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Graduate Thesis Defenses 2014

Andrey Grinshpun

Title: Some Problems in Graph Ramsey Theory
Date: Friday, December 12, 2014
2:00pm, Room: E18-358
Committee: Jacob Fox (chair and thesis advisor), Gábor N. Sárközy, and Jonathan Kelner.

Abstract

A graph $G$ is $r$-Ramsey minimal with respect to a graph $H$ if every $r$-coloring of the edges of $G$ yields a monochromatic copy of $H$, but the same is not true for any proper subgraph of $G$. The study of the properties of graphs that are Ramsey minimal with respect to some $H$ and similar problems is known as graph Ramsey theory; we study several problems in this area.

Burr, Erdős, and Lovász introduced $s(H)$, the minimum over all $G$ that are $2$-Ramsey minimal for $H$ of $\delta(G)$, the minimum degree of $G$. We find the values of $s(H)$ for several classes of graphs $H$, most notably for all $3$-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher.

One natural question when studying graph Ramsey theory is what happens when, rather than considering all $2$-colorings of a graph $G$, we restrict to a subset of the possible $2$-colorings. Erdős and Hajnal conjectured that, for any fixed color pattern $C$, there is some $\varepsilon > 0$ so that every $2$-coloring of the edges of a $K_n$, the complete graph on $n$ vertices, which doesn't contain a copy of $C$ contains a monochromatic clique on $n^{\varepsilon}$ vertices. Hajnal generalized this conjecture to more than $2$ colors and asked in particular about the case when the number of colors is $3$ and $C$ is a rainbow triangle (a $K_3$ where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles.

One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let ${\cal{F}}=\{F_1,F_2,\ldots\}$ be a sequence of graphs such that $F_n$ is a graph on $n$ vertices with maximum degree at most $\Delta$. If each $F_n$ is bipartite, then the vertices of any $2$-edge-colored complete graph can be partitioned into at most $2^{C \Delta}$ vertex disjoint monochromatic copies of graphs from $\cal F$, where $C$ is an absolute constant. This result is best possible, up to the constant $C$.

Joshua Batson

Title: Obstructions to slicing knots and splitting links
Date:Monday, April 7, 2014
11:00am, Room: 34-304
Committee: Peter Ozsvath (Princeton); Tomasz Mrowka; Paul Seidel; Eli Grigsby (Boston College)

Abstract

In this thesis, we use invariants inspired by quantum field theory to study the smooth topology of links in space and surfaces in space-time.

In the first half, we use Khovanov homology to the study the relationship between links in R3 and their components. We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the split union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer- Mrowka and Hedden-Ni to show that Khovanov homology detects the unlink.

In the second half, we consider knots as potential cross-sections of surfaces in R4. We use Heegaard Floer homology to show that certain knots never occur as cross-sections of surfaces with small first Betti number. (It was previously thought possible that every knot was a cross-section of a connect sum of three Klein bottles.) In particular, we show that any smooth surface in R4 with cross-section the (2k, 2k - 1) torus knot has first Betti number at least 2k - 2.

Rosalie Belanger-Rioux

Title: Compressed Absorbing Boundary Conditions for the Helmholtz Equation
Date:Monday, May 5, 2014
10:00am, Room: 12-122
Committee: Laurent Demanet, Steven Johnson, Jacob White

Abstract

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. We obtain a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter. We then obtain a fast algorithm for the application of the absorbing boundary condition using partitioned low rank matrices. The result, modulo a precomputation, is a fast and memory-efficient compression scheme of an absorbing boundary condition for the Helmholtz equation.

Kestutis Cesnavicius

Title: Selmer groups as flat cohomology groups
Date: Thursday, April 10th 2014
11:00am, Room: 3-370
Committee: Bjorn Poonen (chair & advisor), Sug Woo Shin, Benedict Gross (Harvard)

Abstract

Given a prime number $p$, Bloch and Kato showed how the $p^\infty$-Selmer group of an abelian variety $A$ over a number field $K$ is determined by the $p$-adic Tate module. In general, the $p^m$-Selmer group $\mathrm{Sel}_{p^m} A$ need not be determined by the mod $p^m$ Galois representation $A[p^m]$; we show, however, that this is the case if $p$ is large enough. More precisely, we exhibit a finite explicit set of rational primes $\Sigma$ depending on $K$ and $A$, such that $\mathrm{Sel}_{p^m} A$ is determined by $A[p^m]$ for all $p \not \in \Sigma$. In the course of the argument we describe the flat cohomology group $H^1_{\mathrm{fppf}}(\mathcal{O}_K, \mathcal{A}[p^m])$ of the ring of integers of $K$ with coefficients in the $p^m$-torsion $\mathcal{A}[p^m]$ of the N\'{e}ron model of $A$ by local conditions for $p\not\in \Sigma$, compare them with the local conditions defining $\mathrm{Sel}_{p^m} A$, and prove that $\mathcal{A}[p^m]$ itself is determined by $A[p^m]$ for such $p$. Our method sharpens the relationship between $\mathrm{Sel}_{p^m} A$ and $H^1_{\mathrm{fppf}}(\mathcal{O}_K, \mathcal{A}[p^m])$ which was observed by Mazur and continues to work for other isogenies $\phi$ between abelian varieties over global fields provided that $\mathrm{deg} \phi$ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve $11A1$ over certain families of number fields. Standard glueing techniques developed in the course of the proofs have applications to finite flat group schemes over global bases, permitting us to transfer many of the known local results to the global setting.

Lucas Culler

Title: The blowup formula for higher rank Donaldson invariants
Date: Monday, April 28th 2014
10:00AM, Room: 12-122
Committee: Tomasz Mrowka, Paul Seidel, Peter Kronheimer

Abstract

Donaldson's invariants of smooth 4-manifolds were originally defined using prinicipal bundles with structure group SU(2) and SO(3). These invariants have interesting structural features, the simplest case of which is a formula relating the invariants of a 4-manifold and its blowup. The blowup formula can be described in terms of a formal power series called the blowup function, and in 1994 Fintushel and Stern showed that the blowup function is a theta function on a family of elliptic curves over the spectrum of the equivariant cohomology ring of SU(2).

Donaldson invariants admit a natural generalization, to bundles with structure group SU(N) or PU(N). These generalizations have been studied by physicists since the 1990s, using methods from quantum field theory. A rigorous mathematical definition was first given in 2005 by Kronheimer. We prove the existence of a blowup function for the higher rank invariants defined by Kronheimer, and show that it satisfies a special system of ordinary differential equations. In the case N=3, we also confirm a prediction from the physics literature, which in its general form states that the SU(N) blowup function is a theta function on the Jacobian of a family of curves of genus N-1 over the spectrum of the equivariant cohomology ring of SU(N). Finally, we present a formal argument, motivated by ideas from mirror symmetry, which explains the appearance of Abelian varieties and theta functions in 4 dimensional topological field theories.

Alan Deckelbaum

Title: The Structure of Auctions: Optimality and Efficiency
Date: Thursday, April 24, 2014
4:00pm, Room: 8-205
Committee:Constantinos Daskalakis, Michel Goemans, Ankur Moitra

Abstract

The problem of constructing auctions to maximize expected revenue is central to mechanism design and to algorithmic game theory. While the special case of selling a single item has been well understood since the work of Myerson, the multi-item case has been much more challenging, and progress over the past three decades has been sporadic. In the first part of this thesis we develop a mathematical framework for finding and characterizing optimal single-bidder multi-item mechanisms by establishing that revenue maximization has a tight dual minimization problem. This approach reduces mechanism design to a measure-theoretic question involving transport maps and stochastic dominance relations. As an important application, we prove that a grand bundling mechanism is optimal if and only if two particular measure-theoretic inequalities are satisfied. We also provide several new examples of optimal mechanisms and we prove that the optimal mechanism design problem in general is computationally intractable, even in the most basic multi-item setting, unless ZPP contains P^#P.

Another key problem in mechanism design is how to efficiently allocate a collection of goods amongst multiple bidders. In the second part of the thesis, we study this problem of welfare maximization in the presence of unrestricted rational collusion. We generalize the notion of dominant-strategy mechanisms to collusive contexts, construct a highly practical such mechanism for multi-unit auctions, and prove that no such mechanism (practical or not) exists for unrestricted combinatorial auctions. Our results explore the power and limitations of enlarging strategy spaces to incentivize agents to reveal information about their collusive behavior.

Mario DeFranco

Title: The unramified principal series of $p$-adic groups: the Bessel Function
Date: Thursday, April 24, 2014
1:00pm, Room: 24-115
Committee: Benjamin Brubaker, Sug Woo Shin, Ju-Lee Kim

Abstract

Let $G$ be a connected, reductive group with a split maximal torus defined over a non-archimedean local field. I evaluate a matrix coefficient of the unramified principal series of $G$ known as the \textit{Bessel function} at torus elements of dominant coweight. I show that the Bessel function shares many properties with the Macdonald spherical function of $G$, in particular the properties described in Casselman's 1980 evaluation of that function. The analogy I demonstrate between the Bessel and Macdonald spherical functions extends to an analogy between the spherical Whittaker function, evaluated by Cassleman and Shalika in 1980, and a previously unstudied matrix coefficient.

Alexander Dubbs

Title: Beta-Ensembles with Covariance
Date: Wednesday, April 30th 2014
12:15pm, Room: 4-145
Committee: Alan Edelman, Alice Guionnet, Gilbert Strang

Abstract

This thesis presents analytic samplers for the $\beta$-Wishart and $\beta$-MANOVA ensembles with diagonal covariance. These generalize the $\beta$-ensembles of Dumitriu-Edelman, Lippert, Killip-Nenciu, Forrester-Rains, and Edelman-Sutton, as well as the classical $\beta = 1,2,4$ ensembles of James, Li-Xue, and Constantine. Forrester discovered a sampler for the $\beta$-Wishart ensemble around the same time, although our proof has key differences. We also derive the largest eigenvalue pdf for the $\beta$-MANOVA case. In infinite-dimensional random matrix theory, we find the moments of the Wachter law, and the Jacobi parameters and free cumulants of the McKay and Wachter laws. We also present an algorithm that uses complex analysis to solve "The Moment Problem." It takes the first batch of moments of an analytic, compactly-supported distribution as input, and it outputs a fine discretization of that distribution.

David Jackson-Hanen

Title: Symplectic Cohomolgy of Contractible Surfaces
Date: Monday, April 28th 2014
3:00pm, Room: 66-144
Committee: Paul Seidel (advisor), Victor Guillemin and Tomasz Mrowka

Abstract

In 2004 Seidel and Smith proved that the Liouville manifold associated to Ramanujam's surface contains a Lagrangian torus which is not displaceable by Hamiltonian isotopy, and hence that higher products of this manifold provide non-standard symplectic structures on Euclidean space which are convex at infinity. I extend these techniques a wide class of smooth contractible affine surface of log general type to produce a similar torus. I then show that the existence of this torus implies the non-vanishing of the symplectic cohomology of the Liouville manifolds associated to these surfaces, thus confirming a portion of McLean's conjecture that a smooth variety has vanishing symplectic cohomology if and only if it is affine ruled.

Ailsa Keating

Title: Symplectic properties of Milnor fibres
Date: Friday, April 11, 2014
10:00am, Room: 12-122
Committee: Peter Kronheimer (Harvard); Tomasz Mrowka; Paul Seidel

Abstract

We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities.

First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.

In the second half of this thesis, we study exact Lagrangian tori in Milnor fibres. The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to aMorsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones.

Daniel Ketover

Title: Heegaard splittings and min-max minimal surfaces
Date: Tuesday, April 29th 2014
4:00pm, Room: E-17-133
Committee: Toby Colding, Bill Minicozzi, Tomasz Mrowka

Abstract

The min-max technique to construct minimal surfaces in a Riemannian manifold was introduced by Birkhoff (1917) to produce unstable closed geodesics on 2-spheres with arbitrary metric. Later Almgren-Pitts and Smith-Simon used geometric measure theory to show that the technique gives rise to smooth closed embedded minimal surfaces in 3-manifolds. In a 3-manifold the most natural way to run the min-max procedure is with respect to a Heegaard splitting, where a min-max sequence of surfaces will converge as varifolds to a minimal surface, potentially with several connected components and different integer multiplicities. In the 80s Pitts-Rubinstein made a number of conjectures about the geometry and topology of minimal surfaces arising in this fashion. We will give a proof of one of these claims: namely that the min max limit is achieved after neck pinches and then collapsing of sheets with multiplicity. As a corollary we obtain new genus bounds for min-max limits.

John Lesieutre

Title: Negative Answers To Some Positivity Questions
Date: Thursday, April 24th, 2014
1:00pm, Room: 36-155
Committee: James McKernan (advisor), Bjorn Poonen, Francois Charles

Abstract

We construct counterexamples to a number of questions related to positivity properties of line bundles on algebraic varieties. The examples are essentially based on studying the geometry of varieties that admit pseudoautomorphisms of positive entropy, and especially the action of standard Cremona transformations on blow-ups of projective space at configurations of points.
The main examples include the following: nefness is not an open condition in families; the diminished base locus of a divisor is not always a closed set; Zariski decompositions do not necessarily exist in dimension three; asymptotic multiplicity invariants are not always finite in the relative setting; and the number of Fourier-Mukai partners of a variety can be infinite.

Mark Lipson

Title: New statistical genetic methods for elucidating the history and evolution of human populations
Date: Thursday, April 10th 2014
4:00pm, Room: 32-G575
Committee: Bonnie Berger (advisor), David Reich (Harvard), and Ankur Moitra

Abstract

In the last few decades, the study of human history has been fundamentally changed by our ability to detect the signatures left within our genomes by adaptations, migrations, population size changes, and other processes. Rapid advances in DNA sequencing technology have now made it possible to interrogate these signals at unprecedented levels of detail, but extracting more complex information about the past from patterns of genetic variation requires new and more sophisticated models. In this work, I present a suite of sensitive and efficient statistical tools for learning about human history and evolution from large-scale genetic data. I focus first on the problem of admixture inference and describe two new methods for determining the dates, sources, and proportions of ancestral mixtures between diverged populations. These methods have already been applied to a number of important historical questions, in particular that of tracing the course of the Austronesian expansion in Southeast Asia. I also report a new approach for estimating the human mutation rate, a fundamental parameter in evolutionary genetics, and provide evidence that it is higher than has been proposed in recent pedigree-based studies.

Tiankai Liu

Title: On planar rational cuspidal curves
Date: Thursday, April 24, 2014
2:30pm, Room: 1-390
Committee: James McKernan (advisor), Bjorn Poonen, François Charles

Abstract

The Coolidge-Nagata conjecture asserts that every rational curve in the complex projective plane that has only cusps (i.e., for which the normalization map is bijective) can be transformed into a line via a birational automorphism of the plane. We will discuss some progress towards this conjecture, and various techniques for studying cuspidal rational curves.

Anand Uttam Oza

Title: A trajectory equation for walking droplets: hydrodynamic pilot-wave theory
Date: Friday, April 25, 2014
3:30 PM, Room 4-349
Committee: John Bush (thesis advisor), Ruben Rosales (thesis advisor), Steven Johnson

Abstract

Yves Couder and coworkers have demonstrated that millimetric droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic quantum realm, including single-particle diffraction, tunneling, quantized orbits, and wave-like statistics in a corral. We here develop an integro-differential trajectory equation for these walking droplets with a view to gaining insight into their subtle dynamics. The orbital quantization is rationalized by assessing the stability of the orbital solutions. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing. In this limit, the complex walker dynamics give rise to a coherent statistical behavior with wave-like features. We then explore various extensions of our pilot-wave trajectory equation. We describe the dynamics of a weakly-accelerating walker in terms of its wave-induced added mass, which provides rationale for the anomalously large orbital radii observed in experiments. We demonstrate that a walker may execute stable circular orbits in the absence of an external force. When subjected to rotation, these hydrodynamic spin states exhibit a macroscopic analogue of Zeeman splitting.

Jennifer Park

Title: Effective Chabauty for symmetric powers of curves
Date: Tuesday, April 29th 2014
1:00pm, Room: 26-168
Committee: Bjorn Poonen, François Charles, Sug Woo Shin

Abstract

Faltings' theorem states that curves of genus g > 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

Oleksandr Tsymbaliuk

Title: The Affine Yangian of $gl_1$ and the Infinitesimal Cherednik Algebras
Date: Wednesday, April 30th 2014
2:00pm, Room: 1-379
Committee: Pavel Etingof, Roman Bezrukavnikov, Ivan Losev (NEU), Valerio Toledano Laredo (NEU)

Abstract

In the first part of this thesis, we obtain some new results about infinitesimal Cherednik algebras. They have been introduced by Etingof-Gan-Ginzburg as appropriate analogues of the classical Cherednik algebras, corresponding to the reductive groups, rather than the finite ones. Our main result is the realization of those algebras as particular finite W-algebras of associated semisimple Lie algebras with nilpotent 1-block elements.
We also generalize the classification results of such algebras to the $SO_n$ case.
In the second part of the thesis, we discuss the loop realization of the affine Yangian of $gl_1$. Similar objects were recently considered in the work of Maulik-Okounkov on the quantum cohomology theory. We present a purely algebraic realization of these algebras by generators and relations. We discuss some families of their representations. A similarity with the representation theory of the quantum toroidal algebra of $gl_1$ is explained by adapting a recent result of Gautam-Toledano Laredo to the local setting.
We also discuss some aspects of those two algebras such as the degeneration isomorphism, a shuffle presentation, and a geometric construction of the Whittaker vectors.

George Tucker

Title: Statistical methods to infer biological interactions
Date: Thursday, May 1st 2014
10:30am, Room: 32-G575
Committee: Bonnie Berger (advisor), Peter Shor, Norbert Perrimon (Harvard Medical School), Eric Banks (Broad)

Abstract

Biological systems are extremely complex, and our ability to experimentally measure interactions is limited by inherent noise. Technological advances have allowed us to collect unprecedented amounts of raw data, increasing the need for computational methods to disentangle true interactions from noise. In this thesis, we focus on statistical methods to infer two classes of important biological interactions: protein-protein interactions and the link between genotypes and phenotypes. In the first half, we introduce methods to infer protein-protein interactions from luminescence-based mammalian interactome mapping (LUMIER). Our work reveals novel insights into the protein homeostasis machinery. In the second half, we focus on methods to understand the link between genotypes and traits. First, we characterize the effects of related individuals on standard association statistics for genome-wide association studies (GWAS) and introduce a new statistic that corrects for relatedness. Then, we introduce a statistically powerful association testing framework that corrects for confounding from population structure in large scale GWAS.