### Zachary Abel

Title: Folding and Unfolding with Linkages and Origami
Date: Friday, June 3, 2016 | 1:00pm | Room: 2-449
Committee:Henry Cohn (Chair), Erik D. Demaine (Advisor), Jonathan Kelner

#### Abstract

We revisit foundational questions in the kinetic theory of linkages and origami, investigating their folding/unfolding behaviors and the computational complexity thereof.

In Chapter 1, we exactly settle the computational complexity of realizability, rigidity, and global rigidy for graphs or linkages in the plane, even when the graphs in question are either (1) promised to avoid crossings in all of their configurations, or (2) equilateral and required to be drawn without crossings (so-called "matchstick graphs"): these problems are complete for the class $\exists\mathbb{R}$ defined by the Existential Theory of the Reals, or its complement. To accomplish this, we prove a strong form of Kempe's Universality Theorem for linkages that avoid crossings.

Chapter 2 turns to "self-touching" linkage configurations, whose bars are allowed to butt up against each other so long as they do not cross through. We propose an elegant model for representing such configurations in terms of infinitesimal perturbations, working over a field $\mathbb{R}\langle\epsilon\rangle$ that includes formal infinitesimals. Using this model and the powerful Tarski-Seidenberg "Transfer" principle for real closed fields, we prove a self-touching version of the celebrated Expansive Carpenter's Rule Theorem.

We switch to folding polyhedra in Chapter 3: we show a surprisingly simple technique to continuously flatten the surface of any convex polyhedron without distorting intrinsic distances on the surface or letting the surface pierce itself. This origami motion is quite general, and applies to convex polytopes of any dimension. To prove that no piercing occurs, we apply the same infinitesimal techniques from Chapter 2 to formulate a new formal model of self-touching origami that is simpler to work with than existing models.

Finally, Chapter 4 proves that polyhedra are hard to unfold: it is NP-hard to decide whether it is possible to cut a subset of a polyhedron's edges to unfold the surface into a non-overlapping planar net. This general unfolding problem is not known to be solvable in NP due to precision issues, but we show this is not the only obstacle: it remains NP hard (in fact complete) even for orthogonal polyhedra with integer coordinates (all of whose unfolding also have integer coordinates).

### Hannah Alpert

Title: Special gradient trajectories counted by simplex straightening
Date: Tuesday, April 21, 2016 | 10:00am | Room: 2-242
Committee: Larry Guth (advisor), Tomasz S. Mrowka, Emmy Murphy

#### Abstract

We prove three theorems based on lemmas of Gromov involving the simplicial norm on stratified spaces. First, the Gromov singular fiber theorem (with proof originally sketched by Gromov) relates the simplicial norm to the number of maximum-multiplicity critical points of a smooth map of manifolds that drops dimension by 1. Second, the multitangent trajectory theorem (proved with Gabriel Katz) relates the simplicial norm to the number of maximum-multiplicity tangent trajectories of a nowhere-vanishing gradient-like vector field on a manifold with boundary. And third, the Morse broken trajectory theorem relates the simplicial volume to the number of maximally broken trajectories of the gradient flow of a Morse--Smale function. Corollary: a Morse function on a closed hyperbolic manifold must have a critical point of every Morse index.

### Netanel Blaier

Title: The quantum Johnson homomorphism and exotic symplectomorphism of 3-folds
Date: Wednesday, August 3, 2016 | 1:00pm | Room: 2-449
Committee: Paul Seidel, Tom Mrowka, Emmy Murphy

#### Abstract

Given a monotone symplectic manifold and a homology class $A \in H_2(M)$, we introduce an invariant $q\tau_2$ that associates a characteristic class in Hochschild cohomology to every symplectomorphism $\phi : M \to M$ that ''acts trivially" on quantum cohomology. These are analogues to the familiar Johnson kernel and second Johnson homomorphism $\tau_2$ from low-dimensional topology. The method is quite general, and unlike many abstract tools, explicitly computable in certain nice cases. As an application, we prove the existence of an exotic symplectomorphism $\psi_Y$: an element of infinite order in group of symplectomorphism where $Y$ is the blow-up of $\mathbb{P}^3$ at a genus 4 curve. The classical connection between such Fano varieties and cubic 3-folds allows us to factor $\psi_Y$ in $\pi_0 Symp (Y,\omega)$ as a product of six-dimensional generalized Dehn twists.

### Jui-En Chang

Title: The 1-dimensional $\lambda$-self shrinkers in $\mathbb{R}^2$ and the nodal sets of biharmonic Steklov problems
Date: Monday, May 9, 2016 | 3:30pm | Room: 2-449
Committee:William Minicozzi(Advisor), Jared Speck, Xin Zhou

#### Abstract

This thesis contains two of my projects: The behavior of 1-dimensional $\lambda$-self shrinkers, which are also known as $\lambda$-curves in other literature, and the estimation of the asymptotic behavior of the nodal set of biharmonic Steklov problems.

In the defense, I'll focus on the first project. First, we will review the background of mean curvature flow and the importance of self-shrinkers as solitons of the flow equation. From the Gaussian isoperimetric problem, we have the $\lambda$-hypersurface equation which is related to the self-shrinker equation.

After that, we'll examine the solutions of 1-dimensional $\lambda$-self shrinkers. Unlike the result given by Abresch and Langer that the circle is the only close embedded 1-dimension self-shrinker, we show that for certain $\lambda$<$0$, there are some closed, embedded solutions other than circles. For $\frac{-2}{\sqrt{3}}$<$\lambda$<$0$, there are embedded solutions with 2-symmetry. For $\lambda$<$\frac{-7}{2\sqrt{2}}$ , there are embedded solutions with $m$-symmetry, where $m$ is greater than 2. This is such an important result about the classification of $\lambda$-hypersurfaces. For the case $\lambda>0$, we are able to establish the same result as in the self-shrinker case that the circle is the only closed embedded $\lambda$-curve when $\lambda>0$.

### Yakov Berchenko-Kogan

Title: Yang-Mills Replacement
Date: Monday, May 9, 2016 | 1:30pm | Room: 2-449
Committee: Tomasz Mrowka, William Minicozzi, Emmy Murphy

#### Abstract

We develop an analog of the harmonic replacement technique of Colding and Minicozzi in the gauge theory context. The idea behind harmonic replacement dates back to Schwarz and Perron, and the technique involves taking a function $v\colon\Sigma\to M$ defined on a surface $\Sigma$ and replacing its values on a small ball $B^2\subset\Sigma$ with a harmonic function $u$ that has the same values as $v$ on the boundary $\partial B^2$. The resulting function on $\Sigma$ has lower energy, and repeating this process on balls covering $\Sigma$, one can obtain a global harmonic map in the limit. We develop the analogous procedure in the gauge theory context, where we take a connection $B$ on a bundle over a four-manifold $X$, and replace it on a small ball $B^4\subset X$ with a Yang-Mills connection $A$ that has the same restriction to the boundary $\partial B^4$ as $B$. Throughout, we work with connections of the lowest possible regularity $L^2_1(X)$, and so our gauge transformations are in $L^2_2(X)$ and therefore almost but not quite continuous, leading to more delicate arguments than are available in higher regularity.

### John Binder

Title: Fields of Rationality of Cuspidal Automorphic Representations
Date: Thursday, April 7, 2016 | 3:00pm | Room: 2-449
Committee:Sug Woo Shin (Berkeley) (adviser), Julee Kim (committee chair), and David Vogan

#### Abstract

This thesis examines questions related to the growth of fields of rationality of cuspidal automorphic representations in families. Specifically, if F is the family of cuspidal automorphic representations with fixed central character, prescribed behavior at the Archimedean places, and such that the finite component of \pi has a \Gamma-fixed vector, we expect the proportion of \pi in F with bounded field of rationality to be close to zero if $\Gamma$ is small enough. This question was first asked, and proved partially, by Serre for families of classical cusp forms of increasing level. In this thesis, we will answer Serre's question affirmatively by converting the question to a question about fields of rationality in families of cuspidal automorphic representations of GL_2. We will consider the nalogous question for certain sequences of open compact subgroups \Gamma in unitary. A key intermediate result is an equidistribution theorem for the local components of families of cuspidal automorphic representations.

### Dorin Boger

Title: D equivalences
Date: Friday, April 15, 2016 | 2:00pm | Room: 2-146
Committee: Michal Mcbreen, Roman Bezrukavnikov, David Vogan

#### Abstract

We explain a parameterization of D equivalences from Kawamata's conjecture.

### Efrat Engel-Shaposhnik

Title: Antichains of Interval Orders and Semiorders, and Dilworth Lattices of Maximum Size Antichains
Date:Monday, April 11, 2016
4:00pm, Room: 2-449
Committee:Richard Stanley, Michelle Wachs (University of Miami) and Alex Postnikov

#### Abstract

This thesis consists of two parts. In the first part we count antichains of interval orders and in particular semiorders. We associate a Dyck path to each interval order, and give a formula for the number of antichains of an interval order in terms of the corresponding Dyck path. We then use this formula to give a generating function for the total number of antichains of semiorders, enumerated by the sizes of the semiorders and the antichains.

In the second part we expand the work of Liu and Stanley on Dilworth lattices. Let L be a distributive lattice, let \sigma(L) be the maximum number of elements covered by a single element in L, and let K(L) be the subposet of L consisting of the elements that cover \sigma(L) elements. By a result of Dilworth, K(L) is also a distributive lattice. We compute \sigma(L) and K(L) for various lattices L that arise as the coordinate-wise partial ordering on certain sets of semistandard Young tableaux.

### Teng Fei

Title: On the Geometry of the Strominger System
Date:Friday, April 22, 2016 | 2:00pm | Room: 2-429
Committee:Prof. Victor Guillemin (advisor), Prof. Shing-Tung Yau (advisor) and Prof. Paul Seidel

#### Abstract

The Strominger system is a system of partial differential equations describing the geometry of compactifications of heterotic superstrings with flux. Mathematically it can be viewed as a generalization of Ricci-flat metrics on non-Kähler Calabi-Yau 3-folds. In this thesis, I will present some explicit solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds. These spaces include the important local models like \ccc^3 as well as deformed and resolved conifold. Along the way, I also give a new construction of non-Kähler Calabi-Yau 3-folds and prove a few results in complex geometry.

### Darij Grinberg

Title: Studies on Quasisymmetric Functions
Date: Monday, April 11, 2016 | 11:00am | Room: 2-449
Committee: Alexander Postnikov, Pavel Etingof, Ira Gessel (Brandeis)

#### Abstract

In 1983, Ira Gessel introduced the ring of quasisymmetric functions (QSym), an extension of the ring of symmetric functions. It has now become one of the standard examples of a combinatorial Hopf algebra. In my thesis, I elucidate three new aspects of its theory:

1) Gessel has generalized Schur functions to what is now known as P-partition enumerators; these power series are no longer symmetric (just quasisymmetric), but still share various properties of the Schur functions, including a simple formula for the Hopf-algebraic antipode. Malvenuto and Reutenauer have generalized this formula even further, to quasisymmetric functions "associated to a set of equalities and inequalities". I shall reformulate their generalization in the more wieldy terminology of double posets, sketch a new proof and an even further generalization in which a group acts on the double poset by automorphisms.

2) There is a second bialgebra structure on QSym, with the same multiplication but its own "internal" comultiplication and "internal" counit. I shall show how this bialgebra structure can be constructed using the Aguiar-Bergeron-Sottile universal property of QSym through an extension of the base ring; moreover, this approach also constructs the so-called "Bernstein homomorphism", which makes any connected graded commutative Hopf algebra into a comodule over this second bialgebra QSym.

3) Mike Zabrocki has conjectured a recursive formula for the "dual immaculate quasisymmetric functions" (a certain special case of P-partition enumerators, which can be viewed as an analogue of Schur functions). I have proved this formula by introducing a dendriform algebra structure on QSym.

Two other results which have appeared in my thesis (although will probably not be part of its defense for time reasons) are:

4) some generalizations of Whitney's formula for the chromatic polynomial of a graph in terms of broken circuits. These generalizations include one where the broken circuits are assigned weights and, instead of summing over subsets with no broken circuits, we sum over all subsets and weight them using the product of the weights of all broken circuits contained in the subset. A formula for the chromatic symmetric function is also obtained.

5) a proof of a conjecture by Bergeron, Ceballos and Labbé on reduced-word graphs in Coxeter groups (joint work with Alexander Postnikov). This conjecture concerns the reduced expressions of a given element of a Coxeter group. We can form a graph whose vertices are these expressions, and whose edges connect two reduced expressions which are "a single braid move apart". The simplest part of the conjecture says that this graph is bipartite; subtler results can be obtained by labelling the edges (with labels corresponding to the braid move used) and asking whether a cycle always contains an even number of edges with a given label (it does, after "conjugate" labels are identified). There is also a directed version.

### Qiang Guang

Title: Self-shrinkers and translating solitons of mean curvature flow
Date: Thursday, April 21, 2016 | 3:30pm | Room: 2-449
Committee: William Minicozzi, Tobias Colding, Xin Zhou

#### Abstract

We study singularity models of mean curvature flow ("MCF'') and their generalizations. In the first part, we focus on rigidity and curvature estimates for self-shrinkers. We give a rigidity theorem proving that any self-shrinker which is graphical in a large ball must be a hyperplane. This result gives a stronger version of the Bernstein type theorem for shrinkers proved by Ecker-Huisken. One key ingredient is a curvature estimate for almost stable shrinkers. By proving curvature estimates for mean convex shrinkers, we show that any shrinker which is mean convex in a large ball must be a round cylinder. This generalizes a result by Colding-Ilmanen-Minicozzi : no curvature bound assumption is needed. This part is joint work with Jonathan Zhu.

In the second part, we consider $\lambda$-hypersurfaces which can be thought of as a generalization of shrinkers. We first give various gap and rigidity theorems. We then establish the Bernstein type theorem for $\lambda$-hypersurfaces and classify $\lambda$-curves.

In the last part, we study translating solitons of MCF from four aspects: volume growth, entropy, stability, and curvature estimates. First, we show that every properly immersed translator has at least linear volume growth. Second, using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. Third, we estimate the spectrum of the stability operator L for translators and give a rigidity result of L-stable translators. Finally, we provide curvature estimates for L-stable translators, graphical translators and translators with small entropy.

### Ruthi Hortsch

Title: Counting elliptic curves of bounded Faltings height
Date: Friday, April 15, 2016 | 3:00pm | Room: 2-429
Committee: Bjorn Poonen, Drew Sutherland, Davesh Maulik

#### Abstract

Because many invariants and properties of elliptic curves are difficult to understand directly, the study of arithmetic statistics instead looks at what happens "on average", using heights to make this notion rigorous. Previous work on this has primarily used the naive height, which can be calculated easily but is not intrinsically defined.

We give an asymptotic formula for the number of elliptic curves over the rationals with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in an unbounded region in the real plane defined by transcendental equations, and understand this region well enough to give a formula for the number of these lattice points.

### Yin-Tat Lee

Title: Faster Algorithms for Convex and Combinatorial Optimization
Date: Wednesday, April 20, 2016 | 10:30am | Room: 2-131
Committee: Jonathan Kelner, Michel Goemans, Aleksander Mądry

#### Abstract

In this thesis, we revisit three algorithmic techniques: sparsification, ellipsoidal approximation and elimination and use them to obtain the following results on convex and combinatorial optimization:

• Linear Programming: We obtain the first improvement to the running time for linear programming in 25 years. The convergence rate of this algorithm nearly matches the universal barrier for interior point methods. As a corollary, We obtain the first m sqrt(n) time algorithm for solving the maximum flow problem on directed graphs with m edges and n vertices. This improves upon the previous fastest running time of m min (n^(2/3), sqrt(m)), achieved over 15 years ago by Goldberg and Rao.
• Maximum Flow Problem: We obtain one of the first almost-linear-time algorithms for approximating the maximum flow in undirected graphs. As a corollary, We improve the running time of a wide range of algorithms that use the computation of maximum flows as a subroutine.
• Non-Smooth Convex Optimization: We obtain the first nearly-cubic-time algorithm for convex problems under the black box model. As a corollary, this implied a polynomially faster algorithm for three fundamental problems in computer science: submodular function minimization, matroid intersection, and semidefinite programming.
• Graph Sparsification: We obtain the first almost-linear-time algorithm for spectrally approximating any graph by one with just a linear number of edges. This sparse graph approximately preserves all cut values of the original graph and is useful for solving a wide range of combinatorial problems. This algorithm improves all previous linear-sized constructions, which required at least quadratic time.
• Numerical Linear Algebra: Multigrid is a very efficient method for solving large-scale linear systems arising from graphs in low dimensions, and it has been used extensively for 30 years in scientific computing. Unlike the previous approaches that make assumptions on the graphs, We give the first generalization of the multigrid that provably solves Laplacian systems of any graphs in nearly-linear time.

### Francesco Lin

Title: Monopoles and Pin(2)-symmetry
Date: Thursday, March 31, 2016 | 2:00pm | Room: 2-449
Committee: Peter Kronheimer, Tom Mrowka (chair), Paul Seidel

#### Abstract

We introduce an invariant of three manifolds called Pin(2)-monopole Floer homology. This is the analogue from a Morse-theoretic viewpoint of Manolescu's Pin(2)-equivariant Seiberg-Witten Floer homology, and it can be used to provide an alternative disproof of the Triangulation Conjecture. We discuss some algebraic structures that arise in the theory (including a surgery exact triangle) and use them to provide some explicit computations of the invariants.

### Zihan Hans Liu

Title: The Morse index of mean curvature flow self-shrinkers
Date: Friday, April 22, 2016 | 3:30pm | Room: 2-361
Committee: Tobias Colding, William Minicozzi, Tom Mrowka

#### Abstract

In this thesis, we will introduce a notion of index of shrinkers of the mean curvature flow. We will then prove a gap theorem for the index of rotationally symmetric immersed shrinkers in \bbR^3, namely, that such shrinkers have index at least 3, unless they are one of the stable ones: the sphere, the cylinder, or the plane. We also provide a generalization of the result to higher dimensions.

### Oren Mangoubi

Title: Integral Geometry, Hamiltonian Dynamics, and Markov Chain Monte Carlo
Date: Monday, April 25th, 2016
9:30am, Room: 2-361
Committee: Alan Edelman, Jonathan Kelner, Youssef Marzouk (Aero-Astro), and Natesh Pillai (Harvard Statistics department)

#### Abstract

This thesis presents applications of differential geometry and graph theory to the design and analysis of Markov chain Monte Carlo (MCMC) algorithms. MCMC algorithms are used to generate samples from an arbitrary probability density $\pi$ in computationally demanding situations, since their mixing times need not grow exponentially with the dimension of $\pi$. However, if $\pi$ has many modes, MCMC algorithms may still have very long mixing times. It is therefore crucial to understand and reduce MCMC mixing times, and there is currently a need for global mixing time bounds as well as algorithms that mix quickly for multi-modal densities.

In the Gibbs sampling MCMC algorithm, the variance in the size of modes intersected by the algorithm's search-subspaces can grow exponentially in the dimension, greatly increasing the mixing time. We use integral geometry, together with the Hessian of $\pi$ and the Chern-Gauss-Bonnet theorem, to correct these distortions and avoid this increase in the mixing time. In doing so, we prove a new theorem in integral geometry.

Hamiltonian Monte Carlo (HMC) algorithms are some the most widely-used MCMC algorithms. We use the symplectic properties of Hamiltonians to prove global Cheeger-type lower bounds for the mixing times of HMC algorithms, including Riemannian manifold HMC as well as no-U-turn HMC, the workhorse of the popular Bayesian software package /Stan/. One consequence of our work is the impossibility of energy-conserving Hamiltonian Markov chains to search for far-apart sub-Gaussian modes in polynomial time. We then prove a generalization of the Crofton formula that applies to Hamiltonian trajectories, and use our generalized Crofton formula to improve the convergence speed of HMC-based integration on manifolds.

We also present a generalization of the Hopf fibration acting on arbitrary-$\beta$ ghost-valued random variables. For $\beta=4$, the geometry of the Hopf fibration is encoded by the quaternions; we investigate the extent to which the elegant properties of this encoding are preserved when one replaces quaternions with general $\beta>0$ ghosts.

### Alex Moll

Title: Random Partitions and the Quantum Benjamin-Ono Hierarchy
Date: Thursday, April 7, 2016 | 1:00pm | Room: 2-449

#### Abstract

Stanley's Cauchy identity for Jack symmetric functions defines a Jack measure, a model random partitions for every analytic real function v(w) on the unit circle and parameters e_2 < 0 < e_1. Jacks are eigenfunctions of the Hamiltonian Y_3 of the quantum Benjamin-Ono equation with periodic boundary conditions, dispersion and quantization corresponding e_1 + e_2 and -e_1 e_2 respectively. From this point of view, Jack measures are the random energy distribution of a coherent state around a classical configuration v(w). Taking e_2 -> 0 <- e_1 at a comparable rate \beta/2, we prove that the slopes of the profiles of the random partition concentrate on a limit shape independent of \beta, the push-forward of the uniform measure on the circle along v. This is the conserved density of the classical inviscid Hopf-Burgers hierarchy on the circle, following Dubrovin (2015). For the quantum Hopf-Burgers hierarchy (\beta=2), we recover Okounkov's limit shape for Schur measures (2003) as a verification of the correspondence principle. Our main result is the computation of macroscopic fluctuations of random profiles around the limit shape: they converge to the push-forward along v of the restriction to the circle of a Gaussian free field on the upper half-plane whose covariance is independent of \beta. At \beta=2, our result matches Breuer-Duits' central limit theorem (2013) for Borodin's biorthogonal ensembles.

Our limit theorems follow from a diagrammatic all-order convergent expansion of joint cumulants of linear statistics over what we call ribbon paths". This expansion has the same form as the 1/N refined topological asymptotic expansion over ribbon graphs on surfaces for \beta-ensembles on the line in one-cut potentials V due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). We rely on the Lax operator L for the quantum Benjamin-Ono hierarchy introduced by Nazarov-Sklyanin (2013), a Toeplitz operator on the circle whose symbols is an affine Kac-Moody current for gl_1 at level -e_1 e_2 perturbed by (e_1 + e_2) d/dx. We use the spectral shift function of L to construct a generating function Y(u) of local Hamiltonians commuting with Y_3. This explicit Y(u) is a special case of the Y-operator introduced by Nekrasov (2016).

### Oren Rippel

Title: Sculpting Representations for Deep Learning
Date: Friday, April 22, 2016 | 11:00am | Room: 3-270
Committee: Ryan P. Adams (Harvard), Ankur Moitra, Jonathan A. Kelner, Manohar Paluri (Facebook AI Research)

#### Abstract

In machine learning, the choice of space in which to represent our data is of vital importance to their effective and efficient analysis. In this thesis, we develop approaches to address a number of problems in representation learning. We employ deep learning as means of sculpting our representations, and also develop improved representations for deep learning models.

We present contributions that are based on five papers and make progress in several different research directions. First, we present techniques which leverage spatial and relational structure to achieve greater computational efficiency of model optimization and query retrieval. This allows us to train distance metric learning models 5-30 times faster; optimize convolutional neural networks 2-5 times faster; perform content-based image retrieval hundreds of times faster on codes hundreds of times longer than feasible before; and improve the complexity of Bayesian optimization to linear in the number of observations in contrast to the cubic dependence in its naïve Gaussian process formulation.

Furthermore, we introduce ideas to facilitate preservation of relevant information within the learned representations, and demonstrate this leads to improved supervision results. Our approaches achieve state-of-the-art classification and transfer learning performance on a number of well-known machine learning benchmarks.

In addition, while deep learning models are able to discover structure in high dimensional input domains, they only offer implicit probabilistic descriptions. We develop an algorithm to enable probabilistic interpretability of deep representations. It constructs a transformation to a representation space under which the map of the distribution is approximately factorized and has known marginals. This allows tractable density estimation and inference within this alternate domain.

Title: Invariants associated to one-parameter families of curves and their Jacobians
Date: Wednesday, April 6th, 2016 | 11:00am | Room: 2-361
Committee: Bjorn Poonen, Davesh Maulik, Naoki Imai

#### Abstract

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. Conductors are more intrinsic to the geometry of the family, but harder to compute than discriminants are. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus 2 curves, equality no longer holds in general, but the two invariants are related by an inequality. In the first part of this thesis, we investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus. In the second part of this thesis, we give explicit formulae for the sizes of component groups and Tamagawa numbers of the Néron model of the Jacobian, in terms of the combinatorics of the dual graph of the singular fiber of the underlying family of curves. We then provide some applications of these explicit formulae.

### Yi Sun

Title: Quantum intertwiners and integrable systems
Date: Monday, April 25, 2016 | 11:00am | Room: 2-449
Committee: Pavel Etingof (advisor), Alexei Borodin, Valerio Toledano Laredo (Northeastern)

#### Abstract

We present a collection of results on the relationship between intertwining operators for quantum groups and eigenfunctions for quantum integrable systems.

First, we study the Etingof-Kirillov Jr. expression of Macdonald polynomials as traces of intertwiners of quantum groups in the Gelfand-Tsetlin basis. In the quasi-classical limit, we obtain a new Harish-Chandra type integral formula for Heckman-Opdam hypergeometric functions related to an integral formula appearing in recent work of Borodin-Gorin by integration over Liouville tori of the Gelfand-Tsetlin integrable system. At the quantum level, we obtain a new proof of the branching rule for Macdonald polynomials which transparently relates branching of Macdonald polynomials to branching of quantum group representations.

Second, we study traces of intertwiners for quantum affine algebras. In the sl_2 case, we show that such traces valued in the three-dimensional evaluation representation converge in a certain region of parameters and give a representation-theoretic construction of Felder-Varchenko's hypergeometric solutions to the q-KZB heat equation. This gives the first proof that such a trace function converges and resolves the first case of a conjecture of Etingof-Varchenko. As an application, we prove Felder-Varchenko's conjecture that their elliptic Macdonald polynomials are related to Etingof-Kirillov Jr.'s affine Macdonald polynomials. In the general case, we modify the setting of the work of Etingof-Schiffmann-Varchenko to show that traces of such intertwiners satisfy four commuting systems of q-difference equations -- the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations.

### Dmitry Vaintrob

Title: Mirror symmetry and the K theory of p-adic groups
Date: Friday, May 13th, 2016 | 3:00pm | Room: 2-131
Committee: Roman Bezrukavnikov (chair), Ju-Lee Kim, Paul Seidel, Pavel Etingof

#### Abstract

We study the category of (complex-valued) finitely-generated smooth representations of a p-adic group G and its K theory. In particular, we show that every representation has a resolution by representations induced from finitely-generated representations of open compact subgroups. We do this by studying another category, the compactified representation category recently defined by Bezrukavnikov and Kazhdan, and using techniques from toric mirror symmetry to embed it functorially into a geometric category of equivariant constructible sheaves on the Bruhat-Tits building.

### Michael Viscardi

Title: Equivariant quantum cohomology and the geometric Satake equivalence
Date: Thursday, May 12, 2016 | 1:30pm | Room: 2-449
Committee: Roman Bezrukavnikov (advisor), Paul Seidel, Davesh Maulik

#### Abstract

Recent work on equivariant aspects of mirror symmetry has discovered relations between the equivariant quantum cohomology of symplectic resolutions and Casimir-type connections (among many other objects). We provide a new example of this theory in the setting of the affine Grassmannian, a fundamental space in the geometric Langlands program. More precisely, we identify the equivariant quantum connection of certain symplectic resolutions of slices in the affine Grassmannian of a reductive group G with a trigonometric Knizhnik-Zamolodchikov (KZ)-type connection of the Langlands dual group of G. These symplectic resolutions are expected to be symplectic duals of Nakajima quiver varieties, and thus our result can be thought of as a "symplectic dual" to (part of) the work of Maulik and Okounkov on quiver varieties.