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Graduate Thesis Defenses 2015

Inna Entova Aizenbud

Title: Schur-Weyl duality in complex rank
Date: Friday, September 4th, 2015
11:00am, Room: E17-122
Committee:Pavel Etingof (advisor), Roman Bezrukavnikov, Ivan Loseu (Northeastern University)

Abstract

Let V be a finite dimensional vector space. The classical Schur-Weyl duality describes the relation between the actions of the Lie algebra $gl(V)$ and the symmetric group $S_n$ on the tensor power $V^{\otimes n}$. We will discuss Deligne categories $Rep(S_t)$, which are extrapolations to complex t of the categories of finite dimensional representations of the symmetric groups. I will then present a generalization of the classical Schur-Weyl duality in the setting of Deligne categories, which involves a construction of "a complex tensor power of V", and gives us a duality between the Deligne category and a Serre quotient of a parabolic category O for $gl(V)$.

Michael Andrews

Title: The $v_1$-periodic part of the Adams spectral sequence at an odd prime
Date: Monday, April 13th, 2015
5:00pm, Room: E17-128
Committee: Haynes Miller (advisor, MIT), Mark Behrens (Notre Dame), Mike Hopkins (Harvard)

Abstract

We tell the story of the stable homotopy groups of spheres for odd primes at chromatic height $1$, through the lens of the Adams spectral sequence.

We calculate a Bockstein spectral sequence which converges to the $1$-line of the chromatic spectral sequence for the odd primary Adams $E_2$-page. Furthermore, we calculate the associated algebraic Novikov spectral sequence converging to the $1$-line of the $BP$ chromatic spectral sequence. This result is also viewed as the calculation of a direct limit of localized modified Adams spectral sequences converging to the homotopy of the $v_1$-periodic sphere spectrum.

As a consequence of this work, we obtain a thorough understanding of a collection of $q_0$-towers on the Adams $E_2$-page and we obtain information about the differentials between these towers. Moreover, above a line of slope $1/(p^2-p-1)$ we can completely describe the $E_2$ and $E_3$-pages of the mod $p$ Adams spectral sequence, which accounts for almost all the spectral sequence in this range.

Nate Bottman

Title: Pseudoholomorphic quilts with figure eight singularity
Date: Friday, April 17th, 2015
3:00pm, Room: 56-154
Committee: Emmy Murphy, Paul Seidel, Katrin Wehrheim (advisor)

Abstract

In this thesis, I prove several results toward constructing a machine that turns Lagrangian correspondences into A-infinity functors between Fukaya categories. The core of this construction is pseudoholomorphic quilts with figure eight singularity.

In the first part, I establish a collection of strip-width--independent elliptic estimates. The key is function spaces which augment the Sobolev norm with another term, so that the norm of a product can be bounded by the product of the norms in a manner which is independent of the strip-width.

In the second part, which is joint with Katrin Wehrheim, I use my collection of estimates to prove a Gromov compactness theorem for quilts with a strip of (possibly non-constant) width shrinking to zero. This features local C^infinity-convergence away from the points where energy concentrates. At such points, we produce a nonconstant quilted sphere.

In the third part, I prove a removable singularity theorem for the figure eight singularity. Using the Gromov compactness theorem just mentioned, I adapt an argument of Abbas--Hofer to uniformly bound the norm of the gradient of the maps in cylindrical coordinates centered at the singularity. I conclude by proving a "quilted" isoperimetric inequality.

Finally, I propose a blueprint for constructing an algebraic object that binds together the Fukaya categories of many different symplectic manifolds. I call this object the ``symplectic A-infinity 2-category Symp''. The key to defining the structure maps of Symp is the figure eight bubble.

Michael Donovan

Title: Unstable operations in Bousfield-Kan spectral sequences
Date: Monday, April 27th, 2015
5:15pm, Room: E17-128
Committee: Haynes Miller, Paul Goerss, Clark Barwick

Abstract

The homotopy groups of a simplicial commutative algebra or a simplicial Lie algebra support rich algebraic structure. We will explain how this structure is reflected in the Bousfield-Kan (a.k.a. unstable Adams) spectral sequences in these categories, and, time permitting, discuss how calculations of the Bousfield-Kan $E^2$ page can be made.

Jesse Geneson

Title: Bounds on extremal functions of forbidden patterns
Date: Tuesday, February 17, 2015
4:00pm, Room: E17-128
Committee: Henry Cohn, Jacob Fox (Stanford), Peter Shor (chair and advisor).

Abstract

Extremal functions of forbidden sequences and $0-1$ matrices have applications to many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form $(12\ldots l)^{t}$ for $l \geq 2$ and $t\geq 1$, $a b c (a c b)^{t}$ for $t \geq 0$, and $a v a v' a$, 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 $v$. We also prove the existence of infinitely many forbidden $0-1$ matrices $P$ with non-linear extremal functions for which every strict submatrix of $P$ has a linear extremal function. Then we show that for every $d$-dimensional permutation matrix $P$ with $k$ ones, the maximum number of ones in a $d$-dimensional matrix of sidelength $n$ that avoids $P$ is $2^{O(k)}n^{d-1}$.

Saul Glasman

Title: Day convolution and the Hodge filtration on THH
Date: Friday, April 17, 2015
4:00pm, Room: 2-105
Committee: Clark Barwick (advisor), Haynes Miller, Jacob Lurie (Harvard).

Abstract

In the 80s, many mathematicians independently defined a filtration on the Hochschild homology of a commutative algebra A that recovers the Hodge filtration of the de Rham complex of A in the case where A is a smooth Q-algebra. This talk will focus on the latter part of my thesis, which is a refinement of this construction to a filtration by spectra of the topological Hochschild homology of a commutative ring spectrum. If time permits, I'll discuss how to lift this filtration to a filtration of topological cyclic homology using techniques of equivariant stable homotopy theory, and speculate on possible connections with number theory.

Dan Harris

Title: The Pilot-Wave Dynamics of Walking Droplets in Confinement
Date: Friday, May 1, 2015
2:00pm, Room: 3-370
Committee: John Bush (advisor), Ruben Rosales, and Yves Couder (Paris Diderot).

Abstract

A decade ago, Yves Couder and coworkers discovered that millimetric droplets can walk on a vibrated fluid bath, and that these walking droplets or ``walkers'' display several features reminiscent of quantum particles. We first describe our experimental advances, that have allowed for a quantitative characterization of the system behavior and guided the development of our accompanying theoretical models. We then detail our explorations of this rich dynamical system in several settings where the walker is confined, either by boundaries or an external force. Three particular cases are examined: a walker in a corral geometry, a walker in a rotating frame, and a walker passing through an aperture in a submerged barrier. In each setting, as the vibrational forcing is increased, progressively more complex trajectories arise. The manner in which multimodal statistics may emerge from the walker's chaotic dynamics is elucidated.

Benjamin Iriarte

Title: Combinatorics of Acyclic Orientations of Graphs: Algebra, Geometry and Probability
Date: Monday, March 30, 2015
5:00pm, Room: 4-145
Committee: Richard Stanley (advisor and chair), Henry Cohn, and Michelle Wachs (University of Miami).

Abstract

We study aspects of the set of acyclic orientations of a simple undirected graph. Acyclic orientations of a graph may be readily obtained from bijective labellings of its vertex-set with a totally ordered set, and they can be regarded as partially ordered sets. We will study this connection between acyclic orientations of a graph and the theory of linear extensions or topological sortings of a poset, from both the points of view of poset theory and enumerative combinatorics, and of the geometry of hyperplane arrangements and zonotopes. What can be said about the distribution of acyclic orientations obtained from a uniformly random selection of bijective labelling? What orientations are thence more probable? What can be said about the case of random graphs? These questions will begin to be answered during the first part of this talk. Other types of labellings of the vertex-set, e.g. proper colorings, may be used to obtain acyclic orientations of a graph, as well. Motivated by our first results on bijective labellings, in the second part of the talk, we will use eigenvectors of the Laplacian matrix of a graph, in particular, those corresponding to the largest eigenvalue, to label its vertex-set and to induce partial orientations of its edge-set. What information about the graph can be gathered from these partial orientations? Lastly, in the third part, we will delve further into the structure of acyclic orientations of a graph by enhancing our understanding of the duality between the graphical zonotope and the graphical arrangement with the lens of Alexander duality. This will take us to non-crossing trees, which arguably vastly subsume the combinatorics of this geometric and algebraic duality. We will then combine all of these tools to obtain probabilistic results about the number of acyclic orientations of a random graph, and about the uniformly random choice of an acyclic orientation of a graph, among others.

Jiayong Li

Title: A-infinity algebras for Lagrangians via polyfold theory for Morse trees with holomorphic disks
Date: Wednesday, April 22, 2015
1:00pm, Room: 4-231
Committee: Katrin Wehrheim (advisor), Tomasz Mrowka (chair), Paul Seidel

Abstract

For a Lagrangian submanifold, we construct a moduli space of trees of holomorphic disk maps with Morse flow lines as edges. This moduli space is compact and disk bubbling is an interior point. Applying the polyfold machinery to the moduli space yields an A-infinity algebra, and we discuss the invariance of this algebra with respect to choices of almost-complex structures.

Dana Mendelson

Title: Global Behavior and Dynamics of Nonlinear Dispersive Equations with Random Initial Data
Date: Thursday, April 23, 2015
3:00pm, Room: 4-237
Committee: David Jerison, Jared Speck, Gigliola Staffilani (advisor).

Abstract

In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on $\mathbb{R^3}$. We establish an almost sure global existence result with respect to a suitable randomization of the initial data. In particular, this provides examples of initial data of super-critical regularity which lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition and improved averaging effects for the free evolution of the randomized initial~data.

In the second part, we consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\mathbb{T^3}) \times H^{-\frac{1}{2}}(\mathbb{T^3})$. This space is at the critical regularity for this equation and we note that there is no uniform control on the local time of existence for arbitrary initial data. We prove a local in time non-squeezing result and that uniform bounds on the Strichartz norms of solutions implies global in-time non-squeezing. Analogously to the work of Burq and Tzvetkov, we first define a set of full measure with respect to a suitable randomization of the initial data on which the flow of this equation is globally defined. The proof then relies on an approximation result for the flow, which uses probabilistic estimates for the nonlinear component of the flow map, and Gromov's non-squeezing theorem and deterministic stability theory. A key ingredient in the global-in-time result is an approximation result which demonstrates the stability of this equation at low frequencies under high-frequency perturbations of the initial data. The proof relies on multilinear estimates in the $U^p$ and $V^p$ spaces.

Vinoth Nandakumar

Title: Coherent sheaves on varieties arising in Springer theory, and category $\mathcal{O}$
Date: Thursday, April 23, 2015
12:00pm, Room: 32-144
Committee: Roman Bezrukavnikov (advisor), Pavel Etingof, David Vogan

Abstract

In this thesis, we will study three topics related to Springer theory and category $\mathcal{O}$.

In the first part, we examine the geometry of the exotic nilpotent cone. We establish a bijection between the set of dominant weights, and the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of its isotropy group. An analogous statement for the ordinary nilpotent cone was conjectured by Lusztig and Vogan, and proved by Bezrukavnikov.

In the second part, we study the exotic t-structure for two-block Springer fibres (these were originally introduced by Bezrukavnikov and Mirkovic in greater generality to study representations of Lie algebras in positive characteristic). We give a concrete description of the irreducible objects in the heart of this t-structure, and the resulting Ext algebra.

In the third part, we consider certain sub-quotients of category $\mathcal{O}$ and give an example of a "real variation of stability conditions" (which were recently introduced to study Bridgeland stability conditions for Springer fibers). We use the braid group action on these categories, and certain leading coefficient polynomials."

Dimiter Ostrev

Title: The Structure of Optimal and Nearly-Optimal Quantum Strategies for Non-Local XOR Games
Date: Monday, April 27, 2015
1:30pm, Room: 1-132
Committee: Prof. Peter Shor (thesis advisor, committee chair), Prof. Seth Lloyd (MIT Mechanical Engineering), Prof. Jonathan Kelner, Prof. Ankur Moitra

Abstract

We study optimal and nearly-optimal quantum strategies for non-local XOR games. First, we prove the following general result: for every non-local XOR game, there exists a set of relations with the properties: (1) a quantum strategy is optimal for the game if and only if it satisfies the relations, and (2) a quantum strategy is nearly optimal for the game if and only if it approximately satisfies the relations. Next, we focus attention on a specific infinite family of XOR games: the CHSH(n) games. This family generalizes the well-known CHSH game. We describe the general form of CHSH(n) optimal strategies. Then, we adapt the concept of intertwining operator from representation theory and use that to characterize nearly-optimal CHSH(n) strategies.

Aaron Potechin

Title: Analyzing Monotone Space Complexity Via the Switching Network Model
Date: Wednesday, April 29, 2015
4:00pm, Room: E17-136
Committee: Jonathan Kelner, Ankur Moitra, Boaz Barak, and Madhu Sudan

Abstract

Space complexity is the study of how much space/memory it takes to solve problems. Unfortunately, proving general lower bounds on space complexity is notoriously hard. Thus, we instead consider the restricted case of monotone algorithms, which only make deductions based on what is in the input and not what is missing. In this thesis, we develop techniques for analyzing monotone space complexity via a model called the monotone switching network model. Using these techniques, we prove tight bounds on the minimal size of monotone switching networks solving the directed connectivity, generation, and k-clique problems. These results separate monotone analgoues of L and NL and provide an alternative proof of the separation of the monotone NC hierarchy first proved by Raz and McKenzie. We then further develop these techniques for the directed connectivity problem in order to analyze the monotone space complexity of solving directed connectivity on particular input graphs.

Yakov Shlapentokh-Rothman

Title: Mode Stabilities and Instabilities for Scalar Fields on Kerr Exterior Spacetimes
Date: Thursday, May 14, 2015
3:00pm, Room: 3-370
Committee: Igor Rodnianski, William Minicozzi, Jared Speck.

Abstract

In this thesis we study wave and Klein-Gordon equations on Kerr exterior spacetimes. For the wave equation, we give a quantitative refinement and simple proofs of mode stability type statements on Kerr backgrounds in the full sub-extremal range (|a| < M). As an application, we are able to quantitatively control the energy flux along the horizon for solutions to the wave equation in any bounded-frequency regime. This estimate plays a crucial role in the author's recent proof, joint with Mihalis Dafermos and Igor Rodnianski, of boundedness and decay for the solutions to the wave equation on the full range of sub-extremal Kerr spacetimes. For the Klein-Gordon equation, we show that given any Kerr exterior spacetime with non-zero angular momentum, we may find an open family of non-zero Klein-Gordon masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr_+.

Sean Simmons

Title: Preserving Patient Privacy in Biomedical Data Analysis
Date: Tuesday, July 21st 2015
2:00pm, Room: 32-G882
Committee:Bonnie Berger (thesis adviser and chair), Jonathan Kelner, and Vinod Vaikuntanathan

Abstract

The growing stockpiles of patient data found in biomedical repositories and electronic health records promise to be an invaluable resource for improving our understanding of human diseases. Recent work, however, has shown that sharing this data--even when aggregated to produce p-values, regression coefficients, or other study statistics--may compromise patient privacy. This raises a fundamental question: how do we protect patient privacy while still making the most out of their data? In this thesis we introduce various methods for performing privacy preserving analysis on biomedical data. In particular, we focus on methods that achieve a formal notion of privacy known as differential privacy. First, we consider differentially private genome wide association studies (GWAS). We present a new algorithm that substantially improves the runtime and utility of previous methods for privacy preserving GWAS. Building on this result, we demonstrate how to construct differentially private versions of other, more powerful, GWAS statistics. In addition to looking at genomic data analysis, we also study privacy preserving cohort selection for medical studies. More specifically, we introduce an improved method for releasing differentially private answers to medical count queries.

Roberto Svaldi

Title: Log geometry and extremal contractions
Date: Friday, March 27, 2015
4:00pm, Room: E17-122
Committee: Steven Kleiman, J. McKernan (UCSD) (advisor), Bjorn Poonen (chair).

Abstract

The Minimal Model Program (in short, MMP) aims at classifying algebraic varieties at least from a birational point of view, i.e. it is allowed to change the variety under scrutiny as long as its field of rational functions remains unchanged.

In this thesis we study two problems that are inspired by the techniques that were developed in the last 30 years by various mathematicians in an attempt to realize the Minimal Model Program for varieties of any dimension.

In the first part, we consider log smooth pairs $(X, D)$ and we prove that if $K_X+D$ is not nef then there exists a non constant algebraic map from the affine line $\mathbb{A}^1$ to either $X/D$ or to an intersection of components of $D$. We are able to generalize this theorem to the case of a log canonical pair $(X, \Delta)$. In particular this implies a positivity theorem for pairs $(X, \Delta)$ that are Mori hyperbolic, i.e. that do not contain any $\mathbb{A}^1$'s in the strata of the non-klt locus of $\Delta$ or in its complement in $X$.
This kind of result represents a first step towards a log-analogue of Bend and Break.

In the second part, we study certain varieties that naturally arise as possible outcomes of the classification algorithm proposed by the MMP: these are called Mori fiber spaces. A Mori fiber space, is a variety $X$ together with a morphism $f: X \to Y$ such that $\dim X > \dim Y, \; -K_X$ is ample when restricted to every fiber and the relative Picard number is 1. We show that being a general fiber of a Mori fiber space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a $\mathbb{Q}$-factorial Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We apply our criteria to figure out this property up to dimension three and on homogeneous spaces. The smooth toric case is studied and an interesting connection with $K$-semistability is also investigated.

Wuttisak Trongsiriwat

Title: Combinatorics of Permutation Patterns, Interlacing Networks, and Schur Functions
Date: Monday, April 13, 2015
2:00pm, Room: 1-273
Committee: Alex Postnikov (advisor), Richard Stanley, Tom Roby (U of Connecticut)

Abstract

In the first part, we study pattern avoidance and permutation statistics. For a set of patterns $\Pi$ and a permutation statistic st, let $F^\text{st}_n(\Pi;q)$ be the polynomial that counts st on the permutations avoiding all patterns in $\Pi$. Suppose $\Pi$ contains the pattern 312. For a class of permutation statistics (including inversion and descent statistics), we give a formula that expresses $F^\text{st}_n(\Pi;q)$ in terms of these st-polynomials for some subblocks of the patterns in $\Pi$. Using this recursive formula, we construct examples of nontrivial st-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all $\text{inv}$-Wilf equivalences are trivial.

The second part is motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence. We define interlacing networks to be certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to a three-term relations among path weights, which immediately implies the octahedron recurrences. Furthermore, this involution gives some interesting identities of Schur functions generalizing identities by Fulmek-Kleber. Then we study the balanced swap graphs, which encode a class of Schur function identities obtained this way.

Neal Wadhwa

Title: Revealing and Analyzing Imperceptible Deviations in Images and Videos
Date: Thursday, Dec. 3, 2015
9:30am, Room: 56-114
Committee: Professor William T. Freeman (Thesis Supervisor, EECS), Professor Frédo Durand (EECS), Professor John W. M. Bush, Professor Alan Edelman.

Abstract

The world is filled with objects that appear to follow some perfect model. A sleeping baby might look still and a house's roof should be straight. However, both the baby and the roof can deviate subtly from their ideal models of perfect stillness and perfect straightness. These deviations can reveal important information like whether the baby is breathing normally or whether the house's roof is sagging.

In this dissertation, I make the observation that these subtle deviations produce a visual signal that while invisible to the naked eye can be extracted from ordinary and ubiquitous images and videos. I propose new computational techniques to reveal these subtle deviations to the naked eye by producing new images and videos, in which the tiny deviations have been magnified.

I focus on magnifying deviations from two ideal models: perfect stillness and perfect geometries in space. In the first case, I leverage the complex steerable pyramid, a localized version of the Fourier transform, whose notion of local phase can be used to process and manipulate small motions or changes from stillness in videos. In the second case, I find hidden geometric deformations in images by localizing edges to sub-pixel precision.

In both cases, I experimentally validate that the tiny deviations I magnify are indeed real, comparing them to alternative ways of measuring tiny motions and subtle geometric deformations in the world. I also give a careful analysis of how noise in videos impacts our ability to see tiny motions. Additionally, I show the utility of revealing hidden deviations a wide variety of fields, such as biology, physics and structural analysis.

Guozhen Wang

Title: Unstable Chromatic Homotopy Theory
Date: Monday, April 13, 2015
4:00pm, Room: E17-128
Committee: Mark Behrens, Haynes Miller, Clark Barwick.

Abstract

I study unstable homotopy theory with chromatic methods. Using the self maps provided by the Hopkins-Smith periodicity theorem, we can decompose the unstable homotopy groups of a space into its periodic parts, except somelower stems. For fixed n, using the Bousfield-Kuhn functor we can associate to any space a spectrum, which captures the vn-periodic part of its homotopy groups.

I study the homotopy type of the Bousfield-Kuhn functor applied to spheres, which would tell us much about the vn-periodic part of the homotopy groups of spheres provided we have a good understanding of the telescope conjecture. I make use the Goodwillie tower of the identity functor, which resolves the unstable spheres into spectra which are the Steinberg summands of classifying spaces of the additive groups of vector spaces over finite fields.

By understanding the attaching maps of the Goodwillie tower after applying the Bousfield-Kuhn functor, we would be able to determine the homotopy type of its effect on spheres. As an example of how this works in concrete computations, I will compute the $K(2)$ local homotopy groups of the three sphere at primes p>3.

The computations show that the unstable homotopy groups not only have finite p-torsion, their K(2)-local parts also have finite v1-torsion, which indicates there might be a more general finite vn-torsion phenomena in the unstable world, which is conjectured by many people.

Wenzhe Wei

Title: Nuclear Norm Penalized LAD Estimator for Low Rank Matrix Recovery
Date: Friday, May 1, 2015
11:00am Room: 66-160
Committee: Lie Wang(Theis advisor), Laurent Demanet, Peter Kempthorne

Abstract

In the thesis we propose a novel method for low rank matrix recovery. We study the framework using absolute deviation loss function and nuclear norm penalty. An optimal recovery bound is established and proved under certain restricted isometry and restricted eigenvalue assumptions. The estimator is able to recover the underlying matrix with high probability with limited observations that the number of observation is more than the degree of freedom but less than a power of dimension. We then show that the estimator has two advantages. First the theoretical tuning parameter does not depend on the knowledge of the noise level. and the bound can be derived even when noises have fatter tails than normal distribution. The second advantage is that absolute deviation loss function is more robust when compared with square loss function.

Yi Zeng

Title: Mathematical modeling of lithium-ion intercalation particles and their electrochemical dynamics
Date: Thursday, March 19, 2015
10:30am, Room: 66-160
Committee: Martin Bazant (advisor), Hung Cheng, Steven Johnson

Abstract

Lithium-ion battery is a family of rechargeable batteries with increasing importance that is closely related to everyone's daily life. However, despite its enormously wide applications in numerous areas, the mechanism of lithium-ion transport within the battery is still unclear, especially for phase separable battery materials, such as lithium iron phosphate and graphite. Mathematical modeling of the battery dynamics during charging/discharging will be helpful to better understand its mechanism, and may lead to future improvement in the battery technology.

In this thesis, a new theoretical framework, the Cahn-Hilliard reaction (CHR) model, is applied to model the bulk phase separation dynamics of the single intercalated particle in the lithium-ion battery. After a study of the efficient numerical algorithm for solving nonlinear diffusion equations, we numerically investigate the thermodynamics and electrokinetics of the 1D spherical CHR model with different possible material properties in detail. We also extend the CHR model to 2D and briefly study the effects of the surface electron-conducting coating layer.

We also work on the Marcus theory, which is demonstrated to be a better theoretical framework for heterogeneous electron transfer at the surface of intercalated particles in the batteries. We provide simple closed-form approximations to both the symmetric Marcus-Hush-Chidsey (MHC) and the asymmetric-Marcus-Hush (AMH) models by asymptotic technique. By avoiding the numerical evaluations of the improper integral in the old formulae, computing the surface reaction rate with the new approximation is now more than 1000 times faster than before.

Leonardo Andrés Zepeda Núñez

Title: Fast and scalable solvers for the Helmholtz equation
Date: Thursday, May 7, 2015
2:00pm, Room: 54-209
Committee: Alex Barnett, (Dartmouth College-Simons Center for Data Analysis), Laurent Demanet, Maarten de Hoop (Purdue U.- Rice U.), Steven Johnson

Abstract

In this thesis we develop a new family of fast and scalable algorithms to solve the high-frequency Helmholtz equation in heterogeneous medium. The algorithms rely on a layered domain decomposition and a coupling between subdomains using the Green's representation formula, which reduces the problem to a boundary integral system at the interfaces between subdomains. Simultaneously, we introduce a polarization of the waves in up- and down-going components using incomplete Green's integrals, which induces another equivalent boundary integral formulation that is easy to precondition.

The computation is divided in two stages: an offline stage, a computationally expensive but embarrassingly parallel precomputation performed only once; and an online stage, a highly parallel computation with low complexity performed for each right-hand side.

The computational efficiency of the algorithms is achieved by shifting most of the computational burden to an offline precomputation, and by reducing the sequential bottleneck in the online stage using an efficient preconditioner, based on the polarized decomposition, coupled with compressed linear algebra. The resulting algorithms have online runtime $\mathcal{O}(N/P)$, where $N$ is the number of unknowns, and $P$ is the number of nodes in a distributed memory environment; provided that $P = \mathcal{O} \left( N^{\alpha} \right )$. Typically $\alpha = 1/5$ or $1/8$.

Ruixun Zhang

Title: Economic Behavior from an Evolutionary Perspective
Date: Friday, May 15, 2015
10:30am, Room: E17-128
Committee: Andrew W. Lo (advisor, MIT Sloan), Michel Goemans (chair), Peter Kempthorne

Abstract

The conflict between rational models of economic behavior and their systematic de- viations, often referred to as behavioral economics, is one of the most hotly debated issues in social sciences. This thesis reconciles the two opposing perspectives by ap- plying evolutionary principles to economic behavior and deriving implications that cut across species, physiology, and genetic origins.

In the context of a binary-choice model, we first show that risk aversion emerges via natural selection if reproductive risk is “systematic”, i.e., correlated across individ- uals in a given generation. The degree of risk aversion is determined by the stochastic nature of reproductive rates, and different statistical properties lead to different util- ity functions. More generally, irrational behaviors are not just mere divergence from rationality, but seeds necessary for successfully coping with environmental transfor- mations. Furthermore, there is an optimal degree of irrationality in the population depending on the degree of environmental stochasticity.

When applied to evolutionary biology, we show that what appears to be group selection may, in fact, simply be the consequence of natural selection occurring in stochastic environments with “systematic” risks. Those individuals with highly corre- lated risks will appear to form “groups”, even if their actions are totally autonomous, mindless, and, prior to selection, uniformly randomly distributed in the population.

Evolutionary principles can also be used to model the dynamics of financial markets. In a multiperiod model of the contagion of investment ideas, we show that heterogeneous investment styles can coexist in the long run, implying a wider variation of diverse strategies compared to traditional theories. These results may provide new insights to the survival of a wide range of hedge funds. In a model that investors maximize their relative wealth, the initial wealth plays a critical role in determining how the optimal behavior deviates from the Kelly Criterion, regardless of whether the investor is myopic or maximizing the infinite-horizon wealth.

Yufei Zhao

Title: Sparse regularity and relative Szemeredi theorems
Date: Tuesday, March 31, 2015
12:00pm, Room: 32-144
Committee: Jacob Fox (advisor), Michel Goemans, Henry Cohn

Abstract

We extend various fundamental combinatorial theorems and techniques from the dense setting to the sparse setting.

First, we consider Szemeredi's regularity lemma, a fundamental tool in extremal combinatorics. The regularity method, in its original form, is effective only for dense graphs. It has been a long standing problem to extend the regularity method to sparse graphs. We solve this problem by proving a so-called ``counting lemma,'' thereby allowing us to apply the regularity method to relatively dense subgraphs of sparse pseudorandom graphs.

Next, by extending these ideas to hypergraphs, we obtain a simplification and extension of the key technical ingredient in the proof of the celebrated Green--Tao theorem, which states that there are arbitrarily long arithmetic progressions in the primes. The key step, known as a relative Szemeredi theorem, says that any positive proportion subset of a pseudorandom set of integers contains long arithmetic progressions. We give a simple proof of a strengthening of the relative Szemeredi theorem, showing that a much weaker pseudorandomness condition is sufficient.

Finally, we give a short simple proof of a multidimensional Szemeredi theorem in the primes, which states that any positive proportion subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This has been conjectured by Tao and recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler.

Xuwen Zhu

Title: The eleven dimensional supergravity equations, resolutions and Lefschetz fiber metrics
Date: Wednesday, April 1, 2015
4:00pm, Room: E18-466A
Committee: Richard Melrose (advisor), Victor Guillemin, Tomasz Mrowka

Abstract

This thesis consists of three parts. In the first part, we study the eleven dimensional supergravity equations on $\mathbb{B}^7 \times \mathbb{S}^4$ considered as an edge manifold. We compute the indicial roots of the linearized system using the Hodge decomposition, and using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund--Rubin states, is parameterized by three pairs of data on the bounding 6-sphere.

In the second part, we consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the length of the shrinking geodesic. This is joint work with Richard Melrose.

In the third part, the resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial tautological bundle smoothly decomposes into the direct sum of global one-dimensional eigenspaces.