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Graduate Thesis Defenses 2019

Tudor Cristea-Platon

Title: Hydrodynamic analogues of quantum corrals and Friedel oscillations
Date: Friday, April 26th, 2019 | 2:00pm | Room: 4-231
Committee: John W. M. Bush (advisor and chair), Pedro J. Sáenz, and Rodolfo R. Rosales

Abstract

We consider the walking droplet (or ‘walker') system discovered in 2005 by Yves Couder and coworkers. We investigate experimentally and theoretically the behaviour of this hydrodynamic pilot-wave system in both closed and open geometries. First, we consider the dynamics and statistics of walkers confined to corrals. In the elliptical corral, we demonstrate that by introducing a submerged topographical defect, one can create statistical projection effects analogous to the quantum mirage effect arising in quantum corrals. We also report a link between the droplet's statistics and the mean wave-field. In the circular corral, we investigate a parameter regime marked by periodic and weakly aperiodic orbits, then characterise the emergence and breakdown of double quantisation, reminiscent of that arising for walker motion in a harmonic potential. In the chaotic regime, we validate the theoretical result of Durey et al. relating the walker statistics to the mean wave-field. We also rationalise the striking similarity between this mean wave-field and the circular corral's dominant azimuthally-symmetric Faraday mode. Our corral studies underscore the compatibly of the notion of quantum eigenstates and particle trajectories in closed geometries. We proceed by exploring a new hydrodynamic quantum analogue of the Friedel oscillations arising when a walker interacts with a submerged circular well, which acts as a localised region of high excitability. In so doing, we report the first realisation of an open hydrodynamic quantum analogue. We conclude by comparing the hydrodynamic systems to their quantum counterparts. Our work illustrates how, in the closed and open settings considered herein, a pilot-wave dynamics of the form envisaged by de Broglie may lead naturally to emergent statistics similar in form to those predicted by standard quantum mechanics.

Cesar Cuenca

Title: Point Processes of Representation Theoretic Origin
Date: Tuesday, April 23 | 2:30pm | Room 2-449
Committee: Alexei Borodin (chair and advisor), Vadim Gorin, Alexander Postnikov

Abstract

In this thesis, we study certain point processes with origins in the representation theory of the infinite-dimensional orthogonal and symplectic groups. The processes form a 4-parameter family and are called the BC type z-measures.

The first result is that the BC type z-measures are determinantal point processes with explicit correlation kernels written in terms of generalized hypergeometric functions.

The second result, which is joint work with Grigori Olshanski, is that natural q-analogues of the BC type z-measures are the orthogonality measures for the family of q-Racah symmetric functions. This construction is an infinite-variable analogue of a classical picture for orthogonal polynomials of the q-Askey scheme.

Thao Thi Thu Do

Title: Semi-algebraic graphs and hypergraphs in incidence geometry
Date: Wednesday, April 24, 2019 | 1:30pm | Room: 2-449
Committee: Larry Guth (adviser and committee chair), Yufei Zhao and Nike Sun

Abstract

A (hyper)graph is semi-algebraic if its vertices are points in some Euclidean spaces and the (hyper)edge relation is defined by a finite set of polynomial inequalities. Semi-algebraic (hyper)graphs have been studied extensively in recent years, and many classical results in (hyper)graph theory such as Ramsey's theorem and Szemerédi's regularity lemma can be significantly improved in the semi-algebraic setting.

In this defense, we discuss three problems in incidence geometry where the bounds for semi-algebraic (hyper)graphs are generally better than the ones for arbitrary (hyper)graphs : (1) what is the maximum number of hyperedges in a hypergraph forbidding some pattern? (2) what is the most compact way to decompose a graph by complete bipartite subgraphs? and (3) what is the maximum number of edges in a graph where no two neighbor sets have a large intersection?

As most graphs and hypergraphs arising from problems in discrete geometry are semi-algebraic, our results have applications to discrete geometry. The main tools used in our proofs include some version of polynomial partitioning, a Milnor-Thom-type result from topology and a packing-type result in set system theory.

Pavel Galashin

Title: Totally positive spaces: topology and applications
Date: Friday, April 26, 2019 | 3:00pm | Room: 2-146
Committee: Alexander Postnikov, Lauren Williams, Alexei Borodin

Abstract

We study topological spaces arising in total positivity. Examples include Postnikov's totally nonnegative Grassmannian, Lusztig's totally nonnegative part of a partial flag variety, Lam's compactification of the space of electrical networks, and the space of (boundary correlation matrices of) planar Ising networks. We show that all these spaces are homeomorphic to closed balls. In addition, we confirm conjectures of Postnikov and Williams that totally nonnegative partial flag varieties are regular CW complexes. This implies that the closure of each positroid cell inside the Grassmannian is homeomorphic to a closed ball. We discuss the close relationship between the above spaces and the physics of scattering amplitudes, which has served as a motivation for most of our results.

In the second part of the thesis, we investigate the space of planar Ising networks. We give a simple stratification-preserving homeomorphism between this space and the totally nonnegative orthogonal Grassmannian, describing boundary correlation matrices of the planar Ising model by inequalities. Under our correspondence, Kramers--Wannier's high/low temperature duality transforms into the cyclic symmetry of the Grassmannian.

Joint work with Steven Karp, Thomas Lam, and Pavlo Pylyavskyy.

Paul Gallagher

Title: New Progress Towards Three Open Conjectures in Geometric Analysis
Date: Thursday, April 11, 2019 | 3:00pm | Room: 2-255
Committee: Bill Minicozzi (advisor), Toby Colding, Larry Guth

Abstract

This thesis, like all of Gaul, is divided into three parts.

In Part One, I study minimal surfaces in $\mathbb{R}^3$ with quadratic area growth. I give the first partial result towards a conjecture of Meeks and Wolf on asymptotic behavior of such surfaces at infinity. In particular, I prove that under mild conditions, these surfaces must have unique tangent cones at infinity.

In Part Two, I give new results towards a conjecture of Schoen on minimal hypersurfaces in $\mathbb{R}^4$. I prove that if a stable minimal hypersurface $\Sigma$ with weight given by its Jacobi field has a stable minimal weighted subsurface, then $\Sigma$ must be a hyperplane inside of $\mathbb{R}^4$.

Finally, in Part Three, I do an in-depth analysis of the nodal set results of Logonov-Malinnikova. I give explicit bounds for the eigenvalue exponent in terms of dimension, and make a slight improvement on their methodology.

Jan-Christian Huetter

Title: Minimax Estimation with Structured Data: Shape Constraints, Causal Models, and Optimal Transport
Date: Wednesday, April 24, 2019 | 10:00am | Room: 2-361
Committee: Philippe Rigollet (advisor, chair), Elchanan Mossel, Alexander Rakhlin

Abstract

Modern statistics often deals with high-dimensional problems that suffer from the curse of dimensionality.

In this thesis, we study how structural assumptions can be used to overcome this difficulty in several estimation problems, spanning three different areas of statistics: shape-constrained estimation, causal discovery, and optimal transport.

In the context of shape-constrained estimation, we study the estimation of matrices, first under the assumption of bounded total-variation (TV) and second under the assumption that the underlying matrix is Monge, or super-modular.

While the first problem has a long history in image denoising, the latter has so far been mainly investigated in the context of computer science and optimization.

For TV denoising, we provide fast rates that are adaptive to the underlying edge sparsity of the image, as well as generalizations to other graph structures, including higher-dimensional grid-graphs.

For the estimation of Monge matrices, we give near minimax rates for their estimation, including the case where latent permutations act on the rows and columns of the matrix.

In the latter case, we also give two computationally efficient and consistent estimators.

Moreover, we show how to obtain estimation rates in the related problem of estimating continuous totally positive distributions in 2D.

In the context of causal discovery, we investigate a linear cyclic causal model and give an estimator that is near minimax optimal for causal graphs of bounded in-degree.

In the context of optimal transport, we introduce the notion of the transport rank of a coupling and provide empirical and theoretical evidence to show how it can be used to significantly improve rates of estimation of Wasserstein distances and optimal transport plans.

Finally, we give near minimax optimal rates for the estimation of smooth optimal transport maps based on a wavelet regularization of the semi-dual objective.

Yusuf Baris Kartal

Title: Distinguishing open symplectic mapping tori via their wrapped Fukaya categories
Date: Thursday, March 21, 2019 | 10:00am | Room: 2-361
Committee: Paul Seidel (Advisor, Chair), Davesh Maulik, Denis Auroux (Harvard)

Abstract

The main goal of this thesis is to use homological methods as a step towards the classification of symplectic mapping tori. More precisely, we exploit the dynamics of wrapped Fukaya categories to distinguish an open version of symplectic mapping torus associated to a symplectomorphism from the mapping torus of the identity. As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and symplectic cohomology, but that are different as Liouville domains.

This work consists of two parts: in the first part, we define an algebraic model for the wrapped Fukaya category of the open symplectic mapping tori. This construction produces a category- called the mapping torus category- for a given dg-category over $\mathbb{C}$ with an autoequivalence. We then use the continuous dynamics of deformations of these categories to distinguish them under certain hypotheses. More precisely, we construct families of bimodules- analogous to flow lines and use their different periodicity. The construction of the flow uses the geometry of the Tate curve and formal models for the graph of multiplication on $\mathbb{G}_{m,\mathbb{C} ((q))}^{an}$.

The second part focuses on the comparison of mapping torus categories and the wrapped Fukaya categories of the open symplectic mapping tori. For this goal, we introduce the notion of "twisted tensor product" and prove a twisted K\"unneth theorem for the open symplectic mapping tori by using the count of quilted strips. In this part, we also give a large class of Weinstein domains whose wrapped Fukaya category satisfies the conditions for the theorem on mapping torus categories to hold.

Dax Koh

Title: Classical simulation complexity of restricted models of quantum computation
Date: Tuesday, April 23, 2019 | 3:45pm | Room: 1-242
Committee: Peter W. Shor (advisor and chair), Michel X. Goemans, Aram W. Harrow

Abstract

Restricted models of quantum computation are mathematical models which describe quantum computers that have limited access to certain resources. Well-known examples of such models include the boson sampling model, extended Clifford circuits, and instantaneous quantum polynomial-time circuits. While unlikely to be universal for quantum computation, several of these models appear to be able to outperform classical computers at certain computational tasks, such as sampling from certain probability distributions. Understanding which of these models are capable of performing such tasks and characterizing the classical simulation complexity of these models—i.e. how hard it is to simulate these models on a classical computer—are some of the central questions we address in this thesis.

Our first contribution is a classification of various extended Clifford circuits according to their classical simulation complexity. Among these circuits are the conjugated Clifford circuits, which we prove cannot be efficiently classically simulated up to multiplicative or additive error, under certain plausible conjectures in computational complexity theory. Our second contribution is an estimate of the number of qubits needed in various restricted quantum computation models in order for them to be able to demonstrate quantum computational supremacy. Our estimate is obtained by fine-graining existing hardness results for these restricted models. Our third contribution is a new alternative proof of the Gottesman-Knill theorem, which states that Clifford circuits can be efficiently simulated by a classical computer. Our proof uses the sum-over-paths technique and establishes a correspondence between quantum circuits and a class of exponential sums. Our final contribution is a theorem characterizing the operations that can be efficiently simulated using a particular rebit simulator. An application of this result is a generalization of the Gottesman-Knill theorem that allows for the efficient classical simulation of certain nonlinear operations.

Kristin Kurianski

Title: Estimates for solutions to the Dysthe equation and numerical simulations of walking droplets in harmonic potentials
Date: Wednesday, April 24, 2019 | 2:00pm | Room: 2-361
Committee: Gigliola Staffilani (Advisor/Chair), Rodolfo Ruben Rosales, John Bush

Abstract

In this thesis, we study wave-type phenomena both from a numerical point of view and a theoretical one.

We first present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states.

We then recast the integro-differential equation as a coupled system of ordinary differential equations in time. This method is used to simulate droplet lattices in various configurations and in the presence of a harmonic potential, creating structures reminiscent of Wigner molecules. The development of this approach is presented in detail along with its future applications.

We then switch focus to a fluid system described by a modified nonlinear Schrödinger equation. The surface of an incompressible, inviscid, irrotational fluid of infinite depth can be described in two dimensions by the Dysthe equation. Recently, this equation has been used to model extraordinarily large waves occurring on the ocean’s surface called rogue waves. In this thesis, we prove dispersive estimates and Strichartz estimates for the Dysthe equation. We then prove a Kato-type smoothing effect in which we are able to bound uniformly in space the $L^2$ norm in time of a fractional derivative of the linear solution by the $L^2$ norm in space of the initial data. This section of the thesis lays the groundwork for further developments in proving well-posedness via a contraction argument.

Haihao (Sean) Lu

Title: Large-Scale Optimization Methods for Data-Science Applications
Date: Friday, April 26th, 2019 | 2:00pm | Room: E62-550 (at Sloan)
Committee: Robert M. Freund (advisor), Ankur Moitra (chair), Jon Kelner, Rahul Mazumder

Abstract

In this thesis, we present several contributions of large scale optimization methods with the applications in data science and machine learning.

In the first part, we introduce a new functional measure called the growth constant for the convex objective function, based on which we achieve new computational guarantees for both smooth and non-smooth convex optimization, that can improve existing computational guarantees, most notably when the initial iterate is far from the optimal solution set.

In the second part, we develop the notion of relative smoothness, relative continuity and relative strong convexity that is determined relative to a user-specified “reference function”. By doing so, we extend the scope of problems that first-order methods can tackle. We extend the mirror descent algorithm to our new setting, with associated computational guarantees.

In the third part, we propose two new boosting algorithms: the Randomized Gradient Boosting Machine (RGBM) and the Accelerated Gradient Boosting Machine (AGBM). RGBM leads to significant computational gains compared to GBM. AGBM incorporate Nesterov’s acceleration techniques into the design of GBM, which is the first GBM type of algorithm with theoretically-justified accelerated convergence rate.

Lyuboslav Panchev

Title: On the v_1-periodicity of the Moore space
Date: Monday, April 29, 2019 | 3:15pm | Room: 2-131
Committee: Haynes Miller (Thesis advisor), Michael Hopkins (Harvard), Zhouli Xu

Abstract

We present progress in trying to verify a long-standing conjecture by Mark Mahowald on the v1-periodic component of the classical Adams spectral sequence for a Moore space M. The approach we follow was proposed by John Palmieri in his work on the stable category of A-comodules. We improve on Palmieri's work by working with the endomorphism ring of M - End(M), thus resolving some of the initial diculties of his approach and formulating a conjecture of our own that would lead to Mahowald's formulation. We further improve upon a method for calculating differentials via double filration first used by Miller and apply it to our problem.

Kevin Sackel

Title: Getting a Handle on Contact Manifolds
Date: Monday, March 18, 2019 | 3:00pm | Room: 2-449
Committee: Emmy Murphy (advisor, Northwestern), Paul Seidel (chair), Larry Guth

Abstract

In this thesis, we develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact manifolds is guaranteed, a result equivalent to the existence of supporting Weinstein open book decompositions. In the final section, we provide a few words about how this theory is related to the Giroux correspondence between Weinstein open book decompositions and contact structures in three dimensions, as well as providing a framework for possible generalizations to higher dimensions and homotopy data.

Nick Strehlke

Title: The level set equation for mean curvature flow on a convex domain
Date: Tuesday, April 9, 2019 | 12:00pm | Room: 56-159
Committee: Toby Colding (advisor and chair), Tristan Collins, and Bill Minicozzi

Abstract

The level set equation for mean curvature flow is a degenerate elliptic boundary value problem admitting solutions on mean convex domains. On a convex domain, the solution is $C^2$ and has an isolated critical point, and we give asymptotics for solutions near this critical point. Natasa Sesum showed that solutions are not necessarily $C^3,$ and we recover this result and construct non-smooth solutions which are $C^3.$ We also construct solutions having prescribed behavior near the maximum. We do this by analyzing the asymptotics for rescaled mean curvature flow converging to a stationary sphere.

Nicholas Triantafillou

Title: The Chabauty-Coleman Method, Restriction of Scalars, and $\mathbb P^1 \smallsetminus \{0, 1, \infty\}.$
Date: Thursday, April 18, 2019 | 3:00pm | Room: 2-449
Committee: Bjorn Poonen (Advisor and Committee Chair), Wei Zhang, Andrew Sutherland

Abstract

For a number field $K$ and a curve $C/K$, Chabauty's method is a powerful $p$-adic tool for bounding/enumerating the set $C(K)$. The method typically requires that dimension of the Jacobian $J$ of $C$ is greater than the rank of $J(K)$. Since this condition often fails, especially when $[K:\mathbb Q]$ is large, several techniques have been proposed to augment Chabauty's method. For proper curves, Siksek introduced an analogue of Chabauty's method for the restriction of scalars $\text{Res}_{K/\mathbb Q} C$ that can often be used to prove that $C(K)$ is finite (and to compute $C(K)$!) when the rank of $J(\mathcal O_{K,S})$ is as large as $[K:\mathbb Q]\cdot (\dim J - 1)$. However, there are sometimes geometric reasons why Chabauty's method for the restriction of scalars cannot be used to compute $C(K)$.

Developing an analogue of Chabauty's method for restrictions of scalars in the non-proper case, we study the power of this approach together with descent for computing $C = (\mathbb P^1\smallsetminus \{0,1,\infty\})(\mathcal O_{K,S})$. Under mild assumptions on $K$, we show that in this case, descent allows us to avoid all known geometric obstructions to Chabauty's method for restriction of scalars. We also illustrate use of the method with some examples.

Siddharth Venkatesh

Title: Geometry and Representation Theory in the Verlinde Category
Date: Monday, April 29, 2019 | 1:00pm | Room: 2-255
Committee: Pavel Etingof, Roman Bezrukavnikov, George Lusztig

Abstract

In this talk, I will setup the framework for algebraic geometry in symmetric tensor categories in positive characteristic, with a special focus on the Verlinde category, a universal base for semisimple categories. I will give a description of the Verlinde category and then show how you can reduce algebraic geometry in this category to ordinary algebraic geometry. Subsequently, I will apply this to studying affine group schemes and their representations in the Verlinde category. In particular, I will show that the data of a group scheme in this category is the same as a compatible pair of an ordinary group scheme and a Lie algebra in Verlinde. At the end of the talk, I will focus on the example of the group scheme GL(X) for an object X in the Verlinde category, and characterize the representation theory when X is simple.

Isabel Vogt

Title: Some results in the arithmetic and geometry of curves
Date: Monday, April 22, 2019 | 1:00pm | Room: 2-449
Committee: Bjorn Poonen (chair), Joe Harris, Davesh Maulik

Abstract

This thesis consists of four chapters, moving along the spectrum from geometry to arithmetic. In this talk, we will focus on the problems studied in the first two chapters.

In the first chapter, we consider the question of interpolation for Brill--Noether curves. Namely, given $n$ general points in $\mathbb{P}^r$, when does there exist a Brill--Noether curve of degree $d$ and genus $g$ through these points? Via deformation theory, this problem is related to a stability-like condition on the normal bundle of a general such curve. Using this perspective, we answer the question completely when $r=3$ and (joint with Eric Larson) when $r=4$.

In the second chapter, we consider an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer $e$ such that the points of residue degree bounded by $e$ are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to $e$ at least 4. We study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This chapter is joint work with Geoffrey Smith.

In the final chapters we focus more heavily on the arithmetic of elliptic curves. In the third chapter, we study the relationship between the linear algebraic data of (torsion-free, finitely presented) modules over the endomorphism ring of an elliptic curve, and abelian varieties isogenous to a power of the elliptic curve. We generalize a functor attributed to Serre and Tate to the setting where the endomorphism ring of the elliptic curve grows in a Galois extension of the ground field. In the forth chapter, we study a local-global problem for level structure on elliptic curves. Namely, we consider the following problem: if $E$ is an elliptic curve, almost all of whose reductions have a cyclic isogeny of degree $N$, then must $E$ globally have a cyclic $N$-isogeny? Building on work of Sutherland, Anni, and Banwait--Cremona when $N$ is prime, we address the composite degree case. In particular, we classify the level structure giving rise to exceptions and prove finiteness statements.

Jane Wang

Title: The geometry and dynamics of twisted quadratic differentials
Date: Wednesday, April 10, 2019 | 3:00pm | Room: 2-449
Committee: Curt McMullen (advisor), Larry Guth (chair), Tom Mrowka

Abstract

This thesis examines twisted quadratic differentials, also known as dilation surfaces. These are variants of translation surfaces, their more well-studied counterpart. In this work, we study questions about the realizability of mapping class group elements in the affine automorphism groups of dilation surfaces, and how large affine automorphism groups can be. We demonstrate how to construct dilation surfaces with a given pseudo-Anosov map in their affine automorphism group, show the existence of exotic Dehn twists, and construct dilation surfaces with simultaneous Dehn multitwists. The last construction also gives rise to some large affine automorphism groups.

Jonathan Weed

Title: Statistical problems in transport and alignment
Date: Thursday, April 25, 2019 | 10:00am | Room: 2-361
Committee: Rigollet (Chair), Moitra, Justin Solomon (EECS)

Abstract

How should you analyze complicated data? Faced with scans of handwritten digits, noisy snapshots of large biomolecules, three-dimensional LIDAR data, or corrupted social networks, what practical techniques and theoretical guarantees does the statistician have at her disposal? This thesis develops new theory for statistical problems involving data with geometric structure of this kind.

First, we study the Wasserstein distance, a metric on the space of probability measures on an arbitrary metric space. We prove sharp rates of convergence for empirical measures in Wasserstein distance on sufficiently regular compact metric spaces, improving on a line of work going back to Dudley (1969). We give the first nearly-optimal minimax lower bounds for the problem of estimating the Wasserstein distance between two measures, and we prove much better rates can be obtained under three different structural assumptions on the measures. These assumptions, inspired by practice and theory, reveal novel statistical features of the Wasserstein distance.

Second, we consider data corrupted by group transformations. These problems are motivated by cryo-electron microscopy, an important technique in structural biology, the use of which requires reconstructing the structure of biological macromolecules on the basis of noisy, randomly rotated images. We prove the first minimax rates of estimation for a two-dimensional version of this problem. Along the way, we develop a general theory for problems of this kind, applicable to arbitrary compact groups acting on R^d, and provide a novel analysis of the maximum-likelihood estimator for Gaussian mixtures with algebraic structure.

Jingwei Xiao

Title: Germ Expansion, Endoscopic Transfer, and Unitary Periods
Date: Friday, April 19, 2019 | 11:00am | Room: 2-242
Committee: Wei Zhang (advisor and chair), ZhiWei Yun, Omer Offen

Abstract

In this thesis, we study the germ expansions in the Jacquet-Rallis transfer. We prove an identity that relates certain nilpotent orbital integrals for any smooth matching in this transfer. We give two applications of this identity. For the first, we give an elementary local proof of the endoscopic fundamental lemma for unitary groups (theorem of Laumon and Ngo). For the second, we establish a new relative trace formula comparison that is conjectured by Jacquet.