Thesis Defenses
2024

Julius Baldauf
Date: Thursday, March 28, 2024  2:10pm  Room: 2449  Zoom Link
Committee: Bill Minicozzi (Thesis Advisor and Examination Committee Chair), Tristan Collins, Tristan Ozuch
The Ricci Flow on Spin Manifolds
This thesis studies the Ricci flow on manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of SeibergWitten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic HitchinThorpe inequality for simplyconnected, spin 4manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

Adam Block
Date: Tuesday, April 30, 2024  3:00pm  Room: 4149  Zoom Link
Committee: Alexander Rakhlin (advisor), Yury Polyanskiy, Martin Wainwright, Ankur Moitra (chair)
Smoothed Online Learning: Theory and Applications
Many of the algorithms and theoretical results surrounding modern machine learning are predicated on the assumption that data are independent and identically distributed. Motivated by the numerous applications that do not satisfy this assumption, many researchers have been interested in relaxations of this condition, with online learning being the weakest such assumption. In this setting, the learner observes data points one at a time and makes predictions, before incorporating the data into a training set with the goal of predicting new data points as well as possible. Due to the lack of assumptions on the data, this setting is both computationally and statistically challenging. In this thesis, we investigate the statistical rates and efficient algorithms achievable when the data are constrained in a natural way motivated by the smoothed analysis of algorithms. The first part covers the statistical rates achievable by an arbitrary algorithm without regard to efficiency, covering both the fully adversarial setting and the constrained setting in which improved rates are possible. The second part of the thesis focuses on efficient algorithms for this constrained setting, as well as special cases where bounds can be improved under additional structure. Finally, in the third part we investigate applications of these techniques to sequential decicions making, robotics, and differential privacy. We introduce a number of novel techniques, including a Gaussian anticoncentration inequality and a new norm comparison for dependent data.

Murilo Corato Zanarella
Date: Tuesday, April 23, 2024  11:00am  Room: 4370
Committee: Wei Zhang, Zhiwei Yun and Spencer Leslie (Boston College)
First explicit reciprocity law for unitary Friedberg—Jacquet periods
In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin—Selberg, with applications to the Bloch—Kato conjecture and to Iwasawa theory.
This thesis studies the case of Galois representations attached to automorphic representations on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). We prove a new "first explicit reciprocity law" in this setting, which has applications to the rank 0 case of the corresponding Bloch—Kato conjecture.

Gefei Dang
Date: Wednesday, April 24, 2024  3:00pm  Room: 2142
Committee: Wei Zhang, Julee Kim, Zhiwei Yun
Local newforms and spherical characters for unitary groups
We first prove a smooth transfer statement analogous to Jacquet–Rallis’s fundamental lemma and use it to compute the special value of a local spherical character that appears in the Ichino–Ikeda conjecture at a test vector. Then we provide a uniform definition of newforms for representations of both even and odd dimensional unitary groups over padic fields. This definition is compatible with the one given by Atobe, Oi, and Yasuda in the odd dimensional case. Using the nonvanishing of the local spherical character at the test vector, we prove the existence of the representation containing newforms in every tempered Vogan Lpacket. We also show the uniqueness of such representations in Vogan Lpackets and give an explicit description of them using local Langlands correspondence.

Patrik Gerber
Date: Friday, April 26, 2024  9:30am  Room: 2361
Committee: Philippe Rigollet (advisor), Yury Polyanskiy, Martin Wainwright
LikelihoodFree Hypothesis Testing and Applications of the Energy Distance
The first part of this thesis studies the problem of likelihoodfree hypothesis testing: given three samples X,Y and Z with sample sizes n,n and m respectively, one must decide whether the distribution of Z is closer to that of X or that of Y. We fully characterize the problem's sample complexity for multiple distribution classes and with high probability. We uncover connections to twosample, goodness of fit and robust testing, and show the existence of a tradeoff of the form mn ~ k/ε^4, where k is an appropriate notion of complexity and ε is the total variation separation between the distributions of X and Y. We demonstrate that the family of "classifier accuracy" tests are not only popular in practice but also provably nearoptimal, recovering and simplifying a multitude of classical and recent results. We generalize our problem to allow Z to come from a mixture of the distributions of X and Y, and propose a kernelbased test for its solution. Finally, we verify the existence of a tradeoff between m and n on experimental data from particle physics.
In the second part we study applications of the energy distance to minimax statistics. We propose a density estimation routine based on minimizing the generalized energy distance, targeting smooth densities and Gaussian mixtures. We interpret our results in terms of halfplane separability over these classes, and derive analogous results for discrete distributions. As a consequence we deduce that any two discrete distributions are wellseparated by a halfplane, provided their support is embedded as a packing of a highdimensional unit ball. We also scrutinize two recent applications of the energy distance in the twosample testing literature.

Alasdair Hastewell
Date: Thursday, April 18, 2024  12:30pm  Room: 4153  Zoom Link
Committee: Jörn Dunkel (chair), John Bush, Alexander Mietke
Robust spectral representations and model inference for biological dynamics
Current developments in automated experimental imaging allow for highresolution tracking across various scales, from whole animal behavior to tissue scale singlecell trajectories during embryogenesis to spatiotemporal gene expression dynamics or neural dynamics. Transforming these highdimensional data into effective lowdimensional models is an essential theoretical challenge that enables the characterization, comparison, and prediction of the dynamics within and across biological systems. Spectral mode representations have been used successfully across physics, from quantum mechanics to fluid dynamics, to compress and model dynamical data. However, their use in analyzing biological systems has yet to become prevalent. Here, we present a set of noiserobust, geometryaware mathematical tools that enable spectral representations to extract quantitative measurements directly from experimental data. We demonstrate the practical utility of these methods by applying them to the extraction defect statistics in signaling fields on membranes of starfish, the inference of partial differential equations directly from videos of active particle dynamics, and the categorization of emergent patterns in spatiotemporal gene expression during bacterial swarming.
An additional challenge occurs for complex biophysical processes where the underlying dynamics are yet to be entirely determined. Therefore, we would like to use the experimental data to infer effective dynamical models directly that can elucidate the system's underlying biological and physical mechanisms. Building on spectral mode representations, we construct a generic computational framework that can incorporate prior knowledge about biological and physical constraints for inferring the dynamics of living systems through the evolution of their mode representations. We apply this framework first to singlecell imaging data during zebrafish embryogenesis, demonstrating how our framework compactly characterizes developmental symmetry breaking and reveals similarities between panembryo cell migration and Brownian particles on curved surfaces. Next, we apply the framework to the undulatory locomotion of worms, centipedes, robots, and snakes to distinguish between locomotion behaviors. Finally, we present an extension of the framework to the case of nonlinear dynamics when all relevant degrees of freedom are only partially observed, with applications to neuronal and chemical dynamics.

Arun Kannan
Date: Tuesday, April 23, 2024  1:00pm  Room: 1273  Zoom Link
Committee: Pavel Etingof (advisor), Roman Bezrukavnikov, Victor Kac
On Lie Theory in the Verlinde Category
A symmetric tensor category (STC) can be thought of as a “home” to do commutative algebra, algebraic geometry, and Lie theory. They are defined by axiomatizing the basic properties of a representation category of a group (or affine supergroup scheme). Are these the only examples of STCs? In characteristic zero, a famous theorem of Deligne states that, assuming a natural growth condition, representation categories of affine supergroup schemes are the only examples. However, the situation is totally different in positive characteristic, and the Verlinde category Verp is the most fundamental counterexample and appears to play a key role in generalizing the theorem of Deligne to positive characteristic. Moreover, Verp contains the category of supervector spaces. The upshot is that the study of Verp provides new algebraic structures and phenomena beyond that afforded by superalgebra and supergeometry but must also generalize what is already known.
In this thesis defense, we will first survey the theory of symmetric tensor categories. Then, we will discuss new algebraic structures that arise from the Verlinde category, including new constructions of exceptional Lie superalgebras and a generalization of Jordan algebras unique to characteristic 5. Finally, we will turn to progress made on generalizing useful algebraic techniques and machinery from the super setting to the Verp setting, like the Steinberg tensor product theorem and notions of polynomial functors.

Daniil Kliuev
Date: Tuesday, April 16, 2024  2:30pm  Room: 2131
Committee: Pavel Etingof, Roman Bezrukavnikov and Ivan Loseu (Yale)
Positive traces and analytic Langlands correspondence
I will describe my results with coauthors in two directions.
The first direction is the problem of classification of positive traces on quantized Coulomb branches. In the most general setting, this problem generalizes the classical problem of describing irreducible unitary representations of real reductive Lie groups. We consider the case of Kleinian singularities of type $A$ and provide a complete classification of positive traces.
The second direction is analytic Langlands correspondence, which is the following. Let $X$ be a smooth irreducible projective curve over $\mathbb{C}$, $G$ be a complex reductive group. On one side of this conjectural correspondence there are $G^{\vee}$opers on $X$ satisfying a certain topological condition ({\it real} opers), where $G^{\vee}$ is Langlands dual group. On the other side there is joint spectrum of certain operators on $L^2(Bun_G)$, called Hecke operators, where $Bun_G$ is the variety of stable parabolic $G$bundles on $X$ and $L^2(Bun_G)$ is a Hilbert space of squareintegrable halfdensities. We prove most of the main conjectures of analytic Langlands correspondence in the case when $G=\operatorname{PGL}_2(\mathbb{C})$ and $X$ either a genus one curve with points or $X$ is $\mathbb{P}^1$ with higher structures at points.

Vasily Krylov
Date: Monday, April 29, 2024  9:30am  Room: 2143
Committee: Roman Bezrukaunikov (advisor), Zhiwei Yun, and Ivan Loseu (Yale)
Geometry and representation theory of symplectic singularities in the context of symplectic duality
This thesis studies the geometry and representation theory of various symplectic resolutions of singularities from different perspectives. Specifically, we establish a general approach to attack the HikitaNakajima conjecture and illustrate this approach in the example of ADHM spaces. We also study minimally supported representations of the quantizations of ADHM spaces and provide explicit formulas for their characters. Lastly, we describe the monodromy of eigenvalues of quantum multiplication operators for type A Nakajima quiver varieties by examining Bethe subalgebras in Yangians and linking their spectrum with KirillovReshetikhin crystals.

Jae Hee Lee
Date: Monday, April 1, 2024  3:00pm  Room: 2361  Zoom Link
Committee: Prof. Paul Seidel (thesis advisor), Prof. Pavel Etingof, Prof. Denis Auroux (External, Harvard)
Equivariant quantum connections in positive characteristic
In this thesis, we apply techniques from symplectic GromovWitten theory to study the equivariant quantum connections in positive characteristic. The main examples of interest arise from symplectic resolutions. We introduce equivariant generalizations of the quantum Steenrod operations of Fukaya, provide nontrivial computations in the example of the cotangent bundle of the projective line, and explore the relationship with Varchenko's construction of mod $p$ solutions to the quantum differential equation. We then prove the compatibility of the equivariant quantum Steenrod operations with the quantum differential and difference connections. As a consequence, we obtain an identification of our operations for divisor classes with the $p$curvature of the quantum connection in a wide range of examples. 
Ishan Levy
Date: Tuesday, April 23, 2024  1:30pm  Room: 131143
Committee: Davesh Maulik, Michael Hopkins, Haynes Miller, and Jeremy Hahn
The algebraic Ktheory of the chromatic filtration and the telescope conjecture
Chromatic homotopy theory gives a conceptual framework with which to understand the stable homotopy theory, by decomposing the stable homotopy category into monochromatic pieces. There are two variants of these monochromatic pieces, the K(n) and T(n)local categories, the former of which is often quite understandable in terms of formal groups of height n, and the latter of which detects the socalled v_nperiodic part of the stable homotopy groups of spheres. I will explain how algebraic Ktheory has refined our understanding of these monochromatic pieces. On one hand, algebraic Ktheory is an important structural invariant of these categories that 'stably' classifies objects and their automorphisms, and I will explain some tools we have to computationally access the Ktheory of these categories. On the other hand, the algebraic Ktheory of such categories are interesting as spectra: they detect a lot of information about the stable homotopy groups of spheres and have helped us understand the difference between the T(n) and K(n)local categories.

Calder MortonFerguson
Date: Friday, April 26, 2024  1:30pm  Room: 2449  Zoom Link
Committee: Roman Bezrukavnikov (advisor), Zhiwei Yun, Ivan Loseu
KazhdanLaumon categories and representations
In 1988, D. Kazhdan and G. Laumon constructed the \emph{KazhdanLaumon category}, an abelian category $\mathcal{A}$ associated to a reductive group $G$ over a finite field, with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The welldefinedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample for $G = SL_3$.
Since the early 2000s, there has been little activity in the study of KazhdanLaumon categories, despite them being beautiful objects with many interesting properties related to the representation theory of $G$ and the geometry of the basic affine space $G/U$. In the first part of this thesis, we conduct an indepth study of KazhdanLaumon categories from a modern perspective. We first define and study an analogue of the BernsteinGelfandGelfand Category $\mathcal{O}$ for KazhdanLaumon categories and study its combinatorics, establishing connections to BravermanKazhdan's Schwartz space on the basic affine space and the semiinfinite flag variety. We then study the braid group action on $D^b(G/U)$ (the main ingredient in Kazhdan and Laumon's construction) and show that it categorifies the \emph{algebra of braids and ties}, an algebra previously studied in knot theory; we then use this to provide conceptual and geometric proofs of new results concerning this algebra.
After Bezrukavnikov and Polishchuk's counterexample to Kazhdan and Laumon's original conjecture, Polishchuk made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension. He suggested that this conjecture could lead toward a proof that Kazhdan and Laumon's construction is welldefined, and he proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the final chapter of this thesis, we prove Polishchuk's conjecture for all types, and prove that Kazhdan and Laumon's construction is indeed welldefined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

Matthew Nicoletti
Date: Monday, April 29, 2024  2:30pm  Room: 2361  Zoom Link
Committee: Alexei Borodin (Advisor, chair), Scott Sheffield, Lauren Williams (Harvard)
Title: Stochastic Dynamics on Integrable Lattice Models
The purpose of this thesis is to present some new results related to the sixvertex and dimer model. One theme is the construction and analysis of Markov processes which are naturally associated to these lattice models. Certain integrability properties of the sixvertex and dimer model, often related to the YangBaxter equation, allow for the construction of associated Markov chains. In some cases, these are measure preserving Markov chains on configurations of the lattice model. In other cases, they arise via transfer matrices, after choosing a distinguished time coordinate, as a continuous time degeneration of the "time evolution" of the lattice model itself. It is often the case that the integrability of the underlying lattice model provides powerful tools to study the associated Markov chains or their marginals, which are sometimes Markov chains themselves.
In particular together with coauthors, we construct and analyze Markov chains on sixvertex configurations and on dimer model configurations, both of which are models for surface growth in the (2+1)dimensional "Anisotropic KPZ" (or "AKPZ") universality class; we construct a Markov chain generalizing "domino shuffling" which samples exactly from a recently introduced probability measure on tuples of interacting dimer configurations; using a version of the usual domino shuffling algorithm, we construct and analyze deterministic "tembeddings" of certain dimer graphs, which discretize minimal surfaces carrying the conformal structure of the limiting Gaussian free field; we construct stationary measures for several colored interacting particle systems using the Yang—Baxter equation.

Alexander Ortiz
Date: Wednesday, April 24, 2024  1:15pm  Room: 2449  Zoom Link
Committee: Larry Guth (advisor), David Jerison, Gigliola Staffilani
Sparse Fourier restriction for the cone
If the Fourier transform of F(x) is supported near a segment of the lightcone in R^3, what is the shape of the level sets U(N) = {x in R^3 : F(x) > N} for large values of N? In 2000, Thomas Wolff had a creative idea to study a related question based on the method of pointcircle duality, and used it in a pivotal way to prove the first sharp L^pdecoupling estimates for the cone in R^3 for large values of p.
I will discuss new weighted L^2 estimates of F(x) which give us insight into the shape of level sets. I will explain how we use some of the same key ideas introduced by Wolff, together with a few new ones in the same spirit. By Wolff's method, our main theorem will partly be an application of a recent circular maximal function estimate due to Pramanik—Yang—Zahl in 2022 from their study of Kaufmantype restricted projection problems.

Ashwin Sah
Date: Wednesday, April 3, 2024  3:30pm  Room: 2449
Committee: Prof. Yufei Zhao (advisor and chair), Prof. Dor Minzer, and Prof. Philippe Rigollet
Random and exact structures in combinatorics
We aim to show various developments related to notions of randomness and structure in combinatorics and probability. One central notion, that of the pseudorandomnessstructure dichotomy, has played a key role in additive combinatorics and extremal graph theory. In a broader view, randomness (and the pseudorandomness notions which resemble it along various axes) can be viewed as a type of structure in and of itself which has certain typical and global properties that may be exploited to exhibit or constrain combinatorial and probabilistic behavior.
These broader ideas often come in concert to allow the construction or extraction of exact behavior. We look at three particular directions: the singularity of discrete random matrices, thresholds for Steiner triple systems, and improved bounds for Szemerédi's theorem. These concern central questions of the areas of random matrices, combinatorial designs, and additive combinatorics.

Mehtaab Sawhney
Date: Wednesday, April 17, 2024  2:00pm  Room: 2449
Committee: Yufei Zhao, Dor Minzer, and Philippe Rigollet
Probabilistic and Analytic Methods in Combinatorics
The defense will center on fast algorithms for discrepancy theory. Discrepancy theory is broadly concerned with the following problem; given a set of objects, we aim to partition them into pieces which are “roughly equal”. We will focus specifically on vector balancing: given a set of vectors, one seeks to divide them into two parts with approximately equal sum.
Important results in this area, including Spencer’s six standard deviations suffice and Banaszczyk's results towards the Komlós conjecture, were originally purely existential. However, since work of Bansal from 2010, it has become clear that such existential results can often be made algorithmic. We will explain a pair of such results. The first concerns bounds for online vector balancing obtained via a certain Gaussian fixed point random walk. The second gives an algorithmic form of Spencer's theorem that runs in near input sparsity time.

George Stepaniants
Date: Thursday, April 25, 2024  2:30pm  Room: 4149  Zoom Link
Committee: Philippe Rigollet, Jörn Dunkel, Sasha Rakhlin
Inference from Limited Observations in Statistical, Dynamical, and Functional Problems
Observational data in physics and the life sciences comes in many varieties. Broadly, we can divide datasets into crosssectional data which record a set of observations at a given time, dynamical data which follow how observations change in time, and functional data which observe data points over a space (and possibly time) domain. In each setting, prior knowledge of statistical, dynamical systems, and physical theory allow us to constrain the inferences and predictions we make from observational data. This domain knowledge becomes of paramount importance when the data we observe is limited: due to missing labels, small sample sizes, unobserved variables, and noise corruption.
This thesis explores several problems in physics and the life sciences, where the interplay of domain knowledge with statistical theory and machine learning allows us to make inferences from such limited data. We begin in Part I by studying the problem of feature matching or dataset alignment which arises frequently when combining untargeted (unlabeled) biological datasets with low sample sizes. Leveraging the fast numerical methods of optimal transport, we develop an algorithm that gives a stateoftheart solution to this alignment problem with optimal statistical guarantees. In Part II we study the problem of interpolating the dynamics of points clouds (e.g., cells, particles) given only a few sparse snapshot recordings. We show how tools from spline interpolation coupled with optimal transport give efficient algorithms returning smooth dynamically plausible interpolations. Part III of our thesis studies how dynamical equations of motion can be learned from time series recordings of dynamical systems when only partial observations of these systems are captured in time. Here we develop fast routines for gradient optimization and novel tools for model comparison to learn such physically interpretable models from incomplete time series data. Finally, in Part IV we address the problem of surrogate modeling, translating expensive solvers of partial differential equations for physics simulations into fast and easilytrainable machine learning algorithms. For linear PDEs, our prior knowledge of PDE theory and the statistical theory of kernel methods allows us to learn the Green's functions of various linear PDEs, offering more efficient ways to simulate physical systems.

Pu Yu
Date: Wednesday, April 3, 2024  2:00pm  Room: 2255
Committee: Scott Sheffield (advisor), Alexei Borodin, Nike Sun
Conformal welding of random surfaces from Liouville theory
Liouville quantum gravity (LQG) is a natural model describing random surfaces, which arises as the scaling limit for random planar maps. Liouville conformal field theory (LCFT) is the underlying 2D CFT that governs LQG. SchrammLoewner evolution (SLE) is a random planar curve, which describes the scaling limits of interfaces in many statistical physics models. As discovered by Sheffield (2010), one of the deepest results in random geometry is that SLE curves arises as the interfaces under conformal welding of LQG surfaces.
In this thesis, we present some new results on conformal welding of LQG surfaces as well as their applications towards the theory of SLE. We first define a threeparameter family of random surfaces in LQG which can be viewed as the quantum version of triangles. Then we prove the conformal welding result of a quantum triangle and a twopointed quantum disk, and deduce integrability results for chordal SLE with three force points.
The second main result is regarding the conformal welding of a multiple number of LQG surfaces, where under several scenarios, we prove that the output surfaces can be described in terms of LCFT, and the random moduli of the surface is encoded in terms of the partition functions for the SLE curves.
The third part is about the conformal welding of the quantum disks with forested boundary, where we prove that this conformal welding gives a twopointed quantum disk with an independent SLE$_\kappa$ for $\kappa\in(4,8)$. We further extend to the conformal welding of a multiple number of forested quantum disks, where as an application, for $\kappa\in(4,8)$, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for $\kappa \in (6,8)$ and $N\ge 3$ prior to our work.
The conformal loop ensemble (CLE) is a random collection of planar loops which locally look like SLE. For $\kappa \in (4,8)$, the loops are nonsimple and may touch each other and the boundary. As a second application, we derive the probability that the loop surrounding a given point in the nonsimple conformal loop ensemble touches the domain boundary.

Danielle Wang
Date: Tuesday, April 23, 2024  1:00pm  Room: 4265
Committee: Wei Zhang (advisor/chair), Julee Kim, Spencer Leslie (Boston College)
Twisted GanGrossPrasad conjecture for unramified quadratic extensions
The global twisted GGP conjecture is a variant of the GanGrossPrasad conjecture for unitary groups, which considers the restriction of an automorphic representation of GL(V) to its subgroup U(V), for a skewHermitian space V. It relates the nonvanishing of a certain period integral to the central value of an Lfunction attached to the representation.
In this thesis, using a relative trace formula approach, we prove the global twisted GGP conjecture in the unramified case, under some additional local assumptions on the quadratic extension and the automorphic representation. In particular, we reduce the required fundamental lemma to the JacquetRallis fundamental lemma.

Catherine Wolfram
Date: Wednesday, April 24, 2024  3:15pm  Room: 2449  Zoom Link
Committee: Scott Sheffield (thesis advisor), Alexei Borodin, Curtis McMullen
Random geometry in two and three dimensions
A central theme in random geometry is the interplay between discrete models and continuum ones that appear in scaling limits. Surprising structure and symmetry often arises in these scaling limits, leading to an interplay between combinatorics, probability, complex analysis, and geometry.
The dimer model is one of the classical lattice models of statistical mechanics and can be defined in any dimension. In the first half of this thesis, we prove a large deviation principle for dimer tilings in three dimensions. This generalizes a twodimensional result of Cohn, Kenyon, and Propp, and is one of the first results for dimers in any dimension $d>2$. Many ideas and constructions used to study dimers are specific to two dimensions, so our arguments start from a smaller set of tools including Hall's matching theorem, the qualitative description of the Gibbs property, and a double dimer swapping operation.
In the second half of this thesis, we study discrete, geometricallymotivated coordinates called shears on the space of circle homeomorphisms up to M\"obius transformations. The WeilPetersson Teichm\"uller space is a subspace of this which has been of longterm interest in geometry and string theory and has recent connections to SLE curves in probability. We introduce and study natural $\ell^2$ spaces in terms of shears, and obtain sharp results comparing them to H\"older classes of circle homeomorphisms and the WeilPetersson class. We also give some preliminary results about i.i.d. Gaussian random shears.