of decreased performance. This research evaluates the advantages and disadvantages of a kernel written in a higher level language. To do this, the network stack subsystem of the kernel was implemented in Go with the Communicating Sequential Processes (CSP) style. Go is a high level programming language that supports the CSP style, which recommends splitting large tasks into several smaller ones running in independent \threads". Modules for the major networking protocols, including Ethernet, ARP, IPv4, ICMP, UDP, and TCP, were implemented. In this study, the implemented Go network stack, called GoNet, was compared to a representative network stack written in C. The GoNet code is more readable and generally performs better than that of its C stack counterparts. From this, it can be concluded that Go with CSP style is a viable alternative to C for the language of kernel implementations.

80) Xiangyao Yu (MIT), Hongzhe Liu (PRIMES), Ethan Zou (PRIMES), and Srini Devadas (MIT), Tardis 2.0: An Optimized Time Traveling Coherence Protocol (arXiv.org, 27 Nov 2015)

The scalability of cache coherence protocols is a significant challenge in multicore and other distributed shared memory systems. Traditional snoopy and directory-based coherence protocols are difficult to scale up to many-core systems because of the overhead of broadcasting and storing sharers for each cacheline. Tardis, a recently proposed coherence protocol, shows potential in solving the scalability problem, since it only requires O(logN) storage per cacheline for an N-core system and needs no broadcasting support. The original Tardis protocol, however, only supports the sequential consistency memory model. This limits its applicability in real systems since most processors today implement relaxed consistency models like Total Store Order (TSO). Tardis also incurs large network traffic overhead on some benchmarks due to an excessive number of renew messages. Furthermore, the original Tardis protocol has suboptimal performance when the program uses spinning to communicate between threads. In this paper, we address these downsides of Tardis protocol and make it significantly more practical. Specifically, we discuss the architectural, memory system and protocol changes required in order to implement TSO consistency model on Tardis, and prove that the modified protocol satisfies TSO. We also propose optimizations for better leasing policies and to handle program spinning. Evaluated on 20 benchmarks, optimized Tardis at 64 (256) cores can achieve average performance improvement of 15.8% (8.4%) compared to the baseline Tardis and 1% (3.4%) compared to the baseline directory protocol. Our optimizations also reduce the average network traffic by 4.3% (6.1%) compared to the baseline directory protocol. On this set of benchmarks, optimized Tardis improves on a fullmap directory protocol in the metrics of energy, performance and storage, while being simpler to implement.

79) Allison Paul, Spectral Inference of a Directed Acyclic Graph Using Pairwise Similarities (11 Nov 2015)

A gene ontology graph is a directed acyclic graph (DAG) which represents relationships among biological processes. Inferring such a graph using a gene similarity matrix is NP-hard in general. Here, we propose an approximate algorithm to solve this problem efficiently by reducing the dimensionality of the problem using spectral clustering. We show that the original problem can be simplified to the inference problem of overlapping clusters in a network. We then solve the simplified problem in two steps:first we infer clusters using a spectral clustering technique. Then, we identify possible overlaps among the inferred clusters by identifying maximal cliques over the cluster similarity graph. We illustrate the effectiveness of our method over various synthetic networks in terms of both the performance and computational complexity compared to existing methods.

78) Niket Gowravaram, A Variation of nil-Temperley-Lieb Algebras of type A (26 Sep 2015)

We investigate a variation on the nil-Temperley-Lieb algebras of type A. This variation is formed by removing one of the relations and, in some sense, can be considered as a type B of the algebras. We give a general description of the structure of monomials formed by generators in the algebras. We also show that the dimension of these algebras is the sequence ${2n \choose n}$, by showing that the dimension is the Catalan transform of the sequence $2^n$.

77) Caleb Ji, Tanya Khovanova (MIT), Robin Park, and Angela Song, Chocolate Numbers (arXiv.org, 21 Sep 2015), published in Journal of Integer Sequences, vol. 19 (2016)

In this paper, we consider a game played on a rectangular $m \times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along existing grid lines, until just $mn$ individual squares remain.

This paper enumerates the number of ways to break an $m \times n$ bar, which we call chocolate numbers, and introduces four new sequences related to these numbers. Using various techniques, we prove interesting divisibility results regarding these sequences.

76) Niket Gowravaram and Tanya Khovanova (MIT), On the Structure of nil-Temperley-Lieb Algebras of type A (arXiv.org, 1 Sep 2015)

We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a bijection between the monomials and Dyck paths. We show that the distribution of these monomials by degree is the same as the distribution of Dyck paths by the sum of the heights of the peaks minus the number of peaks.

75) Tanya Khovanova (MIT) and Karan Sarkar, P-positions in Modular Extensions to Nim (arXiv.org, 27 Aug 2015), published in International Journal of Game Theory, vol. 46 (2017)

In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in total. We develop a winning strategy for all $m$ with $2$ heaps and for odd $m$ with any number of heaps.

74) Nicholas Diaco and Tanya Khovanova (MIT), Weighing Coins and Keeping Secrets (arXiv.org, 20 Aug 2015), published in Mathematical Intelligencer (September 2016)

In this expository paper we discuss a relatively new counterfeit coin problem with an unusual goal: maintaining the privacy of, rather than revealing, counterfeit coins in a set of both fake and real coins. We introduce two classes of solutions to this problem --- one that respects the privacy of all the coins and one that respects the privacy of only the fake coins --- and give several results regarding each. We describe and generalize 6 unique strategies that fall into these two categories. Furthermore, we explain conditions for the existence of a solution, as well as showing proof of a solution's optimality in select cases. In order to quantify exactly how much information is revealed by a given solution, we also define the revealing factor and revealing coefficient; these two values additionally act as a means of comparing the relative effectiveness of different solutions. Most importantly, by introducing an array of new concepts, we lay the foundation for future analysis of this very interesting problem, as well as many other problems related to privacy and the transfer of information.

73) Luke Sciarappa, Simple commutative algebras in Deligne's categories Rep($S_t$) (arXiv.org, 24 Jun 2015)

We show that in the Deligne categories $\mathrm{Rep}(S_t)$ for $t$ a transcendental number, the only simple algebra objects are images of simple algebras in the category of representations of a symmetric group under a canonical induction functor. They come in families which interpolate the families of algebras of functions on the cosets of $H\times S_{n-k}$ in $S_n$, for a fixed subgroup $H$ of $S_k$.

2014 Research Papers

72) Geoffrey Fudenberg (Harvard), Maxim Imakaev (MIT), Carolyn Lu (PRIMES), Anton Goloborodko (MIT), Nezar Abdennur (MIT), and Leonid Mirny (MIT), Formation of Chromosomal Domains by Loop Extrusion (bioRxiv, 14 Aug 2015), published in Cell Reports 15:9 (31 May 2016): 2038–2049.

Characterizing how the three-dimensional organization of eukaryotic interphase chromosomes modulates regulatory interactions is an important contemporary challenge. Here we propose an active process underlying the formation of chromosomal domains observed in Hi-C experiments. In this process, cis-acting factors extrude progressively larger loops, but stall at domain boundaries; this dynamically forms loops of various sizes within but not between domains. We studied this mechanism using a polymer model of the chromatin fiber subject to loop extrusion dynamics. We find that systems of dynamically extruded loops can produce domains as observed in Hi-C experiments. Our results demonstrate the plausibility of the loop extrusion mechanism, and posit potential roles of cohesin complexes as a loop-extruding factor, and CTCF as an impediment to loop extrusion at domain boundaries.

71) Kavish Gandhi, Maximal Monochromatic Geodesics in an Antipodal Coloring of Hypercube (4 April 2015)

A geodesic in the hypercube is the shortest possible path between two vertices. Leader and Long (2013) conjectured that, in every antipodal $2$-coloring of the edges of the hypercube, there exists a monochromatic geodesic between antipodal vertices. For this and an equivalent conjecture, we prove the cases $n = 2, 3, 4, 5$. We also examine the \emph{maximum} number of monochromatic geodesics of length $k$ in an antipodal $2$-coloring and find it to be $2^{n-1}(n-k+1)\binom{n-1}{k-1}(k-1)!$. In this case, we classify all colorings in which this maximum occurs. Furthermore, we explore the maximum number of antipodal geodesics in a subgraph of the hypercube with a fixed proportion of edges, providing a conjectured optimal configuration as a lower bound, which, interestingly, contains a constant proportion of geodesics with respect to $n$. Finally, we present a series of smaller results that could be of use in finding an upper bound on the maximum number of antipodal geodesics in such a subgraph of the hypercube.

70) Jesse Geneson (MIT) and Peter M. Tian (PRIMES), Sequences of formation width $4$ and alternation length $5$ (arXiv.org, 13 Feb 2015)

Sequence pattern avoidance is a central topic in combinatorics. A sequence
$s$ contains a sequence $u$ if some subsequence of $s$ can be changed into $u$
by a one-to-one renaming of its letters. If $s$ does not contain $u$, then $s$
avoids $u$. A widely studied extremal function related to pattern avoidance is
$Ex(u, n)$, the maximum length of an $n$-letter sequence that avoids $u$ and
has every $r$ consecutive letters pairwise distinct, where $r$ is the number of
distinct letters in $u$.

We bound $Ex(u, n)$ using the formation width function, $fw(u)$, which is the
minimum $s$ for which there exists $r$ such that any concatenation of $s$
permutations, each on the same $r$ letters, contains $u$. In particular, we
identify every sequence $u$ such that $fw(u)=4$ and $u$ contains $ababa$. The
significance of this result lies in its implication that, for every such
sequence $u$, we have $Ex(u, n) = \Theta(n \alpha(n))$, where $\alpha(n)$
denotes the incredibly slow-growing inverse Ackermann function. We have thus
identified the extremal function of many infinite classes of previously
unidentified sequences.

69) William Wu (PRIMES), Nicolaas Kaashoek (PRIMES), Matthew Weinberg (MIT), Christos Tzamos (MIT), and Costis Daskalakis (MIT), Game Theory based Peer Grading Mechanisms for MOOCs, paper for the Learning at Scale 2015 conference, March 14-18, 2015, Vancouver, BC, Canada (4 February 2015)

An efficient peer grading mechanism is proposed for grading the multitude of assignments in online courses. This novel approach is based on game theory and mechanism design. A set of assumptions and a mathematical model is ratified to simulate the dominant strategy behavior of students in a given mechanism. A benchmark function accounting for grade accuracy and workload is established to quantitatively compare eectiveness and scalability of various mechanisms. After multiple iterations of mechanisms under increasingly realistic assumptions, three are proposed: Calibration, Improved Calibration, and Deduction. The Calibration mechanism performs as predicted by game theory when tested in an online crowd-sourced experiment, but fails when students are assumed to communicate. The Improved Calibration mechanism addresses this assumption, but at the cost of more eort spent grading. The Deduction mechanism performs relatively well in the benchmark, outperforming the Calibration, Improved Calibration, traditional automated, and traditional peer grading systems. The mathematical model and benchmark opens the way for future derivative works to be performed and compared.

68) Alexandria Yu, Towards the classification of unital 7-dimensional commutative algebras (19 Jan 2015)

An *algebra* is a vector space with a compatible product operation. An algebra
is called *commutative* if the product of any two elements is independent of the order
in which they are multiplied. A basic problem is to determine how many unital
commutative algebras exist in a given dimension and to find all of these algebras. This
classification problem has its origin in number theory and algebraic geometry. For dimension
less than or equal to 6, Poonen has completely classified all unital commutative
algebras up to isomorphism. For dimension greater than or equal to 7, the situation is
much more complicated due to the fact that there are infinitely many algebras up to
isomorphism. The purpose of this work is to develop new techniques to classify unital
7-dimensional commutative algebras up to isomorphism. An algebra is called *local* if there exists a unique maximal ideal m. Local algebras are basic building blocks for
general algebras as any finite dimensional unital commutative algebra is isomorphic to
a direct sum of finite dimensional unital commutative local algebras. Hence, in order
to classify all finite dimensional unital commutative algebras, it suffices to classify all
finite dimensional unital commutative local algebras. In this article, we classify all unital
7-dimensional commutative local algebras up to isomorphism with the exception
of the special case *k*_{1} = 3 and *k*_{2} = 3, where, for each positive integer* i*, **m**^{i} is the
subalgebra generated by products of *i *elements in the maximal ideal **m** and *k*_{i} is the
dimension of the quotient algebra **m**^{i}/**m**^{i+1}. When *k*_{2} = 1, we classify all finite dimensional
unital commutative local algebras up to isomorphism. As a byproduct of our classification
theorems, we discover several new classes of unital finite dimensional commutative
algebras.

67) Niket Gowravaram and Uma Roy, Diagrammatic Calculus of Coxeter and Braid Groups (arXiv.org, 15 Mar 2015)

We investigate a novel diagrammatic approach to examining strict actions of a Coxeter group or a braid group on a category. This diagrammatic language, which was developed in a series of papers by Elias, Khovanov and Williamson, provides new tools and methods to attack many problems of current interest in representation theory. In our research we considered a particular problem which arises in this context. To a Coxeter group $W$ one can associate a real hyperplane arrangement, and can consider the complement of these hyperplanes in the complexification $Y_W$. The celebrated $K(\pi,1)$ conjecture states that $Y_W$ should be a classifying space for the pure braid group, and thus a natural quotient ${Y_W}/{W}$ should be a classifying space for the braid group. Salvetti provided a cell complex realization of the quotient, which we refer to as the Salvetti complex. In this paper we investigate a part of the $K(\pi,1)$ conjecture, which we call the $K(\pi,1)$ conjecturette, that states that the second homotopy group of the Salvetti complex is trivial. In this paper we present a diagrammatic proof of the $K(\pi,1)$ conjecturette for a family of braid groups as well as an analogous result for several families of Coxeter groups.

66) Arjun Khandelwal, Compact dot representations in permutation avoidance (3 Mar 2015)

A paper published by a Eriksson et. al in 2001 introduced a new form of representing a permutation, referred to as the compact dot representation, with the goal of constructing a smaller superpattern. We study this representation and give bounds on its size. We also consider a variant of the problem, where limitations on the alphabet size are imposed, and obtain lower bounds. Lastly, we consider the Mobius function of the poset of permutations ordered by containment.

65) Suzy Lou and Max Murin, On the Strongly Regular Graph of Parameters (99, 14, 1, 2) (9 Jan 2015)

In an attempt to find a strongly regular graph of parameters (99; 14; 1; 2) or to disprove its existence, we studied its possible substructure and constructions.

64) Shashwat Kishore (PRIMES) and Augustus Lonergan (MIT), Signatures of Multiplicity Spaces in Tensor Products of* sl*_{2} and *U*_{q}(*sl*_{2}) Representations (9 Jan 2015; arXiv.org, 8 Jun 2015)

We study multiplicity space signatures in tensor products of sl2 and *U*_{q}(*sl*_{2}) representations and their applications. We completely classify definite multiplicity spaces for generic tensor products of *sl*_{2} Verma modules. This provides a classification of a family of unitary representations of a basic quantized quiver variety, one of the first such classifications for any quantized quiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic *sl*_{2} master functions. As a corollary of this bound, we obtain a simple and asymptotically correct approximation for the number of real critical points of a generic *sl*_{2} master function. We obtain a formula for multiplicity space signatures in tensor products of finite dimensional simple *U*_{q}(*sl*_{2}) representations. Our formula also gives multiplicity space signatures in generic tensor products of *sl*_{2} Verma modules and generic tensor products of real *U*_{q}(*sl*_{2}) Verma modules. Our results have relations with knot theory, statistical mechanics, quantum physics, and geometric representation theory.

63) Joseph Zurier, Generalizations of the Joints Problem (9 Jan 2015)

In this paper we explore generalizations of the joints problem introduced by B. Chazelle et al.

62) Nathan Wolfe (PRIMES), Ethan Zou (PRIMES), Ling Ren (MIT), and Xiangyao Yu (MIT), Optimizing Path ORAM for Cloud Storage Applications (arXiv.org, 8 Jan 2015)

We live in a world where our personal data are both valuable and vulnerable to misappropriation through exploitation of security vulnerabilities in online services. For instance, Dropbox, a popular cloud storage tool, has certain security flaws that can be exploited to compromise a user's data, one of which being that a user's access pattern is unprotected. We have thus created an implementation of Path Oblivious RAM (Path ORAM) for Dropbox users to obfuscate path access information to patch this vulnerability. This implementation differs significantly from the standard usage of Path ORAM, in that we introduce several innovations, including a dynamically growing and shrinking tree architecture, multi-block fetching, block packing and the possibility for multi-client use. Our optimizations together produce about a 77% throughput increase and a 60% reduction in necessary tree size; these numbers vary with file size distribution.

61) Brice Huang, Monomization of Power Ideals and Generalized Parking Functions (8 Jan 2015)

A power ideal is an ideal in a polynomial ring generated by powers of homogeneous linear forms. Power ideals arise in many areas of mathematics, including the study of zonotopes, approximation theory, and fat point ideals; in particular, their applications in approximation theory are relevant to work on splines and pertinent to mathematical modeling, industrial design, and computer graphics. For this reason, understanding the structure of power ideals, especially their Hilbert series, is an important problem. Unfortunately, due to the computational complexity of power ideals, this is a difficult problem. Only a few cases of this problem have been solved; efficient ways to compute the Hilbert series of a power ideal are known only for power ideals of certain forms. In this paper, we find an efficient way to compute the Hilbert series of a class of power ideals.

60) Kyle Gettig, Linear Extensions of Acyclic Orientations (7 Jan 2015)

Given a graph, an acyclic orientation of the edges determines a partial ordering of the vertices. This partial ordering has a number of linear extensions,* i.e. *total orderings of the vertices that agree with the partial ordering. The purpose of this paper is twofold. Firstly, properties of the orientation that induces the maximum number of linear extensions are investigated. Due to similarities between the optimal orientation in simple cases and the solution to the Max-Cut Problem, the possibility of a correlation is explored, though with minimal success. Correlations are then explored between the optimal orientation of a graph *G *and the comparability graphs with the minimum number of edges that contain *G* as a subgraph, as well as to certain graphical colorings induced by the orientation. Specifically, small cases of non-comparability graphs are investigated and compared to the known results for comparability graphs. We then explore the optimal orientation for odd anti-cycles and related graphs, proving that the conjectured orientations are optimal in the odd anti-cycle case. In the second part of this paper, the above concepts are extended to random graphs, that is, graphs with probabilities associated with each edge. New definitions and theorems are introduced to create a more intuitive system that agrees with the discrete case when all probabilities are 0 or 1, though complete results for this new system would be much more difficult to prove.

59) Shyam Narayanan, Improving the Speed and Accuracy of the Miller-Rabin Primality Test (7 Jan 2015)

In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer *n*.

58) Peter M. Tian, Extremal Functions of Forbidden Multidimensional Matrices (7 Jan 2015)

We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis.

57) Eric Neyman, Cylindric Young Tableaux and their Properties (7 Jan 2015; earlier version on arXiv.org, 19 Oct 2014)

Cylindric Young tableaux are combinatorial objects that first appeared in the 1990s. A natural extension of the classical notion of a Young tableau, they have since been used several times, most notably by Gessel and Krattenthaler and by Alexander Postnikov. Despite this, relatively little is known about cylindric Young tableaux. This paper is an investigation of the properties of this object. In this paper, we extend the Robinson-Schensted-Knuth Correspondence, a well-known and very useful bijection concerning regular Young tableaux, to be a correspondence between pairs of cylindric tableaux. We use this correspondence to reach further results about cylindric tableaux. We then establish an interpretation of cylindric tableaux in terms of a game involving marble-passing. Next, we demonstrate a generic method to use results concerning cylindric tableaux in order to prove results about skew Young tableaux. We finish with a note on Knuth equivalence and its analog for cylindric tableaux.

56) Yilun Du, On the Algorithmic and Theoretical Exploration of Tiling-Harmonic Functions (6 Jan 2015)

In this paper, we explore a new class of harmonic functions defined on a tiling *T*, a square tiling of a region *D*, in * C*. We define these functions as tiling harmonic functions. We develop an efficient algorithm for computing interior values of tiling harmonic functions and graph harmonic functions in a tiling. Using our algorithm, we find that in general tiling harmonic functions are not generally equivalent to graph harmonic functions. In addition, we prove some theoretical results on the structure of tiling harmonic functions and classify one type of tiling harmonic function.

55) Jessica Li, On the Modeling of Snowflake Growth Using Hexagonal Automata (2 Jan 2015; arXiv.org, 8 May 2015; pubished (with Laura P. Schaposnik) in Physical Review E 93:2 (Feb. 2016))

Snowflake growth is an example of crystallization, a basic phase transition in physics. Studying snowflake growth helps gain fundamental understanding of this basic process and may help produce better crystalline materials and benefit several major industries. The basic theoretical physical mechanisms governing the growth of snowflake are not well understood: whilst current computer modeling methods can generate snowflake images that successfully capture some basic features of actual snowflakes, so far there has been no analysis of these computer models in the literature, and more importantly, certain fundamental features of snowflakes are not well understood. A key challenge of analysis is that the snowflake growth models consist of a large set of partial difference equations, and as in many chaos theory problems, rigorous study is difficult. In this paper we analyze a popular model (Reiter’s model) using a combined approach of mathematical analysis and numerical simulation. We divide a snowflake image into main branches and side branches and define two new variables (growth latency and growth direction) to characterize the growth patterns. We derive a closed form solution of the main branch growth latency using a one dimensional linear model, and compare it with the simulation results using the hexagonal automata. We discover a few interesting patterns of the growth latency and direction of side branches. On the basis of the analysis and the principle of surface free energy minimization, we propose a new geometric rule to incorporate interface control, a basic mechanism of crystallization that is not taken into account in the original Reiter’s model.

54) Amy Chou and Justin Kaashoek, __PuzzleJAR: Automated Constraint-based Generation of Puzzles of Varying Complexity__ (arXiv.org, 20 April 2013)

Engaging students in practicing a wide range of problems facilitates their learning. However, generating fresh problems that have specific characteristics, such as using a certain set of concepts or being of a given difficulty level, is a tedious task for a teacher. In this paper, we present PuzzleJAR, a system that is based on an iterative constraint-based technique for automatically generating problems. The PuzzleJAR system takes as parameters the problem definition, the complexity function, and domain-specific semantics-preserving transformations. We present an instantiation of our technique with automated generation of Sudoku and Fillomino puzzles, and we are currently extending our technique to generate Python programming problems. Since defining complexities of Sudoku and Fillomino puzzles is still an open research question, we developed our own mechanism to define complexity, using machine learning to generate a function for difficulty from puzzles with already known difficulties. Using this technique, PuzzleJAR generated over 200,000 Sudoku puzzles of different sizes (9x9, 16x16, 25x25) and over 10,000 Fillomino puzzles of sizes ranging from 2x2 to 16x16. .

53) Tanya Khovanova, Eric Nie, and Alok Puranik, The Sierpinski Triangle and The Ulam-Warburton Automaton (arXiv.org, 25 Aug 2014), published in Math Horizons (September 2015), reprinted in The Best Writing on Mathematics 2016

This paper is about the beauty of fractals and the surprising connections between them. We will explain the pioneering role that the Sierpinski triangle plays in the Ulam-Warburton automata and show you a number of pictures along the way.

52) Tanya Khovanova and Joshua Xiong, Cookie Monster Plays Games (arXiv.org, 6 July 2014)

We research a combinatorial game based on the Cookie
Monster problem called the Cookie Monster game that
generalizes the games of Nim and Wythoff. We also propose
several combinatorial games that are in between the Cookie
Monster game and Nim. We discuss properties of P-positions
of all of these games.

Each section consists of two parts. The first part is a
story presented from the Cookie Monster's point of view, the
second part is a more abstract discussion of the same ideas
by the authors.

51) Tanya Khovanova and Joshua Xiong, Nim Fractals (arXiv.org, 23 May 2014), published in Journal of Integer Sequences, Vol. 17 (2014)

We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters. We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences.

50) Noah Golowich, Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations (arXiv.org, 13 Apr 2014), published in The Electronic Journal of Combinatorics 21:3 (2014)

A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoićič conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoićič's family of equations indeed have a degree of regularity of $n-1$. We also prove a few extensions of this result.

2013 Research Papers

49) Ritesh Ragavender, Odd Dunkl Operators and nilHecke Algebras (30 May 2014)

Symmetric functions appear in many areas of mathematics
and physics, including enumerative combinatorics, the
representation theory of symmetric groups, statistical
mechanics, and the quantum statistics of ideal gases. In the
commutative (or “even”) case of these symmetric functions,
Kostant and Kumar introduced a nilHecke algebra that
categorifies the quantum group *U*_{q} (*sl*_{2})
. This categorification helps to better understand Khovanov
homology, which has important applications in studying knot
polynomials and gauge theory. Recently, Ellis and Khovanov
initiated the program of “oddification” as an effort to
create a representation theoretic understanding of a new
“odd” Khovanov homology, which often yields more powerful
results than regular Khovanov homology. In this paper, we
contribute to- wards the project of oddification by studying
the odd Dunkl operators of Khongsap and Wang in the setting
of the odd nilHecke algebra. Specifically, we show that odd
divided difference operators can be used to construct odd
Dunkl operators, which we use to give a representation of * sl*_{2} on the algebra of skew polynomials and
evaluate the odd Dunkl Laplacian. We then investigate *q*-analogs
of divided difference operators to introduce new algebras
that are similar to the even and odd nilHecke algebras and
act on *q*-symmetric polynomials. We describe such
algebras for all previously unstudied values of *q*. We
conclude by generalizing a diagrammatic method and
developing the novel method of insertion in order to study *q*-symmetric polynomials from the perspective of
bialgebras.

48) Gabriella Studt, Construction of the higher Bruhat order on the Weyl group of type B (27 May 2014)

Manin and Schechtman defined the Bruhat order on the type
A Weyl group, which is closely associated to the Symmetric
group *S _{n}*, as the order of all pairs of
numbers in {1, 2, ..., n} . They proceeded to define a
series of higher orders. Each higher order is an order on
the subsets of {1, 2, ..., n} of size

*k*, and can be computed using an inductive argument. It is also possible to define each of these higher orders explicitly, and therefore know conclusively the lexicographic orders for all

*k*. It is thought that a closely related concept of lexicographic order exists for the Weyl group of type B, and that a similar method can be used to compute this series of higher orders. The applicability of this method is demonstrated in the paper, and we are able to determine and characterize the higher Bruhat order explicitly for certain

*n*and

*k*. We therefore conjecture the existence of such an order for all

*n*>

*k*,as well as its accompanying properties.

47) Jeffrey Cai, Orbits of a fixed-point subgroup of the symplectic group on partial flag varieties of type A (24 May 2014)

In this paper we compute the orbits of the symplectic
group Sp_{2n} on partial flag varieties GL_{2n }/* P* and on partial flag varieties enhanced
by a vector space, C^{2n} x GL_{2n }/* P*. This extends analogous results proved by
Matsuki on full flags. The general technique used in this
paper is to take the orbits in the full flag case and
determine which orbits remain distinct when the full flag
variety GL_{2n }/* B* is projected down
to the partial flag variety GL_{2n }/* P*.

The recent discovery of a connection between abstract algebra and the classical combinatorial Robinson-Schensted (RS) correspondence has sparked research on related algebraic structures and relationships to new combinatorial bijections, such as the Robinson- Schensted-Knuth (RSK) correspondence, the "mirabolic" RSK correspondence, and the "exotic" RS correspondence. We conjecture an exotic RSK correspondence between the or- bits described in this paper and semistandard bi-tableaux, which would yield an extension to the exotic RS correspondence found in a paper of Henderson and Trapa.

46) John Long, Evidence of Purifying Selection in Mammals (9 May 2014)

The Human Genome Project completed in 2003 gave us a reference genome for the human species. Before the project was completed, it was believed that the primary function of DNA was to code for protein. However, it was discovered that only 2% of the genome consists of regions that code for proteins. The remaining regions of the genome are either functional regions that regulate the coding regions or junk DNA regions that do nothing. The distinct ion between these two types of regions is not completely clear. Evidence of purifying selection, the decrease in frequency of deleterious mutations , is likely a sign that a region is functional. The goal of this project was to find evidence of purifying se lection in newly acquired regions in the human genome that are hypothesized to be functional. The mean Derived Allele Frequency of the featured regions was compared to that of control regions to determine the likelihood of selection.

45) Ravi Jagadeesan, A new Gal(Q/Q)-invariant of dessins d'enfants (arXiv.org, 30 March 2014)

We study the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the category of Belyi functions (finite, \'{e}tale covers of $\mathbb{P}^1_{\overline{\mathbb{Q}}}\setminus \{0,1,\infty\}$). We describe a new combinatorial $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-invariant for a certain class of Belyi functions. As a corollary, we obtain that for all $k < 2^{\sqrt{\frac{2}{3}}}$ and all positive integers $N$, there is an $n \le N$ such that the set of degree $n$ Belyi functions of a particular rational Nielsen class must split into at least $\Omega\left(k^{\sqrt{N}}\right)$ Galois orbits. In addition, we define a new version of the Grothendieck-Teichm\"{u}ller group $\widehat{GT}$ into which $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ embeds.

44) Andrey Grinshpun (MIT), Raj Raina (PRIMES), and Rik Sengupta (MIT), Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques (arXiv.org, 26 Jun 2014)

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of
the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey
$H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of
$F$ so that $F'$ is Ramsey for $H$. Burr, Erdos, and Lovasz defined $s(H)$ to
be the minimum degree of $F$ over all Ramsey $H$-minimal graphs $F$. Define
$H_{t,d}$ to be a graph on $t+1$ vertices consisting of a complete graph on $t$
vertices and one additional vertex of degree $d$. We show that $s(H_{t,d})=d^2$
for all values $1<d\le t$; it was previously known that $s(H_{t,1})=t-1$, so it
is surprising that $s(H_{t,2})=4$ is much smaller.

We also make some further progress on some sparser graphs. Fox and Lin
observed that $s(H)\ge 2\delta(H)-1$ for all graphs $H$, where $\delta(H)$ is
the minimum degree of $H$; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs $H$ without
isolated vertices satisfy $s(H)=2\delta(H)-1$. Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that $d$-regular $3$-connected
triangle-free graphs $H$, with one extra technical constraint, satisfy $s(H) =
2\delta(H)-1$; the extra constraint is that $H$ has a vertex $v$ so that if one
removes $v$ and its neighborhood from $H$, the remainder is connected.

43) Boryana Doyle (PRIMES), Geoffrey
Fudenberg (Harvard), Maxim Imakaev (MIT), and Leonid Mirny (MIT), Chromatin Loops as Allosteric Modulators of Enhancer-Promoter Interactions, published in *PLoS Computational Biology* (23 Oct 2014; earlier version in BioRxiv.org, 26 February
2014)

The classic model of eukaryotic gene expression requires direct spatial contact between a distal enhancer and a proximal promoter. Recent Chromosome Conformation Capture (3C) studies show that enhancers and promoters are embedded in a complex network of looping interactions. Here we use a polymer model of chromatin fiber to investigate whether, and to what extent, looping interactions between elements in the vicinity of an enhancer-promoter pair can influence their contact frequency. Our equilibrium polymer simulations show that a chromatin loop, formed by elements flanking either an enhancer or a promoter, suppresses enhancer-promoter interactions, working as an insulator. A loop formed by elements located in the region between an enhancer and a promoter, on the contrary, facilitates their interactions. We find that different mechanisms underlie insulation and facilitation; insulation occurs due to steric exclusion by the loop, and is a global effect, while facilitation occurs due to an effective shortening of the enhancer-promoter genomic distance, and is a local effect. Consistently, we find that these effects manifest quite differently for* in silico* 3C and microscopy. Our results show that looping interactions that do not directly involve an enhancer-promoter pair can nevertheless significantly modulate their interactions. This phenomenon is analogous to allosteric regulation in proteins, where a conformational change triggered by binding of a regulatory molecule to one site affects the state of another site.

42) William Kuszmaul, A New Approach to
Enumerating Statistics Modulo *n* (arXiv.org, 16
February
2014)

We find a new approach to computing the remainder of a polynomial modulo $x^n-1$; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative $\mathbb{Q}$-algebra, our first main result constructs the remainder simply from the coefficients of residues of the polynomial modulo $\Phi_d(x)$ for each $d\mid n$. Since such residues can often be found to have nice values, this simplifies a number of modular enumeration problems; indeed in some cases, such residues are already known while the related modular enumeration problem has remained unsolved. We list six such cases which our technique makes easy to solve. Our second main result is a formula for the unique polynomial $a$ such that $a \equiv f \mod \Phi_n(x)$ and $a\equiv 0 \mod x^d-1$ for each proper divisor $d$ of $n$.

We find a formula for remainders of $q$-multinomial coefficients and for remainders of $q$-Catalan numbers modulo $q^n-1$, reducing each problem to a finite number of cases for any fixed $n$. In the prior case, we solve an open problem posed by Hartke and Radcliffe. In considering $q$-Catalan numbers modulo $q^n-1$, we discover a cyclic group operation on certain lattice paths which behaves predictably with regard to major index. We also make progress on a problem in modular enumeration on subset sums posed by Kitchloo and Pachter.

41) Ajay Saini, Predictive Modeling of Opinion and Connectivity Dynamics in Social Networks (26 January 2014)

Social networks have been extensively studied in recent years with the aim of understanding how the connectivity of different societies and their subgroups influences the spread of innovations and opinions through human networks. Using data collected from real-world social networks, researchers are able to gain a better understanding of the dynamics of such networks and subsequently model the changes that occur in these networks over time. In our work, we use data from the Social Evolution dataset of the MIT Human Dynamics Lab to develop a data-driven model capable of predicting the trends and long term changes observed in a real- world social network. We demonstrate the effectiveness of the model by predicting changes in both opinion spread and connectivity that reflect the changes observed in our dataset. After validating the model, we use it to understand how different types of social networks behave over time by varying the conditions governing the change of opinions and connectivity. We conclude with a study of opinion propagation under different conditions in which we use the structure and opinion distribution of various networks to identify sets of agents capable of propagating their opinion throughout an entire network. Our results demonstrate the effectiveness of the proposed modeling approach in predicting the future state of social networks and provide further insight into the dynamics of interactions between agents in real-world social networks.

40) Rohil Prasad, * Investigating GCD in Z[√2] (1*1
January
2014)

We attempt to optimize the time needed to calculate greatest common divisors in the Euclidean domain Z[√2].

39) Jin-Woo Bryan Oh, Towards Generalizing Thrackles to Arbitrary Graphs (1 January 2014)

In the 1950s, John Conway came up with the notion of * thrackles*, graphs with embeddings in which no edge
crosses itself, but every pair of distinct edges intersects
each other exactly once. He conjectured that |E(G)| ≤ |V(G)|
for any thrackle G, a question unsolved to this day. In this
paper, we discuss some of the known properties of thrackles
and contribute a few new ones.

Only a few sparse graphs can be thrackles, and so it is
of interest to find an analogous notion that applies to
denser graphs as well. In this paper we introduce a
generalized version of thrackles called *near-thrackles*,
and prove some of their properties. We also discuss a large
number of conjectures about them which seem very obvious but
nonetheless are hard to prove. In the final section, we
introduce *thrackleability*, a number between 0 and 1
that turns out to be an accurate measure of how far away a
graph is from being a thrackle..

38) Junho Won, Lower bounds for
the Crossing Number of the Cartesian Product of a
Vertex-transitive Graph with a Cycle* (1* January
2014)

The minimum number of crossings for all drawings of a given graph $G$ on a plane is called its crossing number, denoted $cr(G)$. Exact crossing numbers are known only for a few families of graphs, and even the crossing number of a complete graph $K_m$ is not known for all $m$. Wenping et al. showed that $cr(K_m\Box C_n)\geqslant n\cdot cr(K_{m+2})$ for $n\geqslant 4$ and $m\geqslant 4$. We adopt their method to find a lower bound for $cr(G\Box C_n)$ where $G$ is a vertex-transitive graph of degree at least 3. We also suggest some particular vertex-transitive graphs of interest, and give two corollaries that give lower bounds for $cr(G\Box C_n)$ in terms of $n$, $cr(G)$, the number of vertices of $G$, and the degree of $G$, which improve on Wenping et al.'s result.

37) Ying Gao, On an Extension of Stanley Depth for Refinement-Ordered Posets (30 December 2013)

The concept of Stanley depth was originally defined for graded modules over commutative rings in 1982 by Richard P. Stanley. However, in 2009 Herzog, Vladiou, and Zheng found a property, ndepth, of posets analogous to the Stanley depths of certain modules, which provides an important link between combinatorics and commutative algebra. Due to this link, there arises the question of what this ndepth is for certain classes of posets.

Because ndepth was only recently defined, much remains to
be discovered about it. In 2009, Biro, Howard, Keller,
Trotter and Young found a lower bound for the ndepth of the
poset of nonempty subsets of {1; 2; ...; n} ordered
by inclusion. In 2010, Wang calculated the ndepth of the
product of chains n^{k} \ 0. However, ndepth
has yet to be studied in relation to many other commonly
found classes of posets. We chose to research the properties
of the ndepths of one such well-known class of posets - the
posets which consist of non-empty partitions of sets ordered
by refinement, which we denote as G_{i}.

We use combinatorial and algebraic methods to find the
ndepths for small posets in G_{i}. We show
that for posets of increasing size in G_{i},
new depth is strictly non-decreasing, and furthermore we
show that ndepth[G_{i}] ≥ [8i/29]
for all i. We also find that for all i,
ndepth[G_{i}] ≤ i through the
proof that ndepth[G_{i+1}] ≤ ndepth[G_{i}]
+ 1.

36) Nihal Gowravaram, Enumeration of Subclasses of (2+2)-free Partially Ordered Sets (26 December 2013)

We investigate avoidance in (2+2)-free partially ordered sets, posets that do not contain any induced subposet isomorphic to the union of two disjoint chains of length two. In particular, we are interested in enumerating the number of partially ordered sets of size N avoiding both 2+2 and some other poset α. For any α of size 3, the results are already well-known. However, out of the 15 such α of size 4, only 2 were previously known. Through the course of this paper, we explicitly enumerate 7 other such α of size 4. Also, we consider the avoidance of three posets simultaneously, 2+2 along with some pair (α,β); it turns out that this enumeration is often clean, and has sometimes surprising results. Furthermore, we turn to the question of Wilf-equivalences in (2+2)-free posets. We show such an equivalence between the Y-shaped and chain posets of size 4 via a direct bijection, and in fact, we extend this to show a Wilf-equivalence between the general chain poset and a general Y-shaped poset of the same size. In this paper, while our focus is on enumeration, we also seek to develop an understanding of the structures of the posets in the subclasses we are studying.

35) Yael Fregier (MIT) and Isaac Xia, Lower Central Series Ideal Quotients Over $\mathbb{F}_p$ and $\mathbb{Z}$ (17 November 2013; arXiv.org, 28 Jun 2015)

Given a graded associative algebra $A$, its lower central series is defined by $L_1 = A$ and $L_{i+1} = [L_i, A]$. We consider successive quotients $N_i(A) = M_i(A) / M_{i+1}(A)$, where $M_i(A) = AL_i(A) A$. These quotients are direct sums of graded components. Our purpose is to describe the $\mathbb{Z}$-module structure of the components; i.e., their free and torsion parts. Following computer exploration using *MAGMA*, two main cases are studied. The first considers $A = A_n / (f_1,\dots, f_m)$, with $A_n$ the free algebra on $n$ generators $\{x_1, \ldots, x_n\}$ over a field of characteristic $p$. The relations $f_i$ are noncommutative polynomials in $x_j^{p^{n_j}},$ for some integers $n_j$. For primes * p > 2*, we prove that $p^{\sum n_j} \mid \text{dim}(N_i(A))$. Moreover, we determine polynomials dividing the Hilbert series of each $N_i(A)$. The second concerns $A = \mathbb{Z} \langle x_1, x_2, \rangle / (x_1^m, x_2^n)$. For $i = 2,3$, the bigraded structure of $N_i(A_2)$ is completely described.

34) Steven Homberg, Finding Enrichments of Functional Annotations for Disease- Associated Single-Nucleotide Polymorphisms (10 November 2013)

Computational analysis of SNP-disease associations from GWAS as well as functional annotations of the genome enables the calculation of a SNP set's enrichment for a disease. These statistical enrichments can be and are calculated with a variety of statistical techniques, but there is no standard statistical method for calculating enrichments. Several entirely different tests are used by different investigators in the field. These tests can also be conducted with several variations in parameters which also lack a standard. In our investigation, we develop a computational tool for conducting various enrichment calculations and, using breast cancer-associated SNPs from a GWAS catalog as a foreground against all GWAS SNPs as a background, test the tool and analyze the relative performance of the various tests. The computational tool will soon be released to the scientific community as a part of the Bioconductor package. Our analysis shows that, for R2 threshold in LD block construction, values around 0.8-0.9 are preferable to those with more lax and more strict thresholds respectively. We find that block-matching tests yield better results than peak-shifting tests. Finally, we find that, in block-matching tests, block tallying using binary scoring, noting whether or not a block has an annotation only, yields the most meaningful results, while weighting LD r2 threshold has no influence.

33) Kavish Gandhi, Noah Golowich, and László Miklós Lovász, Degree of Regularity of Linear Homogeneous Equations (arXiv.org, 27 Sept 2013), published in Journal of Combinatorics 5:2 (2014)

We define a linear homogeneous equation to be strongly r-regular if, when a finite number of inequalities is added to the equation, the system of the equation and inequalities is still r-regular. In this paper, we derive a constraint on the coefficients of a linear homogeneous equation that gives a sufficient condition for the equation to be strongly r-regular. In 2009, Alexeev and Tsimerman introduced a family of equations, each of which is (n-1)-regular but not n-regular, verifying a conjecture of Rado from 1933. We show that these equations are actually strongly (n-1)-regular as a corollary of our results.

32) Leigh Marie Braswell and Tanya
Khovanova, __ On the Cookie Monster Problem__ (arXiv.org, 23 Sept 2013), published in Jennifer Beineke & Jason Rosenhouse, The Mathematics of Various Entertaining Subjects: Research in Recreational Math (Princeton University Press, 2015).

The Cookie Monster Problem supposes that the Cookie Monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The Cookie Monster number of a set is the minimum number of moves the Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars. We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for jars containing cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci sequences. We also construct sequences of k jars such that their Cookie Monster numbers are asymptotically rk, where r is any real number between 0 and 1 inclusive.

31) Vahid Fazel-Rezai, Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns (arXiv.org, 18 Sept 2013), published in The Electronic Journal of Combinatorics 21:2 (2014)

We explore a new type of replacement of patterns in permutations, suggested by James Propp, that does not preserve the length of permutations. In particular, we focus on replacements between 123 and a pattern of two integer elements. We apply these replacements in the classical sense; that is, the elements being replaced need not be adjacent in position or value. Given each replacement, the set of all permutations is partitioned into equivalence classes consisting of permutations reachable from one another through a series of bi-directional replacements. We break the eighteen replacements of interest into four categories by the structure of their classes and fully characterize all of their classes.

30) Jesse Geneson (MIT), Rohil Prasad (PRIMES), and Jonathan Tidor (PRIMES), __Bounding sequence extremal functions with formations__ (arXiv.org, 17 Aug 2013), published in The Electronic Journal of Combinatorics 21:3 (2014)

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters.
If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be
the maximum length of any $r$-sparse sequence with $n$ distinct letters which
has no subsequence isomorphic to $u$. For every sequence $u$ define
$\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which
there exists $r$ such that there is a subsequence isomorphic to $u$ in every
$(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on
$\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation
with the same formation width as $u$.

We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that
$\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n)
=n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for every $l \geq
2$ and $t\geq 3$, such that $\alpha(n)$ denotes the inverse Ackermann function.
Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other
papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$
vertices with no pair of edges intersecting in more than $O(1)$ points.

If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter,
$v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is
obtained from $v$ by only moving the first letter of $v$ to another place in
$v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n)
=\Theta(n\alpha(n))$. Furthermore we prove that
$\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) =
n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})}$ for every $t\geq
2$.

29) Jesse Geneson (MIT), Tanya Khovanova (MIT), and Jonathan Tidor (PRIMES), __Convex geometric (k+2)-quasiplanar representations of semi-bar k-visibility graphs__ (arXiv.org, 3 Jul 2013), published in Discrete Mathematics 331 (2014)

We examine semi-bar visibility graphs in the plane and on a cylinder in which sightlines can pass through k objects. We show every semi-bar k-visibility graph has a (k+2)-quasiplanar representation in the plane with vertices drawn as points in convex position and edges drawn as segments. We also show that the graphs having cylindrical semi-bar k-visibility representations with semi-bars of different lengths are the same as the (2k+2)-degenerate graphs having edge-maximal (k+2)-quasiplanar representations in the plane with vertices drawn as points in convex position and edges drawn as segments.

28) Leigh Marie Braswell and Tanya
Khovanova, __ Cookie Monster Devours Naccis__ (arXiv.org, 18 May 2013),
published in the College Mathematics Journal 45:2 (2014)

In 2002, Cookie Monster appeared in *The Inquisitive
Problem Solver*. The hungry monster wants to empty a set
of jars filled with various numbers of cookies. On each of
his moves, he may choose any subset of jars and take the
same number of cookies from each of those jars. The Cookie
Monster number is the minimum number of moves Cookie Monster
must use to empty all of the jars. This number depends on
the initial distribution of cookies in the jars. We discuss
bounds of the Cookie Monster number and explicitly find the
Cookie Monster number for Fibonacci, Tribonacci and other
nacci sequences.

2012 Research Papers

27) William Kuszmaul and Ziling
Zhou, __Equivalence
classes in S _{n} for three families of
pattern-replacement relations__ (arXiv.org, 20 April 2013)

We study a family of equivalence relations in *S _{n}*,
the group of permutations on

*n*letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of

*S*. In particular, we are interested in the number of classes created in

_{c}*S*by each relation and in characterizing these classes. Imposing the condition that the partition of

_{n}*S*has one nontrivial part containing the cyclic shifts of a single permutation, we find enumerations for the number of nontrivial classes. When the permutation is the identity, we are able to compare the sizes of these classes and connect parts of the problem to Young tableaux and Catalan lattice paths. Imposing the condition that the partition has one nontrivial part containing all of the permutations in

_{c}*S*beginning with 1, we both enumerate and characterize the classes in

_{c}*S*. We do the same for the partition that has two nontrivial parts, one containing all of the permutations in

_{n}*S*beginning with 1, and one containing all of the permutations in

_{c}*S*ending with 1.

_{c}

26) William Kuszmaul, __Counting permutations
modulo pattern-replacement equivalences for three-letter
patterns__ (arXiv.org, 20 April 2013), published in the Electronic Journal of Combinatorics 20:4 (2013)

We study a family of equivalence relations in *S _{n}*,
the group of permutations on

*n*letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of

*S*. When the partition is of

_{c}*S*and has one nontrivial part of size greater than two, we provide formulas for the number of classes created in all unresolved cases. When the partition is of

_{3}*S*and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for 13 of the 14 unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.

_{3}

25) Tanya Khovanova and Ziv Scully, __Efficient
Calculation of Determinants of Symbolic Matrices with Many
Variables__ (arXiv.org, 13 April 2013)

Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant algorithms for numerical matrices are often not the fastest for symbolic matrices with many variables. We compare the performance of two algorithms, fraction-free Gaussian elimination and minor expansion, on symbolic matrices with many variables. We show that, under a simplified theoretical model, minor expansion is faster in most situations. We then propose optimizations for minor expansion and demonstrate their effectiveness with empirical data.

24) Michael Zanger-Tishler and Saarik Kalia, __On the
Winning and Losing Parameters of Schmidt's Game__ (8
April
2013)

First introduced by Wolfgang Schmidt, the (*α*,* β*)-game and
its modifications have been shown to be a powerful tool in
Diophantine approximation, metric number theory, and
dynamical systems. However, natural questions about the
winning-losing parameters of most sets have not been studied
thoroughly even after more than 40 years. There are a few
results in the literature showing that some non-trivial
points and small regions are winning or losing, but complete
pictures remain largely unknown. Our main goal in this paper
is to provide as much detail as possible about the global
pictures of winning-losing parameters for some interesting
families of sets.

23) Sheela Devadas and Steven Sam, __Representations
of Cherednik algebras of G (m, r, n) in positive
characteristic__ (arXiv.org, 3 April 2013; forthcoming
in Journal of Commutative Algebra)

We study lowest-weight irreducible representations of
rational Cherednik algebras attached to the complex
reflection groups *G(m, r, n)* in characteristic *p*.
Our approach is mostly from the perspective of commutative
algebra. By studying the kernel of the contravariant
bilinear form on Verma modules, we obtain formulas for
Hilbert series of irreducible representations in a number of
cases, and present conjectures in other cases. We observe
that the form of the Hilbert series of the irreducible
representations and the generators of the kernel tend to be
determined by the value of *n *modulo *p*, and are
related to special classes of subspace arrangements. Perhaps
the most novel (conjectural) discovery from the commutative
algebra perspective is that the kernel can be given the
structure of a "matrix regular sequence" in some instances,
which we prove in some small cases.

22) Christina Chen and Nan Li, __Apollonian Equilateral Triangles__ (arXiv.org,
1 March
2013)

Given an equilateral triangle with a the square of its
side length and a point in its plane with *b, c, d *the
squares of the distances from the point to the vertices of
the triangle, it can be computed that *a, b, c, d * satisfy 3(*a*^{2}+*b*^{2}+*c*^{2}+*d*^{2})
= (*a*+*b*+*c*+*d*)^{2}. This
paper derives properties of quadruples of nonnegative
integers (*a; b; c; d*), called triangle quadruples,
satisfying this equation. It is easy to verify that the
operation generating (*a; b; c; a*+*b*+*c*-*d*)
from (*a; b; c; d*) preserves this feature and that it
and analogous ones for the other elements can be represented
by four matrices. We examine in detail the triangle group,
the group with these operations as generators, and
completely classify the orbits of quadruples with respect to
the triangle group action. We also compute the number of
triangle quadruples generated after a certain number of
operations and approximate the number of quadruples bounded
by characteristics such as the maximal element. Finally, we
prove that the triangle group is a hyperbolic Coxeter group
and derive information about the elements of triangle
quadruples by invoking Lie groups. We also generalize the
problem to higher dimensions.

21) Dhroova Aiylam, __Modified Stern-Brocot
sequences__ (arXiv.org, 29 January 2013)

We present the classical Stern-Brocot tree and provide a new proof of the fact that every rational number between 0 and 1 appears in the tree. We then generalize the Stern-Brocot tree to allow for arbitrary choice of starting terms, and prove that in all cases the tree maintains the property that every rational number between the two starting terms appears exactly once.

20) Nihal Gowravaram and Ravi Jagadeesan, __Beyond
alternating permutations: Pattern avoidance in Young
diagrams and tableaux__ (arXiv.org, 28 January
2013), published in the Electronic Journal of Combinatorics 20:4 (2013)

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have (k - 1) ascents followed by a descent, followed by (k - 1) ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding (k - 1)-ascent permutations in terms of the forbidden pattern. Furthermore, we give inequalities in the sizes of sets of pattern-avoiding permutations in this context that arise from further extensions of shape-equivalence type enumerations.

19) Rohil Prasad and Jonathan Tidor, __Optimal Results in
Staged Self-Assembly of Wang Tiles__ (22 January
2013)

The subject of self-assembly deals with the spontaneous creation of ordered systems from simple units and is most often applied in the field of nanotechnology. The self-assembly model of Winfree describes the assembly of Wang tiles, simulating assembly in real-world systems. We use an extension of this model, known as the staged self-assembly model introduced by Demaine et al. that allows for discrete steps to be implemented and permits more diverse constructions. Under this model, we resolve the problem of constructing segments, creating a method to produce them optimally. Generalizing this construction to squares gives a new flexible method for their construction. Changing a parameter of the model, we explore much simpler constructions of complex monotone shapes. Finally, we present an optimal method to build most arbitrary shapes.

18) Aaron Klein, __On Rank Functions of
Graphs__ (6 January 2013)

We study *rank functions* (also known as graph
homomorphisms onto Z), ways of imposing graded poset
structures on graphs. We rst look at a variation on rank
functions called discrete *Lipschitz functions*. We
relate the number of Lipschitz functions of a graph *G * to the number of rank functions of both *G* and *G* X *E*. We then find generating functions that enable us
to compute the number of rank or Lipschitz functions of a
given graph. We look at a subset of graphs called * squarely generated graphs*, which are graphs whose cycle
space has a basis consisting only of 4-cycles. We show that
the number of rank functions of such a graph is proportional
to the number of 3-colorings of the same graph, thereby
connecting rank functions to the Potts model of statistical
mechanics. Lastly, we look at some asymptotics of rank and
Lipschitz functions for various types of graphs.

17) Andrew Xia, __Integrated Gene Expression
Probabilistic Models for Cancer Staging__ (1 January 2013)

The current system for classifying cancer patients' stages was introduced more than one hundred years ago. With the modern advance in technology, many parts of the system have been outdated. Because the current staging system emphasizes surgical procedures that could be harmful to patients, there has been a movement to develop a new Taxonomy, using molecular signatures to potentially avoid surgical testing. This project explores the issues of the current classification system and also looking for a potentially better way to classify cancer patients’ stages. Computerization has made a vast amount of cancer data available online. However, a significant portion of the data is incomplete; some crucial information is missing. It is logical to attempt to develop a system of recovering missing cancer data. Successful completion of this research saves costs and increases efficiency in cancer research and curing. Using various methods, we have shown that cancer stages cannot be simply extrapolated with incomplete data. Furthermore, a new approach of using RNA Sequencing data is studied. RNA Sequencing can potentially become a cost-efficient way to determine a cancer patient’s stage. We have obtained promising results of using RNA sequencing data in breast cancer staging.

16) Surya
Bhupatiraju, __On the Complexity
of the Marginal Satisfiability Problem__ (18 November
2012)

The marginal satisfiability problem (MSP) asks: Given
desired marginal distributions *D _{S}* for
every subset

*S*of c variable indices from {1, . . . , n}, does there exist a distribution

*D*over n-tuples of values in {1, . . . , m} with those

*S*-marginals

*D*? Previous authors have studied MSP in fixed dimensions, and have classified the complexity up to certain upper bounds. However, when using general dimensions, it is known that the size of distributions grows exponentially, making brute force algorithms impractical. This presents an incentive to study more general, tractable variants, which in turn may shed light on the original problem's structure. Thus, our work seeks to explore MSP and its variants for arbitrary dimension, and pinpoint its complexity more precisely. We solve MSP for

_{S}*n*= 2 and completely characterize the complexity of three closely related variants of MSP. In particular, we detail novel greedy and stochastic algorithms that handle exponentially-sized data structures in polynomial time, as well as generate accurate representative samples of these structures in polynomial time. These algorithms are also unique in that they represent possible protocols in data compression for communication purposes. Finally, we posit conjectures related to more generalized MSP variants, as well as the original MSP.

15) Fengning Ding and Aleksander Tsymbaliuk, __Representations
of Infinitesimal Cherednik Algebras__ (arXiv.org, 17 October
2012), published in Representation Theory 17 (2013)

Infinitesimal Cherednik algebras, first introduced by
Etingof, Gan, and Ginzburg (2005), are continuous analogues
of rational Cherednik algebras, and in the case of gl_{n},
are deformations of universal enveloping algebras of the Lie
algebras sl_{n+1}. Despite these connections, infinitesimal Cherednik algebras are not widely-studied, and basic
questions of intrinsic algebraic and representation
theoretical nature remain open. In the first half of this
paper, we construct the complete center of H_{ζ}(gl_{n}) for the
case of n = 2 and give one particular generator of the
center, the Casimir operator, for general n. We find the
action of this Casimir operator on the highest weight
modules to prove the formula for the Shapovalov determinant,
providing a criterion for the irreducibility of Verma
modules. We classify all irreducible finite dimensional
representations and compute their characters. In the second
half, we investigate Poisson-analogues of the infinitesimal
Cherednik algebras and use them to gain insight on the
center of H_{ζ}(gl_{n}). Finally, we investigate H_{ζ}(sp_{2n}) and
extend various results from the theory of H_{ζ}(gl_{n}), such as
a generalization of Kostant's theorem.

14) Tanya Khovanova and Dai Yang, __Halving Lines and
Their Underlying Graphs__ (arXiv.org, 17 October
2012), published in Involve 11:1 (2018): 1–11

In this paper we study halving-edges graphs corresponding to a set of halving lines. Particularly, we study the vertex degrees, path, cycles and cliques of such graphs. In doing so, we study a vertex-partition of said graph called chains which are equipped with interesting properties.

2011 Research Papers

13) Carl Lian, __Representations of Cherednik Algebras Associated to
Complex Reflection Groups in Positive Characteristic__ (arXiv.org, 1
July
2012)

We consider irreducible lowest-weight representations of
Cherednik algebras associated to certain classes of complex
reflection groups in characteristic *p*. In particular,
we study maximal submodules of Verma modules associated to
these algebras. Various results and conjectures are
presented concerning generators of these maximal submodules,
which are found by computing singular polynomials of Dunkl
operators. This work represents progress toward the general
problem of determining Hilbert series of irreducible
lowest-weight representations of arbitrary Cherednik
algebras in characteristic *p*.

12) Aaron Klein, Joel Brewster Lewis, and Alejandro Morales, __Counting matrices over finite fields with support on skew
Young and Rothe diagrams__ (arXiv.org, 26 March 2012);
published in the Journal of Algebraic Combinatorics (May 2013)

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case when the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.

11) Surya
Bhupatiraju, Pavel Etingof, David Jordan, William Kuszmaul, and Jason
Li, __Lower central series of a free associative algebra over
the integers and finite fields__ (arXiv.org, 8 March
2012), published in the Journal of Algebra (December 2012)

Consider the free algebra A_n generated over Q by n
generators x_1, ..., x_n. Interesting objects attached to A
= A_n are members of its lower central series, L_i = L_i(A),
defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], and
their associated graded components B_i = B_i(A) defined as
B_i=L_i/L_{i+1}. These quotients B_i, for i at least 2, as
well as the reduced quotient \bar{B}_1=A/(L_2+A L_3),
exhibit a rich geometric structure, as shown by Feigin and
Shoikhet and later authors (Dobrovolska-Kim-Ma, Dobrovolska-Etingof,
Arbesfeld-Jordan, Bapat-Jordan).

We study the same problem over the integers Z and finite
fields F_p. New phenomena arise, namely, torsion in B_i over
Z, and jumps in dimension over F_p. We describe the torsion
in the reduced quotient RB_1 and B_2 geometrically in terms
of the De Rham cohomology of Z^n. As a corollary we obtain a
complete description of \bar{B}_1(A_n(Z)) and
\bar{B}_1(A_n(F_p)), as well as of B_2(A_n(Z[1/2])) and
B_2(A_n(F_p)), p>2. We also give theoretical and
experimental results for B_i with i>2, formulating a number
of conjectures and questions based on them. Finally, we
discuss the supercase, when some of the generators are odd (fermionic)
and some are even (bosonic), and provide some theoretical
results and experimental data in this case.

10) David Jordan and Masahiro Namiki, __Determinant
formulas for the reflection equation algebra__ (19
Feb 2012)

In this note, we report on work in progress to explicitly describe generators of the center of the reflection equation algebra associated to the quantum GL(N) R-matrix. In particular, we conjecture a formula for the quantum determinant, and for the quadratic central element, both of which involve the excedance statistic on the symmetric group. Current efforts are directed at proving these formulas, and at finding formulas for the remaining central elements.

9) Ziv Scully, Yan Zhang, and Tian-Yi (Damien) Jiang, __Firing Patterns in the Parallel Chip-Firing Game__ (arXiv.org, 29 Nov 2012), published in *Discrete Mathematics and Theoretical Computer Science (DMTCS)* proc., Nancy, France, 2014

The *parallel chip-firing* game is an automaton on graphs in which vertices “fire” chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study *firing sequences*, which describe each vertex’s interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of *motors*.

8) Sheela Devadas, __Lowest-weight representations of Cherednik algebras in
positive characteristic__ (29 Jan 2012)

We study lowest-weight irreducible representations of
rational Cherednik algebras attached to the complex
reflection groups *G(m, r, n)* in characteristic *p*,
focusing specifically on the case *p* ≤ *n* ,
which is more complicated than the case *p *> *n*.
The goal of our work is to calculate characters (and in
particular Hilbert series) of these representations. By
studying the kernel of the contravariant bilinear form on
Verma modules, we proved formulas for Hilbert series of
irreducible modules in a number of cases, and also obtained
a lot of computer data which suggests a number of
conjectures. Specifically, we find that the shape and form
of the Hilbert series of the irreducible representations and
the generators of the kernel tend to be determined by the
value of *n* modulo *p* .

7) Christina Chen, __Maximizing Volume Ratios for Shadow Covering by
Tetrahedra__ (arXiv.org, 9 Jan 2012)

Define a body A to be able to hide behind a body B if the orthogonal projection of B contains a translation of the corresponding orthogonal projection of A in every direction. In two dimensions, it is easy to observe that there exist two objects such that one can hide behind another and have a larger area than the other. It was recently shown that similar examples exist in higher dimensions as well. However, the highest possible volume ratio for such bodies is still undetermined. We investigated two three-dimensional examples, one involving a tetrahedron and a ball and the other involving a tetrahedron and an inverted tetrahedron. We calculate the highest volume ratio known up to this date, 1.16, which is generated by our second example.

6) Yongyi Chen, Pavel Etingof, David
Jordan, and Michael Zhang, __Poisson traces in positive characteristic__ (arXiv.org,
29 Dec
2011)

We study Poisson traces of the structure algebra A of an affine Poisson variety X defined over a field of characteristic p. According to arXiv:0908.3868v4, the dual space HP_0(A) to the space of Poisson traces arises as the space of coinvariants associated to a certain D-module M(X) on X. If X has finitely many symplectic leaves and the ground field has characteristic zero, then M(X) is holonomic, and thus HP_0(A) is finite dimensional. However, in characteristic p, the dimension of HP_0(A) is typically infinite. Our main results are complete computations of HP_0(A) for sufficiently large p when X is 1) a quasi-homogeneous isolated surface singularity in the three-dimensional space, 2) a quotient singularity V/G, for a symplectic vector space V by a finite subgroup G in Sp(V), and 3) a symmetric power of a symplectic vector space or a Kleinian singularity. In each case, there is a finite nonnegative grading, and we compute explicitly the Hilbert series. The proofs are based on the theory of D-modules in positive characteristic.

5) Saarik Kalia, __The Generalizations of the Golden Ratio: Their Powers,
Continued Fractions, and Convergents__ (23 Dec
2011)

The relationship between the golden ratio and continued fractions is commonly known about throughout the mathematical world: the convergents of the continued fraction are the ratios of consecutive Fibonacci numbers. The continued fractions for the powers of the golden ratio also exhibit an interesting relationship with the Lucas numbers. In this paper, we study the silver means and introduce the bronze means, which are generalizations of the golden ratio. We correspondingly introduce the silver and bronze Fibonacci and Lucas numbers, and we prove the relationship between the convergents of the continued fractions of the powers of the silver and bronze means and the silver and bronze Fibonacci and Lucas numbers. We further generalize this to the Lucas constants, a two-parameter generalization of the golden ratio.

4) Caroline
Ellison, __The Number of Nonzero Coefficients of Powers of a
Polynomial over a Finite Field__ (15 Nov 2011)

Coefficients of polynomials over finite fields often
encode information that can be applied in various areas of
science; for instance, computer science and representation
theory. The purpose of this project is to investigate these
coefficients over the finite field F* _{p}*. We
find four exact results for the number of nonzero
coefficients in special cases of

*n*and

*p*for the polynomial (1 + x + x

^{2})

*. More importantly, we use Amdeberhan and Stanley's matrices to find what we conjecture to be an approximation for the sum of the number of nonzero coefficients of P(x)*

^{n}*over F*

^{n}*. We also relate the number of nonzero coefficients to the number of base*

_{p}*p*digits of

*n*. These results lead to questions in representation theory and combinatorics.

3) Xiaoyu He, __On the
Classification of Universal Rotor-Routers__ (arXiv.org, 6
Nov 2011)

The combinatorial theory of rotor-routers has connections with problems of statistical mechanics, graph theory, chaos theory, and computer science. A rotor-router network defines a deterministic walk on a digraph G in which a particle walks from a source vertex until it reaches one of several target vertices. Motivated by recent results due to Giacaglia et al., we study rotor-router networks in which all non-target vertices have the same type. A rotor type r is universal if every hitting sequence can be achieved by a homogeneous rotor-router network consisting entirely of rotors of type r. We give a conjecture that completely classifies universal rotor types. Then, this problem is simplified by a theorem we call the Reduction Theorem that allows us to consider only two-state rotors. A rotor-router network called the compressor, because it tends to shorten rotor periods, is introduced along with an associated algorithm that determines the universality of almost all rotors. New rotor classes, including boppy rotors, balanced rotors, and BURD rotors, are defined to study this algorithm rigorously. Using the compressor the universality of new rotor classes is proved, and empirical computer results are presented to support our conclusions. Prior to these results, less than 100 of the roughly 260,000 possible two-state rotor types of length up to 17 were known to be universal, while the compressor algorithm proves the universality of all but 272 of these rotor types.

2) Yongyi Chen and Michael Zhang, __On zeroth Poisson homology in positive characteristic__ (30
Sept
2011)

A Poisson algebra is a commutative algebra with a Lie bracket {,} satisfying the Leibniz rule. An important invariant of a Poisson algebra A is its zeroth Poisson homology HP_0(A)=A/A,A}. It characterizes densities on the phase space invariant under all Hamiltonian flows. Also, the dimension of HP_0(A) gives an upper bound for the number of irreducible representations of any quantization of A. We study HP_0(A) when A is the algebra of functions on an isolated quasihomogeneous surface singularity. Over C, it's known that HP_0(A) is the Jacobi ring of the singularity whose dimension is the Milnor number. We generalize this to characteristic p. In this case, HP_0(A) is a finite (although not finite dimensional) module over A^p. We give its conjectural Hilbert series for Kleinian singularities and for cones of smooth projective curves, and prove the conjecture in several cases. (The conjecture has now been proved in general in our follow-up paper with P. Etingof and D. Jordan.)

1) Christina Chen, Tanya Khovanova, and
Daniel A. Klain, __Volume bounds for shadow covering__ (arXiv.org, 8 Sep
2011), published in Transactions of the American Mathematical Society 366 (2014)

For *n* ≥ 2 a construction is given for a large family of compact convex sets *K* and *L* in *n*-dimensional Euclidean space such that the orthogonal projection *L _{u}* onto the subspace

*u*contains a translate of the corresponding projection

^{⊥}*K*for every direction

_{u}*u*, while the volumes of

*K*and

*L*satisfy

*V*>

_{n}(K)*V*. It is subsequently shown that, if the orthogonal projection

_{n}(L)*L*onto the subspace

_{u}*u*contains a translate of

^{⊥}*K*for every direction

_{u}*u*, then the set

*(n/(n−1))L*contains a translate of

*K*. It follows that

*V*≤

_{n}(K)*(n/(n−1))*. In particular, we derive a universal constant bound

^{n}V_{n}(L)*V*≤ 2.942

_{n}(K)*V*, independent of the dimension

_{n}(L)*n*of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.

**Email us:** Primes@math.mit.edu