Thesis Defenses
2025
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Niven Achenjang
Date: Tuesday, April 15, 2025 | 2:15pm | Room: 4-265 | Zoom Link
Committee: Bjorn Poonen, Wei Zhang, and Melanie Matchett Wood
The Average Size of 2-Selmer Groups of Elliptic Curves in Characteristic 2
Given an elliptic curve E over a global field K, the abelian group E(K) is finitely generated, and much effort has been put into trying to understand the behavior of its rank, as E varies. Of note, it is a folklore conjecture that, when all elliptic curves are ordered by a suitably defined height, the average value E[rank E(K)] of their ranks is exactly 1/2, but of course it is not a priori obvious that such an average rank is even finite. In my thesis, I bound E[rank E(K)], for an arbitrary global *function* field K, including those of small characteristic. Combined with previous work of Bhargava-Shankar and Shankar, this completes the proof that E[rank E(K)] is finite for all global fields. -
Evan Chen
Date: Wednesday, December 11, 2024 | 12:30pm | Room: 4-237 | Zoom Link
Committee: Wei Zhang, Zhiwei Yun, Ben Howard
Explicit formulas for weighted orbital integrals for the inhomogeneous and semi-Lie arithmetic fundamental lemmas conjectured for the full spherical Hecke algebra
As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by W. Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by S. Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, there is a recent conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Y. Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above.
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Mo Chen
Date: Tuesday, April 22, 2025 | 10:30am | Room: 2-361 | Zoom Link
Committee: Steven G. Johnson (Chair and Advisor), Laurent Demanet, Marin Soljačić
New Regimes for Topology Optimization in Photonics
Inverse design is a powerful methodology to obtain non-trivial and non-intuitive photonic structures of unprecedent performance. Topology optimization is a particular class of inverse design method that has been increasingly popular in photonics. Numerous topology optimization tools and frameworks have been developed and often yield satisfying results for various engineering problems. This work explores the subtleties involved in the development and application of topology optimization, and presents new regimes for photonic design, where the key to finding the right solutions lies in posing the right questions.
To begin with, we first review the current frameworks for photonic topology optimization. We point out that, as new algorithms emerge, the lack of standardized validation methods presents a challenge for further advancements. To address this, we provide a comprehensive suite of test problems along with a length-scale metric for comparing designs across different algorithms, aiming to facilitate the development and validation of future inverse design approaches.
However, a functioning inverse design algorithm alone is not sufficient to guarantee satisfying designs. We present two case studies highlighting the importance of careful formulation for achieving mathematical robustness and tractability that is crucial to the success of optimization.
The first case examines the inverse design of 3D-printable metalenses with complementary dispersion for terahertz imaging. It illustrates a physical dichotomy between achieving two distinct dispersion behaviors in a thin structure. We demonstrate that a key aspect in making such design tractable is carefully balancing the trade-offs between focal quality and scanning rate in the optimization problem formulation.
The second case focuses on the inverse design of multiresonance filters via quasi-normal mode theory. Traditional filter design approaches have various limitations, and directly applying topology optimization leads to numerically stiff formulations. We propose a new practical high-order-filter design method based on a minimal set of analytical design criteria derived from quasi-normal mode theory. We illustrate our approach by designing 3rd and 4th-order elliptic and Chebyshev dielectric filters. -
Ryan Chen
Date: Thursday, April 17, 2025 | 2:30pm | Room: 3-333 | Zoom Link
Committee: Wei Zhang (advisor and chair), Bjorn Poonen, Zhiwei Yun
Co-rank 1 Arithmetic Siegel--Weil
Theta series for lattices count lattice vectors of fixed norm. Such theta series give some of the first examples of automorphic forms.
It is possible to form "theta series" in other geometric contexts, e.g. for counting problems involving abelian varieties.
It is expected that these theta series again have additional automorphic symmetry.
I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program. These "geometrize" the classical Siegel--Weil formulas, on lattice and lattice vector counting via Eisenstein series.
At these near-central points of functional symmetry, we observe that both the "leading" special value (complex volumes) and the "subleading" first derivative (arithmetic volume) simultaneously have geometric meaning.
The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic, as well as its automorphic counterpart. -
Zihong Chen
Date: Monday, April 28, 2025 | 1:00pm | Room: 2-132 | Zoom Link
Committee: Paul Seidel (Advisor), Denis Auroux, Shaoyun Bai
Quantum Steenrod operations and Fukaya categories
The recent introduction of mod p equivariant operations to symplectic Gromov-Witten theory has fueled exciting developments in the field. In this thesis, we develop new tools for understanding these operations and explore an application to the quantum connection. In one direction, we construct certain operations on the equivariant Hochschild (co)homology of a general A∞-category. We show that when applied to the Fukaya category of a non- degenerated closed monotone symplectic manifold, this construction can be identified with the quantum Steenrod operations via Ganatra’s cyclic open-closed maps. A key ingredient in this identification is a new homotopy theoretic framework for studying various equivariant open-closed maps at once, using a combination of cyclic categories, edgewise subdivision and Abouzaid-Groman-Varolgunes’ operadic Floer theory. In another direction, we utilize quantum Steenrod operations, and Lee’s observation that it is related to the p-curvature of the quantum connection, to study singularities of the quantum connection in characteristic 0, and prove the exponential type conjecture for all closed monotone symplectic manifolds. -
Anlong Chua
Date: Wednesday, April 9, 2025 | 4:00pm | Room: 2-141
Committee: Roman Bezrukavnikov, Pavel Etingof, Zhiwei Yun
Two applications of the Kazhdan-Lusztig map
Let $G$ be a connected reductive group with Lie algebra $\mathfrak{g}$ and Weyl group $W$. Let $P \subset G((t))$ be a parahoric subgroup with Levi quotient $G_P$.
Using the topology of $\operatorname{Lie} P$, Kazhdan and Lusztig define a map from nilpotent orbits in $\operatorname{Lie}G_P$ to conjugacy classes in $W$.
I will explain some compatibilities between Kazhdan-Lusztig maps associated to different parahoric subgroups, as well as the Kazhdan-Lusztig map for the Langlands dual. Time permitting, I will sketch that these compatibilities come from studying the $W$-representation on the cohomology of affine Springer fibers. The main tool is Yun’s Global Springer Theory.
We will explore two applications of these compatibilities. The first is an affine analog of the classical picture relating singular supports of $\operatorname{IC}$ sheaves on the flag variety with special nilpotent orbits. The second is a resolution of Lusztig’s conjecture that strata can be described by fibers of (parahoric) Kazhdan-Lusztig maps.
Seminar webpage: https://math.mit.edu/lg/ -
Alex Cohen
Date: Friday, April 18, 2025 | 10:00am | Room: 2-131 | Zoom Link
Committee: Larry Guth (chair and advisor), Semyon Dyatlov, Henry Cohn
Fractal uncertainty in higher dimensions
A central question in quantum chaos is to understand the behavior of high-frequency eigenfunctions on hyperbolic manifolds. In 2017 Bourgain and Dyatlov proved an uncertainty principle for fractal subsets of the real line, which Dyatlov and Jin later applied to prove new results about eigenfunctions. In this defense, I will discuss an extension of the Bourgain–Dyatlov’s theorem to higher dimensions. The proof uses a connection between harmonic analysis, complex analysis, and potential theory. This connection is used to analyze the decay rate of band-limited functions—a delicate problem in Fourier analysis. My main contribution is constructing certain plurisubharmonic functions on C^d, which is a problem in potential theory. -
Davis Evans
Date: Wednesday, January 8, 2025 | 10:30am | Room: 2-105 | Zoom Link
Committee: John Bush, Jörn Dunkel, Ruben Rosales, Bauyrzhan Primkulov
Ponderomotive Forces in Pilot-Wave Hydrodynamics
Droplets bouncing on a vibrating bath may self-propel (or 'walk') via a resonant interaction with their self-induced pilot wave. In pilot-wave hydrodynamics (PWH), the spontaneous emergence of coherent, wave-like statistics from chaotic trajectories has been reported in several settings. Owing to the similarity of PWH to Louis de Broglie's realist picture of quantum mechanics, the question of how such statistics emerge has received considerable recent attention. A compelling setting where coherent statistics emerge in PWH is the hydrodynamic analog of the quantum corral. When walking droplets are confined to a circular cavity or 'corral', a coherent statistical pattern emerges, marked by peaks in the positional histogram coincident with extrema of the cavity eigenmode. Stroboscopic models that idealize the drop's bouncing dynamics as being perfectly resonant with their Faraday wave field have proven incapable of capturing the emergent statistics.
In this thesis, we present new experimental and theoretical findings in a variety of pilot-wave hydrodynamical settings where non-resonant bouncing plays a key role in the droplet dynamics and emergent statistics. We present an integrated experimental and theoretical study of the hydrodynamic corral, highlighting the role of non-resonant bouncing in the emergent statistics.
Our experimental findings motivate a new theoretical framework that predicts that modulations in the histogram emerge as a consequence of ponderomotive effects induced by non-resonant bouncing. We then connect the ponderomotive drift observed in hydrodynamic corrals to extant theories of quantum mechanics. -
Marisa Gaetz
Date: Tuesday, April 15, 2025 | 2:00pm | Room: 3-442 | Zoom Link
Committee: David Vogan, Pavel Etingof, and George Lusztig
Dual Pairs and Disconnected Reductive Groups
In R. Howe's seminal 1989 paper, "Remarks on classical invariant theory," he introduces the notion of a Lie algebra dual pair (a pair $(\mathfrak{g}_1, \mathfrak{g}_2)$ of reductive Lie subalgebras of a Lie algebra $\mathfrak{g}$ such that $\mathfrak{g}_1$ and $\mathfrak{g}_2$ equal each other's centralizers in $\mathfrak{g}$) and the notion of a Lie group dual pair (a pair $(G_1, G_2)$ of reductive subgroups of a reductive Lie group $G$ such that $G_1$ and $G_2$ are each other's centralizers in $G$). Both notions have since been widely used and studied. This thesis extends what is known about the classifications of complex reductive Lie group and Lie algebra dual pairs, and establishes a step towards a more general framework for understanding complex reductive Lie group dual pairs.
In the first part of this thesis, we classify the reductive dual pairs in the complex classical Lie groups: $GL(n,\mathbb{C})$, $SL(n,\mathbb{C})$, $O(n,\mathbb{C})$, $SO(n,\mathbb{C})$, and $Sp(2n,\mathbb{C})$. We also establish some general relationships between Lie group dual pairs and dual pairs in corresponding Lie algebras and quotient groups. These relationships lead to complete classifications of the reductive dual pairs in the complex classical Lie algebras ($\mathfrak{gl}(n,\mathbb{C})$, $\mathfrak{sl}(n,\mathbb{C})$, $\mathfrak{so}(n,\mathbb{C})$, and $\mathfrak{sp}(2n,\mathbb{C})$) and preliminary progress towards classifying dual pairs in the projective classical groups ($PGL(n,\mathbb{C})$, $PSp(2n,\mathbb{C})$, $PO(n,\mathbb{C})$, and $PSO(n,\mathbb{C})$).
In the second part of this thesis, we complete an explicit classification of the semisimple Lie algebra dual pairs in the complex exceptional Lie algebras, initially outlined by H. Rubenthaler in a 1994 paper. This explicit classification makes Rubenthaler's 1994 result more complete, usable, and understandable.
A major obstacle to understanding reductive Lie group dual pairs is their potential disconnectedness. Inspired in part by this obstacle, in the third part of this thesis we describe the possible disconnected complex reductive algebraic groups $E$ with component group $\Gamma = E/E_0$. We show that there is a natural bijection between such groups $E$ and algebraic extensions of $\Gamma$ by $Z(E_0)$.
Finally, in the last part of this thesis we classify the reductive dual pairs in $PGL(n,\mathbb{C})$. While the connected dual pairs in $PGL(n,\mathbb{C})$ can be easily understood using tools from the first part of this thesis, the classification of the disconnected dual pairs in $PGL(n,\mathbb{C})$ is much more difficult and requires tools from the third part of this thesis. This serves as the first complete classification of dual pairs in a non-classical group and as a step towards understanding how disconnectedness factors into the classification of dual pairs more generally. -
Sarah Greer
Date: Monday, February 10, 2025 | 10:00am | Room: 2-255
Committee: Laurent Demanet, Alan Edelman, John Urschel
Geometrically-informed methods of wave-based imaging
In this thesis, we are interested in understanding and advancing wave-based imaging techniques defined by the adjoint-state method. Wave-based imaging uses wavefield data from receivers on the boundary of a domain to produce an image of the underlying structure in the domain of interest. These images are defined by the imaging condition, derived from the first-order adjoint-state method, which corresponds to the gradient and maps recorded data to their reflection points in the domain. In the first part, we introduce a nonlinear modification to the standard imaging condition that can produce images with resolutions greater than that ordinarily expected using the standard imaging condition. We show that the phase of the integrand of the imaging condition, in the Fourier domain, has a special significance in some settings that can be exploited to derive a super-resolved modification of the imaging condition. Whereas standard imaging techniques can resolve features of a length scale of $\lambda$, our technique allows for resolution level $R < \lambda$, where the super-resolution factor (SRF) is typically $\lambda/R$. We show that, in the presence of noise, $R \sim \sigma$. In the second part, we investigate the Hessian operator, which arises from the second-order adjoint-state method, in the context of full-waveform inversion, a non-linear least-squares problem for estimating material properties within the domain of interest. We analyze the contributions of reflected and transmitted waves to the linearized Hessian operator, demonstrating that reflected waves generally produce a high-rank component with well-distributed eigenvalues, while transmitted waves contribute to a low-rank component with poorly distributed eigenvalues. This decomposition of the Hessian, motivated by the underlying physical system, provides insights that can be used to improve inversion strategies. The advancements in both parts of this thesis leverage the underlying structure and geometry of the domain of interest, providing the foundation for the zero-phase imaging condition in the first part and informing the decomposition of the Hessian operator in the second part. -
Mitchell Harris
Date: Wednesday, February 5, 2025 | 2:00pm | Room: 2-132
Committee: Pablo Parrilo, Steven Johnson, Ankur Moitra
Computational Tradeoffs and Symmetry in Polynomial Nonnegativity
Understanding when a polynomial is nonnegative on a region is a fundamental problem in applied mathematics. Although exact conditions for nonnegativity are computationally intractable, there has been a surge of recent work giving sufficient conditions for nonnegativity to address its many practical applications. A major trend in this direction has been the use of convex optimization to characterize polynomials that are sums of squares (SOS); nevertheless, this well-studied condition can be computationally intensive for polynomials of moderate degree and dimension.
This thesis addresses the challenge of balancing computational cost against the strength of sufficient conditions for nonnegativity. We make progress towards bridging the gap between simple but crude sufficient conditions, and the more powerful but expensive SOS approach.
In the first part, we introduce new certificates of nonnegativity that may be used when SOS is too expensive yet cheaper sufficient conditions are too conservative. For this, we leverage different features of the polynomials, including its Bernstein coefficients, a lower-degree interpolant, or its harmonic decomposition.
In the second part, we construct coordinate-invariant sufficient conditions for nonnegativity and study the symmetry properties of the space of Gram matrices. By considering it as a representation of GL(n, R) and combining this module structure with classical invariant theory, we construct an explicit equivariant map for nonnegativity certification. We further introduce an alternative approach using equivariant neural networks, analyzing their benefits and limitations. -
Andrey Khesin
Date: Thursday, December 5, 2024 | 10:00am | Room: 2-449 | Zoom Link
Committee: Peter Shor (chair and advisor), Isaac Chuang, Aram Harrow, Jonathan Kelner
Quantum Computing from Graphs
Many are familiar with the notion that quantum computers are fundamentally different to classical ones. One of these differences is the fact that performing quantum measurements can change the underlying quantum state. Additionally, quantum information is difficult to transmit and store, so algorithms for quantum error-correction and fault-tolerance are of much interest. While the most common representations of error-correcting codes have proven exceptionally useful as a descriptive tool, they otherwise offer little guidance for concrete constructions or coding algorithm analysis. We introduce a representation of certain quantum error-correcting codes as graphs with certain structures. With these graphs we can convert efficiently between various code representations, gain insight into how the codes propagate information, and discuss properties of codes by examining analogous properties in the codes' graphs. In particular, we show that one such graph property puts lower bounds on its code's distance, as well as gives us a simple and efficient decoding procedure for the code. This procedure is very similar to playing a quantum version of the children's game Lights Out. This change in perspective has already led to discovering several new codes and proving general results about typical graph codes, extending results on best known bounds. This defense will include a general introduction to quantum error-correction, a showcase of various graph codes, both old and new, as well as an explanation of the quantum Lights Out game and its relationship to decoding. -
Cameron Krulewski
Date: Monday, April 28, 2025 | 5:30pm | Room: 2-135 | Zoom Link
Committee: Michael Hopkins, Haynes Miller, Jeremy Hahn
Invertible Functorial Field Theory for Symmetry Breaking and Interactions in Quantum Field Theory
I will discuss two applications of invertible field theories to quantum field theory. Functorial field theories, which are functors from a bordism category to a target category, are invertible when they factor through the Picard groupoid of the target. After additionally imposing reflection positivity, such theories are classified, due to results of Freed-Hopkins, by Anderson-dual bordism groups. The first application we study is toward a certain form of spontaneous symmetry breaking. We model three physical processes using a twisted Gysin sequence of Anderson-dual bordism groups. Using generalized Euler classes, we study the “Smith maps” of Madsen-Tillmann spectra that underlie the sequence, and use them to draw physical predictions.
The second application we study is toward fermionic symmetry-protected topological phases (SPTs). Generalizing work of Freed-Hopkins, we define and compute twisted Atiyah-Bott-Shapiro maps from twisted spin bordism to shifts of K-theory in order to compare two models of SPTs. This talk represents several joint projects with Antolín Camarena, Debray, Devalapurkar, Liu, Pacheco-Tallaj, Sheinbaum, Stehouwer, and Thorngren. -
David Lee
Date: Monday, April 14, 2025 | 4:30pm | Room: 2-131
Committee: Jeremy Hahn (Advisor), Michael Hopkins (Harvard), and Haynes Miller
On the p-local homotopy type of truncated Brown—Peterson spectra
It is a result of Adams and Priddy that $BSU$ admits a unique infinite loop space structure after localization at a prime $p$. On the way to proving this statement, they prove that the $p$-localized connective complex $K$-theory spectrum $ku_{(p)}$ is characterized by its $\mathbb F_p$-cohomology as a Steenrod module. I will talk about a generalization of this part to truncated Brown—Peterson spectra $BP\langle n\rangle$ at odd primes, which can be thought of as higher chromatic analogues of the connective complex $K$-theory. In particular, since the $\mathbb F_p$-cohomology depends only on the $p$-completion, a part of this result is that we can recover the $p$-local homotopy type of $BP\langle n\rangle$ from its $p$-completion. Finally, I will describe some applications and open questions, including the status of the problem at $p=2$. -
Tang-Kai Lee
Date: Thursday, April 17, 2025 | 3:30pm | Room: 4-149
Committee: Bill Minicozzi (advisor), Larry Guth, Tristan Ozuch
Uniqueness problems in mean curvature flow
We investigate uniqueness phenomena in mean curvature flow. In the defense, we talk about separate joint work with Alec Payne and Jingze Zhu, focusing on two central problems: the behavior of the flow near singularities and the extension of the flow beyond singular times. These questions have significant applications in geometry, topology, and analysis.
For the first problem, given a generic closed mean curvature flow of surfaces, we formulate a canonical way to study the limit model near a singularity. Using this framework, we establish a uniqueness result for singularity models. As a consequence, we resolve a uniqueness problem for gradient flow lines in ordinary differential equation theory, related to a question posed by Thom and Arnold and revisited by Colding--Minicozzi.
For the second problem, we examine the level set flow as a weak formulation that ensures long-time existence of mean curvature flow past singularities. This approach, however, can lead to fattening, a phenomenon reflecting "genuine non-uniqueness" of the extended flow. While genuine uniqueness cannot always be expected, we address this challenge by establishing an intersection principle for comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness. -
Matthew Lerner-Brecher
Date: Monday, April 28, 2025 | 2:30pm | Room: 2-135
Committee: Alexei Borodin (Advisor), Nike Sun, Jacopo Borga
The Fourier-Bessel Series and Hard Edge Limits
The universality classes defined by the Airy and Bessel kernels are two of the most fundamental in random matrices and growth models more generally. Broadly speaking, one often encounters the Airy kernel when studying models where the relevant eigenvalues or particles are unbounded, and the Bessel kernel when examining their constrained counterparts. In this thesis, we analyze two recent problems where the relevant expressions involve a variant of the Airy functions known as the Fourier-Airy series. In both cases, we find that the constrained versions have natural analogues expressible in terms of the Fourier-Bessel series echoing the relationship between the Airy and Bessel kernels.
In the first part, we study the hard-edge limit of a multilevel extension of the Laguerre beta-ensemble at zero temperature. In particular, we show that asymptotically the ensemble is given by Gaussians with covariance matrix expressible in terms of the Fourier-Bessel series. These Gaussians also have an explicit representation as the partition functions of additive polymers arising from a random walk on roots of the Bessel functions. Our approach builds off of techniques introduced by Gorin and Kleptsyn and is rooted in using the theory of dual and associated polynomials to diagonalize transition matrices relating levels of the ensemble. Like the corresponding soft edge limit in the Hermite case studied by Gorin and Kleptsyn, the object we introduce should represent a new universality class for zero temperature random matrices.
In the second part, we introduce a new diffusion process which arises as the n goes to infinity limit of a Bessel process of dimension d >= 2 conditioned upon remaining bounded below one until time n. In addition to being interesting in its own right, we argue that the resulting diffusion process is a natural hard edge counterpart to the Ferrari-Spohn diffusion. -
Weixiao Lu
Date: Thursday, April 17, 2025 | 1:00pm | Room: 3-333 | Zoom Link
Committee: Prof. Wei Zhang (advisor), Prof. Zhiwei Yun, Prof. Ju-Lee Kim
A relative trace formula approach to the stable trace formula on unitary group
We develop a relative trace formula on GL_n which can be compared to the stable trace formula on the unitary group. Locally, we prove the fundamental lemma and transfer and also derive a character identity based on the transfer. Globally, we develop a (simple) relative trace formula and compared it to the (simple) stable trace formula on the unitary group. -
Nitya Mani
Date: Friday, March 21, 2025 | 4:00pm | Room: Eisenbud Auditorium, Simons Laufer Mathematical Sciences Institute (17 Gauss Way, Berkeley, CA 94720) | Zoom passcode: 930629 | Zoom Link
Committee: Ankur Moitra, Pablo Parillo, Yufei Zhao, Jacob Fox
A probabilistic perspective on graph coloring
Graph coloring problems are among the most widely studied in combinatorics, with many basic questions in the field still widely open. Such questions often have wide ranging applications across fields as diverse as statistical physics, theoretical computer science, transportation, epidemiology, cybersecurity, circuit design, and network science more broadly. We study graph coloring from a probabilistic perspective, focusing on two different graph coloring problems that share an underlying theme: given an exponentially large family of objects derived from a graph vertex-coloring, can we understand what a typical or random object in this large family looks like without manually searching through exponentially many alternatives?
We start by seeking to better understand when it is possible to efficiently sample a random proper coloring from a given graph, i.e. a uniformly random vertex coloring such neighboring vertices receive different colors. We then proceed to study the structure of monochromatic shapes that arise in a uniformly random vertex coloring when we do not impose any constraints on the colors of neighboring vertices. -
Elia Portnoy
Date: Thursday, April 3, 2025 | 3:00pm | Room: 4-149 | Zoom Link
Committee: Larry Guth (advisor), Peter Shor, Alex Lubotzky
Quantitative embeddings with applications
In this thesis, we discuss quantitative embeddings that generalize a theorem of Kolmogorov and Barzdin. The theorem says that any bounded degree graph with V vertices can be mapped into a 3-dimensional ball of radius V^(1/2), so that at most a constant number of edges intersect any unit ball. In one generalization we describe how much freedom we have in placing the vertices of the graph, and in the other we prove a similar result for higher dimensional simplicial complexes. We also discuss applications of these quantitative embeddings to an isoperimetric problem in metric geometry and a problem in quantum error-correcting codes.
The metric geometry problem asks the following. Given a set U in n-dimensional space which is diffeomorphic to a ball, whose boundary has volume at most 1, is there a diffeomorphism from the unit n-dimensional ball to U, which increases the volume of every (n-1)-dimensional submanifold by at most a constant? We show that the answer is positive for n at least 4 and negative for n=3.
The quantum codes problem is about finding codes for which we can position the qubits on a lattice in 3-dimensional space, so that any two qubits involved in some check are at most a constant distance from each other. We build on several recent breakthroughs to construct such codes with the best possible parameters. -
Zhi (Robert) Ren
Date: Wednesday, April 30, 2025 | 2:00pm | Room: 45-500A
Committee: Youssef Marzouk, Philippe Rigollet, Alan Edelman
Theoretical Foundations of Flow-based Methods for Sampling and Generative Modeling
Sampling from an arbitrary probability distribution is a central problem in computational statistics and machine learning. Transportation of measure offers a useful approach to this problem: the idea is to construct a measurable map that pushes forward a relatively simple source distribution to the target probability distribution. One can then simulate from the target distribution by drawing samples from the source distribution and evaluating the transport map. This construction is applicable to both generative modeling and variational inference; when the map is invertible, one can also estimate the density of the target measure by evaluating the density of the pushforward of the source distribution under the inverse transport map.
Over the past decade, various parameterizations of such transports have been proposed. Generally speaking, they fall into two categories: the static approach, where the displacement from x to T(x) is represented directly and the dynamic approach that employs evolution of measures by some differential equation over some fictitious time. While many of these models have achieved enormous success in practical applications, their theoretical underpinnings remain largely unexplored. In this thesis, we provide a theoretical foundation for flow-based methods for sampling and generative modeling, and a unified view of both continuous and discrete-time approaches.
In the first part of the thesis, we address the approximation theory of flow based methods. In particular, we show how the regularity of the underlying ODE velocity field relates to the regularity of densities and prove related neural network approximation bounds. In addition, we show how introduction of a time-reparameterized schedule can dramatically improve the regularity of the velocity, helping resolve potential singularities. In the second part of the thesis, we focus on the interplay between flow-based models and nonparametric statistics. In particular, we consider pullback density estimators under these flow based models from likelihood-based objectives. The estimators we consider arise from both discrete and continuous-time parameterizations of the transport, and the underlying function classes we consider include H\"older balls and neural networks. In all these cases, we show they achieve minimax optimal rate for learning s-smooth densities. -
Xinrui Zhao
Date: Monday, April 28, 2025 | 3:30pm | Room: 2-132 | Zoom Link
Committee: Prof. Tobais Colding, Prof. Christoph Kehle and Prof. Tristan Ozuch-Meersseman
Geometry and analysis of Ricci curvature and mean curvature flows
In this thesis, we study the geometry and analysis of spaces with Ricci curvature bounded below from the following three perspectives and the asymptotically conical singularities of mean curvature flows in the following two perspectives.
For the spaces with Ricci curvature bounded below, firstly we study the unique continuation problem on RCD spaces, which is a long-standing open problem, with little known even in the setting of Alexandrov spaces. Together with Qin Deng, we proved that on RCD(K,2) spaces both harmonic functions and caloric functions satisfy weak unique continuation properties. Furthermore we constructed counter-examples showing that strong unique continuation in general fails for harmonic and caloric functions on RCD(K,N) spaces where N is greater or equal to 4.
Secondly, we consider constructing a canonical diffeomorphism between the n-sphere and a n-dimensional space with Ricci curvature bounded from below by n-1 which is close to the n-sphere in the Gromov-Hausdorff sense. Together with Bing Wang we proved that the first (n+1)-eigenfunctions of Laplacian provides a Bi-Holder diffeomorphism and we further give a counter-example showing that the Bi-Holder estimate is sharp and cannot be improved to a bi-Lipschitz estimate.
Thirdly, we study the Margulis Lemma on RCD spaces. Together with Qin Deng, Jaime Santos-Rodríguez and Sergio Zamora, we extend the Margulis Lemma for manifolds with lower Ricci curvature bounds to the RCD setting. As one of our main tools, we obtain improved regularity estimates for Regular Lagrangian flows on these spaces.
For the asymptotically conical singularities of mean curvature flows, firstly together with Tang-Kai Lee, we proved asymptotically conical self-shrinkers as tangent flows of MCFs are unique, generalizing the result in the case of hypersurface proven by Chodosh-Schulze.
Secondly, together with Tang-Kai Lee we prove that given any asymptotically conical shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a type I singularity at time 1 at the origin modeled on the given shrinker.