# Graduate Thesis Defenses 2012

### Nikhil Savale

Title: Spectral Asymptotics for Coupled Dirac Operators
Date: Tuesday, July 17th 2012
10:00am Room: 2-132
Committee: Tomasz Mrowka (advisor), Victor Guillemin, Paul Seidel

#### Abstract

In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an $n$ dimensional manifold is of order $r^{\frac{n+1}{2}}$ followed by a remainder of $O(r^{\frac{n}{2}})$. We perform computations of spectral flow on the sphere which show that $O(r^{\frac{n-1}{2}})$ is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show that its odd functional trace exhibits cancellations in its first $\frac{n+1}{2}$ terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained.

### Peter Speh

Title: A Classification of Real and Complex Nilpotent Orbits of Reductive Groups in Terms of Complex Even Nilpotent Orbits
Date: Monday, April 30th 2012
4:00pm Room: 2-143
Committee: David Vogan (advisor), Sigurdur Helgason, George Lusztig

#### Abstract

Let $\mathfrak{g}$ be a complex, reductive Lie algebra. We prove a theorem parametrizing the set of nilpotent orbits in $\mathfrak{g}$ in terms of even nilpotent orbits of subalgebras of $\mathfrak{g}$ and show how to determine these subalgebras and how to explicitly compute this correspondence. We prove a theorem parametrizing strong real forms of nilpotent orbits in $\mathfrak{g}$ in terms of even nilpotent orbits of subalgebras of $\mathfrak{g}$ and show how to explicitly compute this correspondence.

### Tsao-Hsien Chen

Title: Geometric Langlands in Prime Characteristic
Date: Thursday, April 26th 2012
4:30pm Room: 2-142
Committee: Roman Bezrukavnikov (advisor), Pavel Etingof, Zhiwei Yun

#### Abstract

Let $C$ be a smooth projective curve over an algebraically closed field $k$ of sufficiently large characteristic. Let $G$ be a semisimple algebraic group over $k$ and let $G^{\vee}$ be its Langlands dual group over $k$. Denote by ${Bun}_G$ the moduli stack of $G$-bundles on $C$ and ${LocSys}_{G^{\vee}}$ the moduli stack of $G^{\vee}$-local systems on $C$. Let $D_{{Bun}_G}$ be the sheaf of crystalline differential operators on ${Bun}_G$. In this thesis I construct an equivalence between the derived category $D({QCoh}({LocSys}_{G^\vee}^0))$ of quasi-coherent sheaves on some open subset ${LocSys}_{G^{\vee}}^0\subset{LocSys}_{G^{\vee}}$ and derived category $D(D_{{Bun}_G}^0-{mod})$ of modules over some localization $D_{{Bun}_G}^0$ of $D_{{Bun}_G}$. This generalizes the work of Bezrukavnikov-Braverman in the $GL_n$ case.

### Olga Stroilova

Title: $t_k$ and $L(n)$
Date: Tuesday, April 24th 2012
3:30pm Room: 2-146
Committee: Haynes Miller (advisor), Mark Behrens, Charles Rezk (University of Illinois)

#### Abstract

The generalized Tate construction $t_k$ reduces chromatic level.

Computing the dual of $L(n)=L(n)_1$.

Towards the duals of Steinberg summands in corresponding Thom spectra of negative representations, $L(n)_{-q}$. An equivariant loopspace machine. The base case $t_k$ of a sphere.

Results parallel work of A. Cathcart, B. Guillou and P. May, and N. Stapleton, or could benefit from parallel.

### Alejandro Morales

Title: Combinatorics of Colored Factorizations, Flow Polytopes and of Matrices over Finite Fields
Date: Monday, April 23rd 2012
3:00pm Room: 2-143
Committee: Alexander Postnikov (thesis advisor), Richard Stanley, Olivier Bernardi

#### Abstract

In the first part of this thesis we study factorizations of the permutation $(1,2,\ldots,n)$ into $k$ factors of given cycle type. Using representation theory, Jackson obtained for each $k$ an elegant formula for counting these factorizations according to the number of cycles of each factor. For the case $k=2$, Bernardi gave a bijection between these factorizations and tree-rooted maps; certain graphs embedded on surfaces with a distinguished spanning tree. This type of bijection also applies to all $k$ and we use it to show a symmetry property of a refinement of Jackson's formula first exhibited in the case $k=2,3$ in joint work with E. Vassilieva.

We then give applications of this symmetry property. First, we study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,\ldots,k$ are in distinct cycles of the random permutation of $\{1,2,\ldots,n\}$ obtained as product of two uniformly random $n$-cycles. Second, we use the symmetry to give a short bijective proof of the number of planar trees and cacti with given vertex degree distribution calculated by Goulden and Jackson.

In the second part we establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied combinatorially by Postnikov and Stanley and by Baldoni and Vergne using residues. As a special family of flow polytopes, we study the Chan-Robbins-Yuen polytope whose volume is the product of the consecutive Catalan numbers. We introduce generalizations of this polytope and give intriguing conjectures about their volume.

In the third part we consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a $q$-analogue of permutations with restricted positions (i.e., rook placements). Extending a result of Haglund, we show that when the set of entries is a skew Young diagram, the numbers, up to a power of $q-1$, are polynomials with nonnegative coefficients. We apply this result to the case when the set of entries is the Rothe diagram of a permutation. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.

### Ramis Movassagh

Title: Eigenvalues and Low Energy Eigenvectors of Quantum Many-Body Systems
Date: Monday, April 23rd 2012
11:00am Room: 4-159
Committee: Peter W. Shor (advisor), Alan Edelman, Jeffrey Goldstone

#### Abstract

This thesis is divided into two parts. The first part is devoted to eigenvalues of quantum many-body systems (QMBS). I introduce Isotropic Entanglement (IE) and show that the distribution of QMBS with generic interactions can be accurately obtained using IE. Next, I discuss the eigenvalue distribution of the Anderson model in one dimension from free probability theory.

The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free (FF) Hamiltonians, real and imaginary time evolution of quantum spin systems on a line within MPS representation and the numerical implementation. I then give the degeneracy and unfrustration condition for quantum spin chains with generic local interactions. Next I discuss an interesting measure zero example where the FF Hamiltonian is carefully constructed to give a unique ground state with high entanglement S ∼ log(L) and a polynomial gap.

### Anatoly Preygel

Title: Thom-Sebastiani and Duality for Matrix Factorizations, and Results on the Higher-Structures of the Hochschild Invariants
Date: Friday, April 20th 2012
12:00pm Room: 2-143
Committee: Paul Seidel (chair), Jacob Lurie (advisor), Clark Barwick

#### Abstract

The 2-periodic dg-category MF of matrix factorizations is a categorical singularity invariant, appearing on the algebro-geometric side of mirror symmetry. Though it lives in commutative algebra, and the first definitions were very explicit, it turns out to admit a few descriptions very much in the spirit of homotopy theory. After recalling some categorical background, we'll state the results in the title and mention how they're useful for establishing basic properties of MF and for relating its Hochschild invariants to classical linear-algebraic invariants of singularities. Finally, we'll give a description of MF in terms of "derived algebraic geometry," and see how this allows one to give conceptual proofs of the results in the title.

### Jethro Van Ekeren

Title: Modular Invariance for Vertex Superalgebras
Date: Thursday, April 19th 2012
4:00pm Room: 2-142
Committee: Victor Kac (advisor), Pavel Etingof, David Vogan

#### Abstract

We generalize Zhu's theorem on modular invariance of characters of vertex algebras to the setting of vertex superalgebras with rational, rather than integer, conformal weights. To recover $SL_2(\mathbb{Z})$-invariance it turns out to be necessary to consider characters of twisted $V$-modules. Initially we assume $V$ to be rational, then we remove this assumption as well. Then $SL_2(\mathbb{Z})$-invariance is regained by including certain 'logarithmic' characters.

### Niels Martin Møller

Title: Mean Curvature Flow Self-Shrinkers with Genus and Asymptotically Conical Ends
Date: Thursday, April 19th 2012
3:00pm Room: 3-333
Committee: Tobias H. Colding (advisor), Richard B. Melrose, Tomasz Mrowka

#### Abstract

On the theory of minimal surfaces and singularities in mean curvature flow, including:

(1) New asymptotically conical self-shrinkers with a symmetry, in $\mathbb{R}^{n+1}$.

(1') Classification of complete embedded self-shrinkers with a symmetry, in $\mathbb{R}^{n+1}$, and of asymptotically conical ends with a symmetry.

(2) Construction of complete, embedded self-shrinkers $\Sigma_g^2\subseteq\mathbb{R}^3$ of genus $g$, with asymptotically conical infinite ends (as conjectured by Ilmanen in the early 1990's), via minimal surface gluing.

(3) Construction of closed embedded self-shrinkers $\Sigma_g^2\subseteq \mathbb{R}^3$ of genus $g$.

For (2) it was necessary to develop a stability theory in a setting with unbounded geometry, via a Schauder theory in weighted Hölder spaces for Schrödinger operators of Ornstein-Uhlenbeck type. The results in (1)-(1') appeared in a joint paper with Stephen Kleene, and the results in (2) were proven in collaboration with Kleene-Kapouleas.

### Sheel Ganatra

Title: Symplectic Cohomology and Duality for the Wrapped Fukaya Category
Date: Thursday, April 19, 2012
11:00am Room: 4-153
Committee: Denis Auroux, U.C. Berkeley (advisor), Tomasz Mrowka, Paul Seidel (chair)

#### Abstract

Consider the wrapped Fukaya category $\mathcal{W}$ of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of $\mathcal{W}$ to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of $\mathcal{W}$ are isomorphisms, in a manner compatible with the ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal, (2) Fourier-Mukai theory for the wrapped Fukaya category via holomorphic quilts, (3) a geometric bimodule duality map, coming from discs with two negative punctures and many positive punctures, and (4) a generalization of the Cardy condition.

### Nick Sheridan

Title: Homological mirror symmetry for a Calabi-Yau hypersurface in projective space
Date: Thursday, April 19th 2012
9:00am Room: 2-142
Committee: Paul Seidel (advisor), Denis Auroux, Tom Mrowka

#### Abstract

This thesis is concerned with Kontsevich's Homological Mirror Symmetry conjecture. We consider the generalized pair of pants, which is de fined to be the complement of $n$ generic hyperplanes in $(n-2)$-dimensional projective space. The pair of pants is conjectured to be mirror to the Landau-Ginzburg model $(\mathbb{C}^n,W)$, where $W = z_1 \ldots z_n$. We construct an immersed Lagrangian sphere in the pair of pants, and show that its endomorphism $A_{\infty}$ algebra in the Fukaya category is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror, giving some evidence for the Homological Mirror Symmetry conjecture in this case. We build on these results to prove Homological Mirror Symmetry for a smooth $d$-dimensional Calabi-Yau hypersurface in projective space, for any $d \ge 3$.

### Roman Travkin

Title: Quantum geometric Langlands correspondence in positive characteristic: the GL(N) case
Date: Wednesday, April 18th 2012
4:30pm Room: 2-131
Committee: Roman Bezrukavnikov, Dennis Gaitsgory, Zhiwei Yun

#### Abstract

Let $C$ be a smooth connected projective curve of genus $>1$ over an algebraically closed field $k$ of characteristic $p>0$, and $c\in k\setminus\mathbb F_p$. Let $\operatorname{Bun}_N$ be the stack of rank $N$ vector bundles on $C$ and $\mathcal{L}_{\det}$ the line bundle on $\operatorname{Bun}_N$ given by determinant of derived global sections. We construct an equivalence of derived categories of modules for certain localizations of the twisted crystalline differential operator algebras $\mathcal{D}_{\operatorname{Bun}_N,\mathcal{L}_{\det}^c}$ and $\mathcal{D}_{\operatorname{Bun}_N,\mathcal{L}_{\det}^{-1/c}}$.

The first step of the argument is the same as that of Bezrukavnikov--Braverman for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier--Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of $p$-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de~Rham local systems on $C$.

### Nikola Kamburov

Title: A free boundary problem inspired by a conjecture of De Giorgi
Date: Wednesday, April 18th 2012
2:00pm Room: 56-114
Committee: David Jerison (advisor), Tobias Colding, Gigliola Staffilani

#### Abstract

We study global monotone solutions of the free boundary problem that arises from minimizing the energy functional $I(u) = \int |\nabla u|^2 + V(u)$, where $V(u)$ is the characteristic function of the interval $(-1,1)$. This functional is a close relative of the scalar Ginzburg-Landau functional $J(u) = \int |\nabla u|^2 + W(u)$, where $W(u) = (1-u^2)^2/2$ is a standard double-well potential. According to a famous conjecture of De Giorgi, global critical points of $J$ that are bounded and monotone in one direction have level sets that are hyperplanes, at least up to dimension $8$. Recently, Del Pino, Kowalczyk and Wei gave an intricate fixed-point-argument construction of a counterexample in dimension $9$, whose level sets "follow" the entire minimal non-planar graph, built by Bombieri, De Giorgi and Giusti (BdGG). In this thesis, we turn to the free boundary variant of the problem and we construct the analogous example; the advantage here is that of geometric transparency as the interphase $\{|u|$ < $1\}$ will be contained within a unit-width band around the BdGG graph. Furthermore, we avoid the technicalities of Del Pino, Kowalczyk and Wei's fixed-point argument by using barriers only.

### Steven Sam

Title: Free resolutions, combinatorics, and geometry
Date: Friday, April 13th 2012
10:00am Room: 2-146
Committee: Steven Kleiman, Richard Stanley (advisor), Jerzy Weyman (co-advisor, Northeastern University)

#### Abstract

Recent results of Boij-Söderberg and Eisenbud-Schreyer describe the cone of graded Betti tables for finitely generated modules over a polynomial ring. In this talk, I will discuss three results related to this cone.

1. Eisenbud-Schreyer showed that there is a "dual" cone given by cohomology tables of vector bundles on projective space. The description of these cones over other (embedded) projective varieties is related to the existence of "Ulrich sheaves" which I construct on Schubert degeneracy loci.

2. The cone is naturally triangulated by a poset structure on the extremal rays. I will explain a module-theoretic interpretation of this poset structure.

3. I will explain the analogous statements for regular local rings and some partial results for local hypersurface rings.

### Joel Lewis

Title: Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux
Date: Tuesday, April 10th 2012
2:00pm Room: 4-145
Committee: Alex Postnikov (advisor), Richard Stanley, Tom Roby

#### Abstract

We consider a variety of questions related to pattern avoidance in alternating permutations and generalizations thereof. We give bijective enumerations of alternating permutations avoiding patterns of length $3$ and $4$, of permutations with descent set $k \mathbb{Z}$ avoiding the identity permutation of length $k + 1$ or $k + 2$, and of the reading words of Young tableaux of any skew shape avoiding any of the patterns $132$, $213$, $312$, or $231$. Our bijections include a simple bijection involving binary trees, variations on the Robinson-Schensted-Knuth correspondence, and recursive bijections established via isomorphisms of generating trees.

### Inna Zakharevich

Title: Scissors Congruence and K-theory
Date: Monday, March 19th 2012
4:30pm Room: 2-131
Committee: Michael Hopkins, Clark Barwick, Haynes Miller

#### Abstract

Hilbert's third problem asks the following question: given two polyhedra with the same volume, is it possible to dissect one into finitely many polyhedra and rearrange it into the other one? The answer (due to Dehn in 1901) is no: there is another invariant that must also be the same. Further work in the 60s and 70s generalized this to other geometries by constructing groups which encode scissors congruence data. Though most of the computational techniques used with these groups related to group homology, the algebraic K-theory of various fields appears in some very unexpected places in the computations. We will give a different perspective on this problem by examining it from the perspective of algebraic K-theory: we construct the K-theory spectrum of a scissors congruence problem and relate some of the classical structures on scissors congruence groups to structures on this spectrum.