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Graduate Thesis Defenses 2022

Araminta Amabel

Title: Genera via Deformation Theory and Supersymmetric Mechanics
Date: Monday, April 11, 2022 | 4:30pm | Room: 2-131
Committee: Mike Hopkins, Haynes Miller, Jeremy Hahn

Abstract

We will discuss naturally occurring genera (i.e. cobordism invariants) inspired by the deformation theory inspired by supersymmetric quantum mechanics. First, we construct a canonical deformation quantization for symplectic supermanifolds. Secondly, we prove a super-version of Nest-Tsygan’s algebraic index theorem, generalizing work of Engeli. This work is inspired by the appearance of the same genera in three related stories: index theory, trace methods in deformation theory, and partition functions in quantum field theory. Using the trace methodology, we compute the genus appearing in the story for supersymmetric quantum mechanics. This involves investigating supertraces on Weyl-Clifford algebras and deformations of symplectic supermanifolds.

Morris Ang

Title: Integrability in random conformal geometry
Date: Tuesday, April 5, 2022 | 1:30pm | Room: 2-361
Committee: Scott Sheffield (advisor), Alexei Borodin, Nike Sun

Abstract

Liouville quantum gravity (LQG) is a random surface arising as the scaling limit of random planar maps. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in many statistical physics models. Liouville conformal field theory (LCFT) is the quantum field theory underlying LQG. Each of these satisfies conformal invariance or covariance. This thesis proves exact formulas in random conformal geometry; we highlight a few here.

The Brownian annulus describes the scaling limit of uniform random planar maps with the annulus topology, and is the canonical annular 𝛾-LQG surface with 𝛾 =√︀8/3. We obtain the law of its modulus, which is as predicted from the ghost partition function in bosonic string theory.

The conformal loop ensemble (CLE) is a random collection of loops in the plane which locally look like SLE, corresponding to the scaling limit of all interfaces in several important statistical mechanics models. We derive the three-point nesting statistic of simple CLE on the sphere. It agrees with the imaginary DOZZ formula of Zamolodchikov (2005) and Kostov-Petkova (2007), which is the three-point structure constant of the generalized minimal model conformal field theories.

We compute the one-point bulk structure constant for LCFT on the disk, thereby proving the formula proposed by Fateev, Zamolodchikov and Zamolodchikov (2000). This is a disk analog of the DOZZ constant for the sphere. Our result represents the first step towards solving LCFT on surfaces with boundary via the conformal bootstrap.

Our arguments depend on the interplay between LQG, SLE and LCFT. Firstly, LQG behaves well under conformal welding with SLE curves as the interfaces. Secondly, LCFT and LQG give complementary descriptions of the same geometry.

Robert Burklund

Title: Multiplicative structures on Moore spectra
Date: Monday, April 25, 2022 | 4:30pm | Room: 2-131
Committee: Mike Hopkins, Haynes Miller, Davesh Maulik, Arpon Raksit

Abstract

One of the distinguishing features of higher algebra is the difficulty of constructing quotients. In this talk I will explain a new technique for constructing algebra structures on quotients. This technique allows us to prove that $\mathbb{S}/8$ is an $\mathbb{E}_1$-algebra, $\mathbb{S}/32$ is an $\mathbb{E}_2$-algebra, $\mathbb{S}/p^{n+1}$ is an $\mathbb{E}_n$-algebra at odd primes and, more generally, for every $h$ and $n$ there exist generalized Moore spectra of type $h$ which admit an $\mathbb{E}_n$-algebra structure.

Yibo Gao

Title: Symmetric structures in the weak and strong Bruhat orders
Date: Thursday, April 7, 2022 | 3:00pm | Room: 2-131
Committee: Alexander Postnikov (advisor), Richard Stanley, Lauren Williams

Abstract

The weak and strong Bruhat orders are classical and rich combinatorial objects, with connections to Schubert calculus, Lie algebras, hyperplane arrangements, sorting networks and so on. In this thesis, we study various new symmetries within these structures, including the balance constant and the hull metric property of the weak order, and the self-dual intervals and boolean elements in the strong order. Much of the work involved is joint with Christian Gaetz.

Linus Hamilton

Title: Applications and limits of convex optimization
Date: Monday, April 25, 2022 | 10:00am | Room: 2-449
Committee: Ankur Moitra, John Kelner, Philippe Rigolet

Abstract

To word it as confusingly as possible, the Paulsen problem asks: if a set of vectors is close to isotropic and equal-norm, then is it close to an isotropic equal-norm set of vectors? This question gained notoriety as "one of the most intractable problems in [operator theory]." In 2017, it was finally resolved by Kwok et al in a triumphant 103 pages of smoothed analysis and operator scaling. Unfortunately we won't have enough time to deliver their proof, so I will instead exhibit a new one. It is short enough to present unabridged, and still have time left over to graze on two other problems related to convex optimization: graphical model structure learning, and the impossibility of accelerated gradient descent in hyperbolic space.

Kai Huang

Title: K-stability of Log Fano Cone Singularities
Date: Thursday, May 12, 2022 | 10:00am | Zoom: https://mit.zoom.us/j/95027529410
Committee: Chenyang Xu, Davesh Maulik, Zhiwei Yun

Abstract

In this thesis, we define the delta-invariant for log Fano cone singularities, and show that the necessary and sufficient condition for K-semistability is 𝛿 ≥ 1. This generalizes the result of C. Li and K. Fujita.

We also prove that on any log Fano cone singularity of dimension 𝑛 whose 𝛿-invariant is less than n+1/n, any valuation computing the delta-invariant has a finitely generated graded ring. This shows a log Fano cone is K-polystable if and only if it is uniformly K-stable. Together with earlier works, this implies the Yau-Tian-Donaldson Conjecture for Fano cone.

Andrei Ionov

Title: Tilting sheaves for real groups and Koszul duality
Date: Wednesday, April 13, 2022 | 4:00pm | Room: 2-142
Committee: Roman Bezrukavnikov, David Vogan, Zhiwei Yun

Abstract

For a real form of an algebraic group acting on the flag variety we define a t-structure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form we construct an analog of a Soergel functor, which fully-faithfully embeds the subcategory of tilting objects to the category of coherent sheaves on a block variety. We apply the results to give a new, purely geometric, proof of the Soergel's conjecture for quasi-split groups.

Pakawut (Pro) Jiradilok

Title: Inequalities and Asymptotic Formulas in Algebraic Combinatorics
Date: April 26, 2022 | 3:00pm | Room: 2-255
Committee: Alex Postnikov (advisor), Alexei Borodin, Thomas Roby (University of Connecticut)

Abstract

This thesis concerns certain inequalities and asymptotic formulas in algebraic combinatorics. It consists of two separate parts. The first part studies inequalities concerning triangular-grid billiards and plabic graphs of Lam-Postnikov essential dimension 2. The material in this part is based on joint work with Colin Defant. The second part studies inequalities and asymptotic formulas concerning large-scale rook placements.

Chun Hong Lo

Title: Gromov-Witten Invariants of Blow Ups of $\mathbb{P}^2$ using Logarithmic Geometry
Date: Monday, June 6, 2022 | 10:30am | Room: 2-449 (Also on Zoom)
Committee: Davesh Maulik, Andrei Negut, Ziquan Zhuang

Abstract

We show that Parker's recursion for computing Gromov-Witten invariants of blow ups of $\mathbb{P}^2$ can be derived using logarithmic Gromov-Witten theory and punctured maps. We extend the recursion to compute Gromov-Witten invariants with Hodge class insertions.

Ashwin Narayan

Title: Distortion Metrics for Biological Data
Date: Wednesday, January 19, 2022 | 2:00pm | Room: 2-255
Committee: Bonnie Berger (advisor), Jon Kelner, Bryan Bryson

Abstract

Advances in experimental methods in biology have allowed researchers to gain an unprecedentedly high-resolution view of the molecular processes within cells, using so-called single-cell technologies. Every cell in the sample can be individually profiled- the amount of each type of protein or metabolite or other molecule of interest can be counted. Understanding the molecular basis that determines the differentiation of cell fates is thus the holy grail promised by these data.

However, the high-dimensional nature of the data, replete with correlations between features, noise, and heterogeneity means the computational work required to draw insights is significant. In particular, understanding the differences between cells requires a quantitative measure of similarity between the single-cell feature vectors of those cells. A vast array of existing methods, from those that cluster a given dataset to those that attempt to integrate multiple datasets or learn causal effects of perturbation, are built on this foundational notion of similarity.

In this dissertation, we delve into the question of similarity metrics for high-dimensional biological data generally, and single-cell RNA-seq data specifically. We work from a global perspective - where we find a distance function that applies across the entire dataset - to a local perspective - where each cell can learn its own similarity function. In particular, we first present SCHEMA, a method for combining similarity information encoded by several types of data, which has proven useful in analyzing the burgeoning number of datasets which contain multiple modalities of information. We also present DENsV1s, a package of algorithms for visualizing single-cell data, which improve upon existing dimensionality-reduction methods that focus on local structure by accounting for density in high-dimensional space. Lastly, we zoom in on each datapoint, and show a new method for learning k-nearest neighbors graphs based on local decompositions.

Altogether, the works demonstrate the importance - through extensive validation on existing datasets - of understanding high-dimensional similarity.

Minjae Park

Title: Random surface interpretations of two-dimensional Liouville quantum gravity and Yang-Mills theory
Date: Thursday, April 28, 2022 | 2:00pm | Room: Room 2-449
Committee: Scott Sheffield, Nike Sun, Promit Goshal

Abstract

The theory of random surfaces (or "sums over surfaces") has its historical roots in quantum gravity, string theory, statistical physics, and combinatorics. This thesis explores random surfaces in two settings: one related to Liouville quantum gravity, and one related to Euclidean Yang-Mills theory in two dimensions.

The first part introduces a specific regularization of Liouville quantum gravity surfaces. It also establishes the Polyakov-Alvarez formula on non-smooth surfaces with Brownian loops instead of the zeta-regularized Laplacian determinant. Consequently, "weighting by a Brownian loop soup" changes the so-called central charge of the regularized random surfaces, as expected in physic literature. This result justifies a definition of Liouville quantum gravity surfaces in the supercritical regime where the central charge is greater than 1.

The second part describes continuum Wilson loop expectations on the plane as sums over surfaces, an example of gauge string duality. In contrast to the Gross-Taylor expansion, our weight is explicit as ±Nᵡ where χ is the Euler characteristic, for any gauge group U(N), SO(N), Sp(N/2). Based on the well-established continuum theory in two dimensions, we provide a probabilistic treatment for Wilson loop expectations, also leading to various applications like an alternative proof for the Makeenko-Migdal equation and a connection with a random walk on permutations.

Chengyang Shao

Title: Long Time Dynamics of Spherical Objects Governed by Surface Tension
Date: Monday, April 4, 2022 | 11:00AM | Room: 2-449 (Also on Zoom)
Committee: Gigliola Staffilani, David Jerison, Andrew Lawrie

Abstract

This thesis is devoted to the study of evolutionary partial differential equations describing the motion of spherical objects in two different physical scenarios. They share the common feature of involving the mean curvature operator, hence relating to motions governed by surface tension. The mean curvature operator makes both problems highly nonlinear.

In the first part, we study the long time behavior of an idealistic model of elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, and also includes a damping term that models the dissipative nature of genuine physical systems. With the presence of damping, a small perturbation of the sphere converges exponentially in time to the sphere, and without the damping the evolution that is $\varepsilon$-close to the sphere has life span longer than $\varepsilon^{-1/6}$. Both results are proved using a new Nash-Moser-Hörmander type theorem proved by Baldi and Haus.

In the second part, we derive a differential equation that describes the nonlinear vibration of a spherical water droplet under zero gravity. The equation is legitimately referred as the water waves equation for a sphere. We develop a toolbox for para-differential calculus on curved manifolds and prove the local existence for this equation by para-linearizing the equation. This approach avoids using Nash-Moser type iterations, and sets up stage for study of longer time behavior of spherical water waves. For the longer time behavior, we discuss the resonance problem related to this equation, pointing out that it is a highly nontrivial problem of Diophantine analysis in the realm of number theory.

Dominic Skinner

Title: Thermodynamic and topological characterization of living systems
Date: Thursday, April 14, 2022 | 9:30am | Room: 1-379
Committee: Jörn Dunkel (thesis advisor), Philippe Rigollet, Nikta Fakhri (MIT Physics)

Abstract

Recent advances in microscopy techniques make it possible to study the growth, dynamics, and response of complex biophysical systems at single-cell and subcellular resolution, from bacterial communities and tissues to intra-cellular signaling and expression dynamics of genes. Despite such progress in data collection, major theoretical challenges remain to find structure and organizing principles in these data, which are often noisy and represent only a partial observation of the system. One such challenge is to estimate the rate at which a system is consuming free energy. All living systems must consume free energy to maintain or increase local order, and theoretical models can provide insights into the thermodynamic efficiency of important cellular processes. In experiments however, many degrees of freedom typically remain hidden to the observer, making thermodynamic inference challenging. Here, we introduce a framework to infer improved bounds on the rate of entropy production, by reformulating the problem of inference as a problem of optimization. We demonstrate the broad applicability of our approach by providing improved bounds on the energy consumption rates in a diverse range of biological systems including bacterial flagella motors, gene regulatory dynamics, and intracellular calcium oscillations. Another challenge is to distinguish two amorphous yet structurally different cellular materials, where in contrast to crystals, cellular structures are somewhat disordered. Here, we use information contained in the local topological structure to define a distance between disordered multicellular systems. Our metric allows an interpretable reconstruction of equilibrium and non-equilibrium phase spaces and embedded pathways from static system snapshots alone. Applied to cell-resolution imaging data, the framework recovers time-ordering without prior knowledge about the underlying dynamics, revealing that fly wing development solves a topological optimal transport problem, and enables comparisons across a wide range of different systems from zebrafish brains to bacterial colonies.

Jonathan Tidor

Title: Higher-order Fourier analysis with applications to additive combinatorics and theoretical computer science
Date: Wednesday, April 20, 2022 | 1:00pm | Room: TBD
Committee: Yufei Zhao (advisor), Larry Guth, Lisa Sauermann

Abstract

Fourier analysis has been used for over one hundred years as a tool to study certain additive patterns. For example, Vinogradov used Fourier-analytic techniques (known in this context as the Hardy-Littlewood circle method) to show that every sufficiently-large odd integer can be written as the sum of three primes, while van der Corput similarly showed that the primes contain infinitely-many three-term arithmetic progressions.

Over the past two decades, a theory of higher-order Fourier analysis has been developed to study additive patterns which are not amenable to classical Fourier-analytic techniques. For example, while three-term arithmetic progressions can be studied with Fourier analysis, all longer arithmetic progressions require higher-order techniques. These techniques have led to a new proof of Szemerédi's theorem in addition to results such as counts of $k$-term arithmetic progressions in the primes.

This thesis contains five results in the field of higher-order Fourier analysis. In the first half, we use these techniques to give applications in additive combinatorics and theoretical computer science. We prove an induced arithmetic removal lemma first in complexity 1 and then for patterns of all complexities. This latter result solves a central problem in property testing known as the classification of testable arithmetic properties. We then study a class of multidimensional patterns and show that many of them satisfy the popular difference property analogously to the one-dimensional case. However there is a surprising spectral condition which we prove necessarily appears in higher dimensions that is not present in the one-dimensional problem.

In the second half of this thesis, we further develop the foundations of higher-order Fourier analysis. We determine the set of higher-order characters necessary over $\mathbb{F}_p^n$, showing that classical polynomials suffice in the inverse theorem for the Gowers $U^k$-norm when $k\leq p+1$, but that non-classical polynomials are necessary whenever $k>p+1$. Finally, we prove the first quantitative bounds on the $U^4$-inverse theorem in the low-characteristic regime $p$ < $5$.

Alexandra Utiralova

Title: Harish-Chandra bimodules in complex rank
Date: Friday, April 22, 2022 | 3:00pm | Room: 2-135
Committee: Pavel Etingof, Roman Bezrukavnikov, Andrei Negut

Abstract

The Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogs of classical objects provide insights into their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in the Deligne categories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of the Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define Harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however, in the Deligne categories, it is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of the Deligne categories.

Sahana Vasudevan

Title: Large genus bounds for the distribution of triangulated surfaces in moduli space
Date: Thursday, April 14, 2022 | 2:00pm | Room: 2-449
Committee: Larry Guth (advisor), Tomasz Mrowka, Scott Sheffield

Abstract

A triangulated surface is a compact surface formed by gluing together unit equilateral triangles. When the genus is at least two, it admits a unique conformal hyperbolic metric. Brooks and Makover started the study of random triangulated surfaces in the large genus limit, and proved results about their systole, diameter and Cheeger constant. Subsequently, Mirzakhani proved analogous results about random hyperbolic surfaces. These results motivate the question: how are triangulated surfaces distributed in the moduli space of hyperbolic surfaces, as the genus becomes large? In my thesis, I proved that triangulated surfaces are well distributed in moduli space in a fairly strong sense.

Ruoxuan Yang

Title: Stable and unstable shock formation of the Burgers-Hilbert equation
Date: Wednesday, April 6, 2022 | 10:30am | Room: 2-361 (Also on Zoom)
Committee: Gigliola Staffilani (advisor), David Jerison, Andrew Lawrie

Abstract

The study of singularities has been an important part in the analysis of PDEs. One key type of singularities is shock. In many cases the shock has a self-similarity structure. Recently, the modulated self-similarity technique has achieved success in fluid dynamic equations. In this thesis, we apply this technique to establish finite time shock formation of the Burgers-Hilbert equation. The shocks are asymptotic self-similar at one single point. The shocks can be stable or unstable, both of which have an explicitly computable singularity profile, and the shock formation time and location are described by explicit ODEs. For the stable shock, the initial data are in an open set in the H^5-norm, and the shock profile is a cusp with Holder 1/3 continuity. For the unstable shock, the initial data are in a co-dimension 2 subset of the H^9 space, and the shock profile is of Holder 1/5 continuity. Both cases utilize a transformation to appropriated self-similar coordinates, the quantitative properties of the corresponding self-similar solution to the inviscid Burgers' equation, and transport estimates. In the case of unstable shock, we, in addition, control the two unstable directions by Newton's iteration.