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Graduate Thesis Defenses 2021

Nilin Abrahamsen

Title: Improved tools for local Hamiltonians
Date: Friday, April 30, 2021 | 2:00pm | https://mit.zoom.us/j/92960583583
Committee: Peter Shor, Jon Kelner, Anand Natarajan (CSAIL)

Abstract

In this thesis we consider computational problems related to many-body spin systems with a structured energy operator, a local Hamiltonian. We begin with the most structured setting where the Hamiltonian has a spectral gap and spatial locality. This setting is widely studied using approximate ground space projectors (AGSPs). In chapter 1 ee give an improved analysis of AGSPs in the setting of local Hamiltonians with a degenerate ground space. This implies a direct generalization of the AGSP=>entanglement bound implication of [Arad, Landau, and Vazirani ’12] from unique to degenerate ground states. We use the improved analysis to give a particularly simple algorithm for frustration-free spin systems provided an AGSP with structure as a matrix product operator. We apply our tools to a recent 2D area law of [Anshu, Arad, and Gosset ’21], giving a sub-exponential-time classical algorithm to compute the ground states. This time complexity cannot be improved beyond sub-exponential assuming the randomized exponential time hypothesis, even for the special case of classical constraint satisfaction problems on the 2D grid. In chapter 2 we consider frustrated systems and extend results for spin chains to certain trees with intrinsic dimension <2. This condition is met for generic trees in the plane and for certain models of hyperbranched polymers in 3D. In chapter 3 we relax the conditions on the Hamiltonian and no longer require a spectral gap or geometric locality, and we consider an approximation problem for the spectrum of the local Hamiltonian. We give a simple proof of a Chernoff bound for the spectrum of a k-local Hamiltonian based on Weyl’s inequalities. The complexity of estimating the spectrum’s e(n)-th quantile up to constant relative error thus exhibits the following dichotomy: For e(n) = d^n the problem is NP-hard and maybe even QMA-hard, yet there exists constant a > 1 such that the problem is trivial for e(n) = a^n.

Alexey Balitskiy

Title: Bounds on Urysohn width
Date: Tuesday, April 20, 2021 | 1:00pm | https://mit.zoom.us/j/4206650769
Committee: Larry Guth, Lisa Piccirillo, William Minicozzi

Abstract

The notion of the Urysohn d-width of a metric space quantifies how close it can be approximated by a d-dimensional simplicial complex. Namely, the d-width of a space is at most W if it admits a continuous map to a d-complex with all fibers of diameter at most W. This notion was introduced in the context of dimension theory, used in approximation theory, appeared in the work of Gromov on systolic geometry, and nowadays it is a metric invariant of independent interest. The main results of this thesis establish bounds on the width, relating local and global geometry of Riemannian manifolds in two contexts. One of them is bounding the global width of a manifold in terms of the width of its unit balls. The other one is waist-like inequalities, when a manifold is sliced into a family of (singular) surfaces, and the global width is related to the supremal width of the slices.

Aleksandr Berdnikov

Title: Lipschitz homotopies of mappings from S^3 to S^2
Date: Friday, April 23, 2021 | 3:00pm | https://mit.zoom.us/j/7147913127?pwd=UldMd3RreXhaSlZBOW85Wk5UbXEyQT09
Committee: Larry Guth, Andrew Lawrie and Paul Seidel

Abstract

Is it always "easy" to homotope one mapping into another? The answer depends on the spaces being mapped. For a large class of spaces X — scalable spaces — the answer is yes: if mappings f,g: S^n \to X are homotopic, one can find a homotopy between them that is "as simple" as f and g themselves. We focus on the example S^3 \to S^2 that, while being visually simple, lead to the discovery of scalable spaces.

Yongyi Chen

Title: Self-intersection of Manin-Drinfeld Cycles and Taylor expansion of L-functions
Date: Wednesday, May 5, 2021 | 6:00pm | Room: https://mit.zoom.us/j/96851980074
Committee: Wei Zhang, Zhiwei Yun, Ben Howard (Boston College)

Abstract

A rising philosophy in the theory of automorphic representations in number theory is that higher central derivatives of $L$-functions of automorphic forms should correspond to the intersection numbers of special cycles on moduli spaces. A classic early result along this philosophy was achieved by Gross and Zagier, who proved that the derivative of the $L$-function of an elliptic curve is equal, up to a constant, to the Néron-Tate height pairing of a special point called a \emph{Heegner point} on the elliptic curve.

A more recent result was proven in the function field case by Yun and Zhang which showed that higher derivatives of the base change $L$-function of an unramified automorphic representation over $\mathrm{PGL}_2$ over a function field are equal, up to a constant, to the self-intersection number, inside the moduli stack of $\mathrm{PGL}_2$-shtukas, of the moduli stack of shtukas for an anisotropic torus.

We prove in the function field case that the higher derivatives of the square of the $L$-function of unramified automorphic representations over $\mathrm{PGL}_2$ are equal, up to a constant, to the self-intersection number, inside the moduli stack of $\mathrm{PGL}_2$-shtukas, of the moduli stack of shtukas for the split torus.

Julien Edward Clancy

Title: Interpolating Spline Curves of Measures
Date: Tuesday, April 27, 2021 | 9:00am | https://mit.zoom.us/j/94157063207
Committee: Philippe Rigollet, Bill Minicozzi, Justin Solomon

Abstract

When dealing with classical, Euclidean data, the statistician's toolkit is enviably deep: from linear and nonlinear regression, to dealing with sparse or structured data, to interpolation techniques, most any problem dealing with vector or matrix data is amenable to several different statistical methods. Yet modern data is often not Euclidean in nature. The semantic content of natural images does not have a vector structure; shifting an image one pixel to the right does not perceptibly change it, but its vector representation is very different. For model cross-validation or bootstrapping, each data point is a dataset in its own right, and one might want to consider an "average dataset". In an ensemble method, experts may express their beliefs as prior distributions; how would we perform a statistical analysis of these?

Recently much attention has been paid to a framework which subsumes all of these problems: the Wasserstein space of measures with finite second moment. Works on point estimation, generalized means, and linear regression have appeared, as have some on smooth interpolation, greatly expanding the statistical toolkit for modern data. In this vein, the present work is broadly a theoretical and computational exploration of curves of measures which in some sense minimize curvature while interpolating data, as splines do in Euclidean space. We answer several questions about the relationship between the intrinsic Wasserstein-Riemannian curvature of such curves and a particle flow-based, "fluid-dynamical" formulation, and provide fast and accurate smooth interpolation techniques. We also study a related probabilistic interpolation problem unique to the measure setting, which asks for particle trajectories that satisfy certain interpolation constrains and minimize a KL divergence, in analogy with the Schrodinger bridge problem. We conclude with an extension of our methods to the case of unbalanced measures in the Wasserstein-Fisher-Rao space.

Robin Elliott

Title: Quantitative Topology of Loop Space
Date: Wednesday, April 28, 2021 | 1:00pm | https://mit.zoom.us/j/96925391353
Committee: Larry Guth (thesis advisor), Haynes Miller, Tomasz Mrowka

Abstract

This thesis investigates how the size of a cycle in the based loop space of a simply connected Riemannian manifold controls its topology. Analogous to Gromov’s notion of distortion of higher homotopy groups of underlying Riemannian manifold, we define notions of distortion for (co)homology classes in the loop space with real coefficients, and study the asymptotics of these distortions. Upper bounds for co-homological distortion are obtained using K.-T. Chen’s theory of iterated integrals to set up differential forms on the loop space. Lower bounds, matching the upper bounds up to a log factor, are given by exhibiting an efficient family of cycles built out of the cells of a cell decomposition the underlying manifold.

Jesse Freeman

Title: The Surgery Exact Triangle in Monopole Floer Homology with Z[i] Coefficients
Date: Thursday, April 22, 2021 | 3:00pm | https://mit.zoom.us/j/8084644271
Committee: Tomasz Mrowka, Paul Seidel, Irving Dai

Abstract

In their seminal paper, Kronheimer, Mrowka, Ozsváth and Szabó, establish the existence of a surgery exact triangle for Monopole Floer homology. The triangle was a theoretical breakthrough and allowed us to answer questions about the surgery correspondence between knots and 3-manifolds that were previously mysterious. A serious limitation of their triangle was that it only holds over characteristic two. In this paper, we establish the existence of a surgery exact triangle using local coefficients valued in Z[i].

Christian Gaetz

Title: New combinatorics of the weak and strong Bruhat orders
Date: Tuesday, February 23, 2021 | 11:00am | Room: Remote
Committee: Alexander Postnikov (advisor), Richard Stanley, David Speyer

Abstract

This thesis describes a line of work, much of it joint with Yibo Gao, which began with our proof of Stanley’s conjecture that the weak order on the symmetric group is Sperner. Further developments—either directly related or related in our thinking at the time—involve weighted path enumeration in the weak and strong Bruhat orders, specializations of Schubert polynomials, separable elements in finite Weyl groups, and inequalities for linear extensions of finite posets.

Peter Haine

Title: On the homotopy theory of stratified spaces
Date: Monday, May 3, 2021 | 4:30pm | https://mit.zoom.us/j/132540375
Committee: Clark Barwick, Jeremy Hahn, Haynes Miller

Abstract

A natural question arises when working with intersection cohomology and other stratified invariants of singular manifolds: what is the correct stable homotopy theory for these invariants to live in? But before answering that question one first has to identify the correct unstable homotopy theory of stratified spaces. The exit-path category construction of MacPherson Treumann and Lurie provides functor from suitably nice stratified topological spaces to "abstract stratified homotopy types” — ∞-categories with a conservative functor to a poset. Work of Ayala–Francis–Rozenblyum even shows that their conically smooth stratified topological spaces embed into the ∞-category of abstract stratified homotopy types. In this talk, we explain some of our work which goes further and produces an equivalence between the homotopy theory of all stratified topological spaces and these abstract stratified homotopy types.

Daniil Kalinov

Title: Construction of Deligne categories through ultrafilters and its applications
Date: Monday, April 12, 2021 | 12:00pm | Room: Remote
Committee: Pavel Etingof, Roman Bezrukavnikov, Valerio Toledano Laredo (NEU)

Abstract

The present thesis is concerned with the study of Deligne categories and their application to various representation-theoretic problems. The lens that is used to view Deligne categories in this study is the one of ultrafilters and ultraproducts. As will be shown in our work, this approach turns out to be a very powerful one. Especially if one wants to solve such representation-theoretic problems as presented by P.Etingof in his papers on "Representation theory in complex rank".

The results are presented in two parts. In the first one an introduction to the theory of ultrafilters is given, along with the construction of the Deligne categories through ultrafilters. This also allows us to understand how one can make sense of Deligne categories as a limit in rank and characteristic.

The later part describes two applications of this construction to actual representation-theoretic problems. The first application consists of the full classification of simple commutative, associative and Lie algebras in $\textbf{Rep}(S_\nu)$ for $\nu \notin \mathbb Z_{\ge 0}$. The second one concerns the construction of deformed double current algebras as a space of endomorphisms of a certain ind-object of $\textbf{Rep}(S_\nu)$. It is also proven that this construction agrees with Guay's deformed double current algebra of type $A$ if the rank $r\ge 4$ (Guay's algebra is presently only defined for such rank), and the presentation by generators and relations for the case of $r=1$ is given.

Frederic Koehler

Title: Provable Algorithms for Learning and Variational Inference in Undirected Graphical Models
Date: Tuesday, April 27, 2021 | 1:00pm | https://mit.zoom.us/j/99064696005
Committee: Ankur Moitra, Elchanan Mossel, Guy Bresler

Abstract

Graphical models are a general-purpose tool for modeling complex distributions in a way which facilitates probabilistic reasoning, with numerous applications across machine learning and the sciences. This thesis deals with algorithmic and statistical problems of learning a high-dimensional graphical model from samples, and related problems of performing inference, both areas of research which have been the subject of continued interest over the years. Our main contributions are the first computationally efficient algorithms for provably (1) learning a (possibly ill-conditioned) walk-summable Gaussian Graphical Model from samples, (2) learning a Restricted Boltzmann Machine (or other latent variable Ising model) from data, and (3) performing naive mean-field variational inference on an Ising model in the optimal density regime. These different problems illustrate a set of key principles, such as the diverse algorithmic applications of "pinning" variables in graphical models. We also show in some cases that these results are nearly optimal due to matching computational/cryptographic hardness results.

Hyuk Jun Kweon

Title: Bounds on the Torsion Subgroups of Néron-Severi Groups
Date: Monday, May 3, 2021 | 2:00pm | https://mit.zoom.us/j/92629961965?pwd=SUltVnVBNDZtL0k1UW9LUjFFNU8vUT09
Committee: Bjorn Poonen, Steven Kleiman, Chenyang Xu

Abstract

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$, and the N\'eron--Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_\et(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements.

Tim Large

Title: Spectral Fukaya categories of Liouville manifolds
Date: Thursday, April 29, 2021 | 1:00pm | https://mit.zoom.us/j/99107888285
Committee: Paul Siedel (thesis advisor), Tom Mrowka, Denis Auroux

Abstract

This thesis constructs stable homotopy types underlying symplectic Floer homology, realizing a program proposed by Cohen, Jones and Segal twenty-five years ago. We work in the setting of Liouville manifolds with a stable symplectic trivialization of their tangent bundles, where we prove that the moduli spaces of Floer trajectories are smooth and stably framed. We then develop a basic TQFT formalism, in the stable homotopy category, for producing operations on these Floer homotopy types from families of punctured Riemann surfaces. As a byproduct, we can generalize many familiar algebraic constructions in traditional Floer homology over the integers to Floer homotopy theory: among them symplectic cohomology, wrapped Floer cohomology, and the Donaldson-Fukaya category.

Yau Wing Li

Title: Endoscopy for affine Hecke categories
Date: Monday, May 3, 2021 | 2:00pm | Room: https://mit.zoom.us/j/98261138739​
Committee: Zhiwei Yun (advisor & committee chair), Roman Bezrukavnikov, George Lusztig

Abstract

We show that the neutral block of the affine monodromic Hecke category for a reductive group is monoidally equivalent to the neutral block of the affine Hecke category for the endoscopic group. The semisimple complexes of both categories can be identified with the generalized Soergel bimodules via the Soergel functor. We extend this identification of semisimple complexes to the neutral blocks of the affine Hecke categories by the technical machinery developed by Bezrukavnikov and Yun.

Zhulin Li

Title: Unstable modules with only the top k Steenrod operations
Date: Monday, April 12, 2021 | 4:30pm | Room: Remote
Committee: Haynes Miller, Jeremy Hahn, Birgit Richter

Abstract

In this talk, I will introduce unstable modules with only the top k Steenrod operations at the prime 2. I'll show that they have projective dimension at most k. Then I'll establish forgetful functors, suspension functors, loop functors and Frobenius functors between such modules. The forgetful functors induce an inverse system of Ext groups, and the inverse system stabilizes when the covariant module is bounded above. In addition, I will talk about a generalization of the Lambda algebra which computes the Ext group from such modules to suspensions of the base field.

Hans Emil Oscar Mickelin

Title: Themes in numerical tensor calculus
Date: Tuesday, April 13, 2021 | 9:00am | Room: Remote
Committee: Sertac Karaman, Laurent Demanet, Pablo Parrilo

Abstract

This thesis studies several distinct, but related, aspects of numerical tensor calculus. First, we introduce a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, the format captures low-rank structures on each grid-scale, which leads to an increase in compression for a fixed accuracy.

Secondly, we consider phase retrieval problems for signals that exhibit a low-rank tensor structure. This class of signals naturally includes a wide set of multidimensional spatial and temporal signals, as well as one- or two-dimensional signals that can be reshaped to higher-dimensional tensors. For a tensor of order $d$, dimension $n$ and rank $r$, we present a provably correct, polynomial-time algorithm that can recover the tensor-structured signals using a total of $\mathcal{O}(dnr)$ measurements, far lower than the $\mathcal{O}(n^d)$ measurements required by dense methods.

Thirdly, we consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, i.e., with nearby components. This deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low-rank assumptions of typical low-rank tensor and matrix completion settings. Our approach enables recovery even with a substantial number of missing entries, for instance for $n$-dimensional tensors of rank $n$ with up to $40\%$ missing entries.

Lastly, we study properties and algorithms for low storage-cost representations in two constrained tensor formats. We study algorithms for computing with the tensor ring format, which is an extension of the tensor train format with variable end-ranks, as well as properties of orthogonally decomposable symmetric tensors.

Vishal Patil

Title: Geometry, topology and mechanics of twisted elastic fibers
Date: Thursday, April 22, 2021 | 2:00pm | https://mit.zoom.us/j/99259649777
Committee: Jörn Dunkel (advisor), John Bush, Mathias Kolle

Abstract

Elastic rods and fibers are ubiquitous in physical and biological systems across a range of length scales, from microtubules to construction beams. In this thesis, we explore the impact of twist on the failure and stability of elastic rods by studying fragmentation and knot dynamics. We begin with a famous phenomenon in elastic rod fragmentation observed by Feynman, who discovered that dry spaghetti typically breaks into three or more pieces when exposed to large bending stresses. Combining theory, experiments and analytic scaling arguments, we demonstrate that twist may be used to achieve binary fracture of brittle elastic rods. Additionally, we show that quenching allows for robust control of the fragmentation cascade. In the second half of this thesis, we use twist to investigate the stability of softer fibers in knotted configurations. We identify twist-based topological counting rules that explain the relative stability of bend knots, which are used to tie two ropes together. These counting rules reveal an underlying stability phase diagram which agrees with numerical simulations and experimental testing of several climbing and sailing knots. Combining the notions of structural and topological stability, we then investigate the energy discharge dynamics of a knotted elastic fiber after it is broken. We show that this class of topological batteries contains special topologically resonant states for which energy release is superslow. Finally, we apply our topological model to surgical knots. Through numerical simulation, we show that topology can be used to identify mechanically stable and balanced suture knots.

Andrew Senger

Title: Multiplicative Structures on Brown–Peterson Spectra at Odd Primes
Date: Monday, April 26, 2021 | 4:30pm | https://mit.zoom.us/j/132540375
Committee: Haynes Miller, Mike Hopkins, Jeremy Hahn

Abstract

We show that the odd-primary Brown-Peterson spectrum does not admit the structure of an E_{2(p^2 +2)} ring spectrum and that there can be no map MU → BP of E_{2p+3} ring spectra for odd primes p. This extends results of Lawson at the prime 2.

Boya Song

Title: Computational Modeling of Bacterial Biofilms
Date: Monday, April 26, 2021 | 12:30pm | https://mit.zoom.us/j/94301675338?pwd=Q3o0U3lYbFlaVnBPdnRxUjRJenYzUT09
Committee: Jörn Dunkel (advisor), Knut Drescher (Max Planck Institute for Terrestrial Microbiology & Philipps-Universität Marburg), Rodolfo Ruben Rosales

Abstract

With recent advances in experimental imaging and image analysis techniques, highly time-resolved measurements of complex bacterial communities at single-cell resolution are now possible to obtain. Guided by these rich experimental data sets, we improve a recently proposed three-dimensional individual-based simulation framework to uncover governing microscopic dynamics at single-cell level that drive the structural developments in growing biofilms. Our individual-based model incorporates the essential biophysical processes of cell growth and division, viscous drag, attractive-repulsive cell-surface interactions, attractive-repulsive cell-cell interactions and external forces and torques (e.g.\ from surrounding flow field). Codes employing graphics processing units (GPUs) are developed to perform simulations to achieve a high degree of parallelization. To validate our simulations with single-cell experimental data, we develop quantitative methods to effectively summarize biofilm architectural properties by a feature vector. With this simulation framework, we investigate the collective dynamics of Vibrio cholerae biofilm formation in various flow intensities. Our experimental and numerical results imply that mechanical cell-cell interactions, combined with the effect of flow when flow intensity is high, account for the emergence of order and structure seen in growing biofilms. In addition, this framework is used to identify the single-cell level mechanisms in the breakdown of Vibrio cholerae biofilm architecture during exposure to antibiotics. We further apply this framework to identify universal mechanical properties that determine early-stage biofilm architectures of four widely studied bacterial species.This work shows an enhanced understanding of the microscopic physics governing biofilm development, which is essential to control and inhibit bacterial populations.

Paxton Turner

Title: Combinatorial methods in statistics
Date: Thursday, April 22, 2021 | 10:00am | https://mit.zoom.us/j/99191136897
Committee: Michel Goemans, Ankur Moitra, Philippe Rigollet (thesis advisor)

Abstract

This thesis explores combinatorial methods in random vector balancing, nonparametric estimation, and network inference. First, motivated by problems from controlled experiments, we study random vector balancing from the perspective of discrepancy theory, a classical topic in combinatorics, and give sharp statistical results along with improved algorithmic guarantees. Next, we focus on the problem of density estimation and investigate the fundamental statistical limits of coresets, a popular framework for obtaining algorithmic speedups by replacing a large dataset with a representative subset. In the third chapter, motivated by the problem of fast evaluation of kernel density estimators, we demonstrate how a multivariate interpolation scheme from finite-element theory based on the combinatorial-geometric properties of a certain mesh can be used to significantly improve the storage and query time of a nonparametric estimator while also preserving its accuracy. Our final chapter focuses on pedigree reconstruction, a combinatorial inference task of recovering the latent network of familial relationships of a population from its extant genetic data.

In this talk I focus on discrepancy theory and coresets.

John Urschel

Title: Topics in Applied Linear Algebra
Date: Thursday, August 5, 2021 | 9:00am | https://mit.zoom.us/j/91831499879
Committee: Michel Goemans (Thesis advisor), Alan Edelman, Philippe Rigollet

Abstract

This thesis considers four independent topics: determinantal point processes, extremal problems in spectral graph theory, force-directed layouts, and eigenvalue algorithms, all tied together by the backdrop of linear algebra. For determinantal point processes (DPPs), we consider the classes of symmetric and signed DPPs, respectively, and in both cases connect the problem of learning the parameters of a DPP to a related matrix recovery problem. Next, we consider two conjectures in spectral graph theory regarding the spread of a graph, and resolve both. In addition, for force-directed layouts of graphs, we connect the layout of the boundary of a Tutte spring embedding to trace theorems from the theory of elliptic PDEs, and provide a rigorous theoretical analysis of the popular Kamada-Kawai objective, proving hardness of approximation and structural results regarding optimal layouts, and providing a polynomial time approximation scheme for this problem. Finally, we consider the Lanczos method for computing extremal eigenvalues of a symmetric matrix and produce new error estimates for this algorithm.

Kaavya Valiveti

Title: The Fock-Schwartz spin representation space
Date: Thursday, July 22, 2021 | 10:00am | Room: 2-449 | Zoom Link
Committee: Richard Melrose (advisor), Tomasz Mrowka, Pavel Etingof

Abstract

In this thesis, we define and study a family of Sobolev-like subspaces (the "Fock-Sobolev spaces") and the corresponding Schwartz-like space (the "Fock-Schwartz space") arising from the infinite-dimensional spin representation constructed by Pressley and Segal. In particular, we study the infinitesimal actions of the group of orientation-preserving diffeomorphisms, Diff^+(S^1), and the loop group LSpin(2n), as well as the action of an infinite-dimensional Clifford algebra on the Fock-Sobolev spaces and Fock-Schwartz space. All of this work is motivated by the goal of constructing the Dirac-Ramond operator on the loop space of a string manifold.

Donghao Wang

Title: Monopoles and Landau-Ginzburg Models
Date: Friday, April 30, 2021 | 3:30 | https://mit.zoom.us/j/94752075761
Committee: Tomasz Mrowka (Advisor), Paul Seidel, Clifford Taubes (Harvard)

Abstract

In this thesis, we define the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is any oriented compact 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form on $Y$, generalizing the construction of Kronheimer-Mrowka for closed 3-manifolds. The basic setup is borrowed from the seminal paper of Meng-Taubes. This thesis will be divided into three parts:

Part I is concerned with the geometry of planar ends. We exploit the framework of the gauged Landau-Ginzburg models to address two model problems for the (perturbed) Seiberg-Witten moduli spaces on either $\mathbb{C}\times\Sigma$ or $\mathbb{H}^2_+\times\Sigma$, where $\Sigma$ is any compact Riemann surface of genus $\geq 1$. These results will lead eventually to the compactness theorem in the second part;

In Part II, we supply the analytic foundation for this Floer theory based on the results from Part I. The Euler characteristic of this Floer homology recovers the Milnor-Turaev torsion invariant of $Y$ by a classical theorem of Meng-Taubes and Turaev.

In Part III, more topological properties of this Floer theory are explored in the special case that the boundary $\partial Y$ is disconnected and the 2-form $\omega$ is non-vanishing on $\partial Y$. Using Floer's excision theorem, we establish a gluing result for this Floer homology when two such 3-manifolds are glued suitably along their common boundary. As applications, we construct the monopole Floer 2-functor and the generalized cobordism maps. Using results of Kronheimer-Mrowka and Ni, we prove that for any such irreducible $Y$, this Floer homology detects the Thurston norm on $H_2(Y,\partial Y;\R)$ and the fiberness of $Y$. Finally, we show that our construction recovers the monopole link Floer homology for any link inside a closed 3-manifold.

Richard Zhang

Title: Analytic Solutions to the Laplace, Poisson, and Biharmonic Equations with Internal Boundaries: Theory and Application to Microfluidic Dynamics
Date: Wednesday, April 28, 2021 | 9:00am | https://mit.zoom.us/j/92576861199
Committee: Rodolfo Ruben Rosales, Gilbert Strang, and Pedro Saenz (Assistant Professor of Mathematics, UNC Chapel-Hill)

Abstract

This dissertation focuses on developing analytical methods for elliptic partial differential equations with conditions imposed on internal boundaries. Internal boundaries are formed where materials with different properties meet to form interfaces. These interfaces arise in a variety of physical and engineering contexts such as in the evaporation of water droplets, dielectric double-spheres, and soft-material Janus drops. The solutions to problems with interfaces are often singular where the interfaces meet the boundaries or two interfaces meet. This causes difficulties when attempting to solve these problems solely with numerical approaches. In contrast analytical approaches (while limited to relatively simple geometries) lend significantly insight into the nature of the singularities with full resolutions in some cases. Potentially this knowledge can then be used to improve the quality of numerical solutions for more generic situations.

We will focus here on four important elliptic PDE problems: Laplace, Poisson, biharmonic and Stokes flow. First we introduces our main analytic result known as the Parity Split Method (PSM) developed in the context of the Laplace and the Poisson equation. The method is then applied to the problem of a thermally driven evaporative liquid bridge in a long V-shaped channel. The problem involves solving acoupled temperature-concentration system of the Laplace equation. Complex analysis based analytic solutions to the concentration equation are also developed along the way. Finally, we extend the PSM for biharmonic equation and addresses several numerical issues in regards to solving for the fluid flow around a soft-material Janus drop.

Yu Zhao

Title: K-theoretic Hall algebra on surfaces and categorification
Date: Tuesday, April 27, 2021 | 9:30am | https://mit.zoom.us/j/3555435016
Committee: Andrei Negut (Chair), Pavel Etingof, Olivier Schiffmann

Abstract

In the present thesis, we construct the K-theoretic Hall algebra corresponding to smooth algebraic surfaces and prove its associativity. We also construct a weak categorification of the elliptic Hall algebra.