# Summer 2011

### Martina Balagovic

Title: On Representations of Quantum Groups and Cherednik Algebras
Date: Wednesday, August 3, 2011
11:00am, Room 2-105
Committee: Pavel Etingof (thesis advisor), George Lusztig, David Vogan

#### Abstract

In the first part of the thesis, we study quantum groups associated to a semisimple Lie algebra $\mathfrak{g}$. The classical Chevalley theorem states that for $\mathfrak{h}$ a Cartan subalgebra and $W$ the Weyl group of $\mathfrak{g}$, the restriction of $\mathfrak{g}$-invariant polynomials on $\mathfrak{g}$ to $\mathfrak{h}$ is an isomorphism onto the $W$-invariant polynomials on $\mathfrak{h}$, Res : $\mathbb{C} [\mathfrak{g}]^\mathfrak{g} \rightarrow \mathbb{C} [\mathfrak{h}]^W$. A recent generalization to the case when the target space $\mathbb{C}$ of the polynomial maps is replaced by a finite dimensional representation $V$ of $\mathfrak{g}$ shows that the restriction map Res: $( \mathbb{C} [\mathfrak{g}] \otimes V)^\mathfrak{g} \rightarrow \mathbb{C}[\mathfrak{h}] \otimes V$ is injective, and that the image can be described by three simple conditions. We further generalize this to the case when a semisimple Lie algebra $\mathfrak{g}$ is replaced by a quantum group. We provide the setting for the generalization, prove that the restriction map
Res: $(O_{q}(G) \otimes V)^{U_{q^{(\mathfrak{g})}}} \rightarrow O(H) \otimes V$ is injective and describe the image.

In the second part we study rational Cherednik algebras $H_{1,c}(W,\mathfrak{h})$ over the field of complex numbers, associated to a finite reflection group $W$ and its reflection representation $\mathfrak{h}$. We calculate the characters of all irreducible representations in category $\mathcal{O}$ of the rational Cherednik algebra for $W$ the exceptional Coxeter group $H_3$ and for $W$ the complex reflection group $G_{12}$. In particular, we determine which of the irreducible representations are finite dimensional, and compute their characters.

In the third part, we study rational Cherednik algebras $H_{t,c}(W,\mathfrak{h})$ over the field of finite characteristic $p$. We first prove several general results about category $\mathcal{O}$, and then focus on rational Cherednik algebras associated to the general and special linear group over a finite field of the same characteristic as the underlying algebraically closed field. We calculate the characters of irreducible representations with trivial lowest weight of the rational Cherednik algebra associated to $GL \:_n(\mathbb{F} \: p^r)$ and $SL \:_n(\mathbb{F}\: p^r)$, and characters of all irreducible representations of the rational Cherednik algebra associated to $GL \:_2(\mathbb{F} \:_p)$.

### Yoonsuk Hyun

Title: On Affine Embeddings of Reductive Groups
Date: Tuesday, July 26th, 2011
5:30pm, Eckerstraße 1 Room 218, Albert Ludwig University of Freiburg
Committee: James McKernan, Chenyang Xu, Stefan Kebekus

#### Abstract

In this thesis, we study the properties and the classification of embeddings of homogeneous spaces, especially the case of affine normal embeddings of reductive groups. We might guess that as in the case of toric varieties, some specific subset of one-parameter subgroups may contribute to the classification of affine embeddings of general reductive group. To check this, we review the theory of affine normal SL(2)-embeddings, and prove that the classification cannot be solved entirely based on one-parameter subgroups. We can also show that even though this set does not give a complete answer to the classification problem, but still contains useful information about varieties. We will also give examples of GL(2)-embeddings which had not previously been constructed in detail, which might be helpful in understanding the general classification of affine normal G-embeddings.

### Suho Oh

Title: Combinatorics related to the Totally nonnegative Grassmannian
Date: Wednesday, June 1, 2011
11:00am, Room 2-135
Committee: Alexander Postnikov, Richard Stanley, Michel Goemans

#### Abstract

Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroids. For studying these cells, Postnikov also introduced Plabic graphs, which are generalizations of double wiring diagrams. We will first go over various combinatorial aspects of positroids and plabic graphs.

Then I will introduce a joint work with Alexander Postnikov and David Speyer. While studying quantum minors, Leclerc and Zelevinsky made a conjecture on certain combinatorial collections called maximal weakly separated collections. We solve the conjecture in a much more general setting using plabic graphs.

# Spring

Title: Geometric Algorithms for Reconfigurable Structures
Date: Tuesday, May 17, 2011
10:00am, Room 2-135
Committee: Erik Demaine, Michel Goemans (chair), Jacob Fox

#### Abstract

In this thesis, we study three problems related to geometric algorithms of reconfigurable structures.

In the first problem, strip folding, we present two universal hinge patterns for a strip of material that enable the folding of any integral orthogonal polyhedron of only a constant factor smaller surface area, and using only a constant number of layers at any point. These geometric results offer a new way to build programmable matter that is substantially more efficient than what is possible with a square N x N sheet of material, which can achieve all integral orthogonal shapes only of surface area O( N ) and may use Θ(N 2) layers at one point. To achieve these results, we develop new approximation algorithms for milling the surface of an integral orthogonal polyhedron, which simultaneously give a 2-approximation in tour length and an 8/3-approximation in the number of turns. Both length and turns consume area when folding strip, so we build on previous approximation algorithms for these two objectives from 2D milling.

In the second problem, maxspan, the goal is to maximize the distance between the endpoints of a fixed-angle chain. We prove a necessary and sufficient condition for characterizing maxspan configurations. The condition states that a fixed-angle chain is in maxspan configuration if and only if the configuration is line-piercing (that is, the line through each of the links intersects the line segment through the endpoints of the chain in the natural order). We call this the Line-Piercing Theorem. The Line-Piercing Theorem was originally proved by Borcea and Streinu in 2008 using Morse-Bott theory and Mayer-Vietoris sequences, but we give an elementary proof based on purely geometric arguments. The Line-Piercing Theorem also leads to efficient algorithms for computing the maxspan for certain classes of fixed-angle chains.

In the third problem, efficient reconfiguration of pivoting tiles, we present an algorithmic framework for reconfiguring a modular robot consisting of identical 2D tiles, where the basic move is to pivot one tile around another at a shared vertex. The robot must remain connected and avoid collisions throughout all moves. For square tiles, and hexagonal tiles on either a triangular or hexagonal lattice, we obtain optimal O(n 2)-move reconfiguration algorithms. In particular, we give the first proofs of universal reconfigurability for the first two cases, and generalize a previous result for the third case. We also consider a strengthening of a model analyzed by Dumitrescu and Pach (SoCG 2004) where tiles slide instead of pivot (making it easier to avoid collisions), and obtain an optimal O(n 2)-move reconfiguration algorithm, improving their O(n 3) bound.

### Kartik Venkatram

Title: Rational Curves in Homogeneous Varieties
Date: Thursday, May 12, 2011
2:00pm, Room 8-205
Committee: James McKernan (thesis advisor), Chenyang Xu, Davesh Maulik

#### Abstract

We investigate the space of rational curves in various homogeneous spaces, with a focus on the quasi-map compactification induced by the Quot functors. We study its birational geometry via techniques from the Mori program, describing its associated cones of ample and effective divisors as well as Mori chambers within the latter. We compute the base loci of all effective divisors, and give a conjectural description of the induced models.

### Chris Dodd

Title: Equivariant Coherent Sheaves, Soergel Bimodules, and Categorification of Affine Hecke Algebras
Date: Wednesday, May 11, 2011
4:00pm, Room 2-142
Committee: Roman Bezrukavnikov (thesis advisor), Pavel Etingof, Zhiwei Yun

#### Abstract

We shall discuss three different versions of "categorification" of the affine hecke algebra and related objects: The first is by equivariant coherent sheaves on the Grothendieck resolution (and related objects), the second is by certain classes on bimodules over polynomial rings, called Soergel Bimodules, and the third is by certain categories of constructible sheaves on the affine flag manifold (for the dual group). We shall explain results relating all three of these, and use them to deduce nontrivial equivalences of categories.

### Timothy Nguyen

Title: The Seiberg-Witten Equations on Manifolds with Boundary
Date: Monday, May 2, 2011
1:00pm, Room 5-134

#### Abstract

In this thesis, we undertake an in-depth study of the Seiberg-Witten equations on manifolds with boundary. We divide our study into three parts.

In Part One, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Here, we study the solution space of these equations without imposing any boundary conditions. We show that the boundary values of this solution space yield an infinite dimensional Lagrangian in the symplectic configuration space on the boundary. One of the main difficulties in this setup is that the three-dimensional Seiberg-Witten equations, being a dimensional reduction of an elliptic system, fail to be elliptic, and so there are resulting technical difficulties intertwining gauge-fixing, elliptic boundary value problems, and symplectic functional analysis.

In Part Two, we study the Seiberg-Witten equations on a 3-manifold with cylindrical ends. Here, Morse-Bott techniques adapted to the infinite-dimensional setting allow us to understand topologically the space of solutions to the Seiberg-Witten equations on a semi-infinite cylinder in terms of the finite dimensional moduli space of vortices at the limiting end. By combining this work with the work of Part One, we make progress in understanding how cobordisms between Riemann surfaces may provide Lagrangian correspondences between their respective vortex moduli spaces. Such an understanding would provide analytic groundwork for Donaldson's topological quantum field theoretic approach to the Seiberg-Witten invariants of closed 3-manifolds.

Finally, in Part Three, we study analytic aspects of the Seiberg-Witten equations on a cylindrical 4-manifold supplied with Lagrangian boundary conditions of the type coming from the first part of this thesis. The resulting equations constitute a nonlinear infinite-dimensional free boundary value problem and is highly nontrivial. We prove fundamental elliptic regularity and compactness type results for the corresponding equations, so that these results may therefore serve as foundational analysis for constructing a monopole Floer theory on 3-manifolds with boundary.

### Tova Brown

Title: Bordered Heegaard Floer homology and four-manifolds with corners
Date: Friday, April 29, 2011
3:00pm, Room 2-139
Committee: Denis Auroux (thesis advisor), Peter Ozsváth, Paul Seidel

#### Abstract

The Heegaard Floer hat invariant is defined on closed 3-manifolds, with a related invariant for 4-dimensional cobordisms, forming a 3+1 topological quantum field theory. Bordered Heegaard Floer homology generalizes this invariant to parametrized Riemann surfaces and to cobordisms between them, yielding a 2+1 TQFT. We discuss an approach to synthesizing these two theories to form a 2+1+1 TQFT, by defining Heegaard Floer invariants for Lefschetz fibrations with corners.

### Michael Schnall-Levin

Title: RNA: Algorithms, Evolution and Design
Date: Friday, April 29, 2011
12:15pm, Room 32G-575
Committee: Bonnie Berger (thesis advisor), Peter Shor, Norbert Perrimon (Harvard Medical School)

#### Abstract

Modern biology is being remade by a dizzying array of new technologies, a deluge of data, and an increasingly strong reliance on computation to guide and interpret experiments. In two areas of biology, computational methods have become central: predicting and designing the structure of biological molecules and inferring function from molecular evolution. In this thesis, I develop a number of algorithms for problems in these areas and combine them with experiment to provide biological insight.

First, I study the problem of designing RNA sequences that fold into specific structures. To do so I introduce a novel computational problem on Hidden Markov Models (HMMs) and Stochastic Context Free Grammars (SCFGs). I show that the problem is NP-hard, resolving an open question for RNA secondary structure design, and go on to develop a number of approximation approaches.

I then turn to the problem of inferring function from evolution. I develop an algorithm to identify regions in the genome that are serving two simultaneous functions: encoding a protein and encoding regulatory information. I first use this algorithm to find microRNA targets in both Drosophila and mammalian genes and show that conserved microRNA targeting in coding regions is widespread. Next, I identify a novel phenomenon where an accumulation of sequence repeats leads to surprisingly strong microRNA targeting, demonstrating a previously unknown role for such repeats.

Finally, I address the problem of detecting more general conserved regulatory elements in coding DNA. I show that such elements are widespread in Drosophila and can be identified with high confidence, a result with important implications for understanding both biological regulation and the evolution of protein coding sequences.

### Nick Rozenblyum

Title: Connections on Conformal Blocks
Date: Friday, April 29, 2011
12:00pm, Room 2-135
Committee: Jacob Lurie (thesis advisor), Dennis Gaitsgory (Harvard), Clark Barwick, Pavel Etingof

#### Abstract

For an algebraic group G and a projective curve X, we study the category of D-modules on the moduli space Bun_G of principal G-bundles on X using ideas from conformal field theory. We describe this category in terms of the action of infinitesimal Hecke functors on the category of quasi-coherent sheaves on Bun_G. This family of functors, parametrized by the Ran space of X, act by averaging a quasi-coherent sheaf over infinitesimal modifications of G-bundles at prescribed points of X. We show that sheaves which are, in a certain sense, equivariant with respect to infinitesimal Hecke functors are exactly D-modules, i.e. quasi-coherent sheaves with a flat connection. This gives a description of flat connections on a quasi-coherent sheaf on Bun_G which is local on the Ran space.

### HwanChul Yoo

Title: Combinatorics in Schubert Varieties and Specht Modules
Date: Thursday, April 28, 2011
2:00pm, Room 56-180
Committee: Alexander Postnikov (thesis advisor), Richard Stanley, David Vogan

#### Abstract

In this defense, we discuss two results connecting geometric/algebraic objects with combinatorial objects. In the first part, we link Schubert varieties in the full flag variety with hyperplane arrangements. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the Schubert variety if and only if the Schubert variety is rationally smooth. For classical types, the arrangements are constructed from signed graphs as signed graphical arrangements. Using this description we also find an explicit combinatorial formula for the Poincare polynomial in type A. In the second part, we relate Specht modules of general diagram with a new class of polytopes. We conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module. We give evidences to this conjecture, and calculate the volume for toric staircase diagram cases. We also define a new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For toric ribbon diagrams, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.

### Fucheng Tan

Title: Families of p-adic Galois representations
Date: Wednesday, April 27, 2011
1:00pm, Room 8-205
Committee: Barry Mazur (thesis advisor, Harvard), Kiran Kedlaya, Bjorn Poonen

#### Abstract

In this article, I first generalize Kisin's theory of finite slope subspaces to arbitrary p-adic fields, and then apply it to the generic fibera of (any) Galois deformation space of 2-dimensional Galois representations. We are able to show that any indecomposible de Rham point lies in the finite slope subspace.

It follows from the construction of finite slope subspace that the complete local ring of a point in the finite slope subspace is closely related to the finite slope deformation ring at the same point. As a consequence, we manage to show the flatness of the weight map near de Rham points, and accumulation and smoothness of de Rham points. In particular, we have a precise dimension formula for the finite slope subspace. Taking into account twists by characters, we define the nearly finite slope subspace, which is believed to serve as the local eigenvariety as is suggested by Colmez's theory of trianguline representation. Following Gouvêa-Mazur and Kisin, we construct an infinite fern in the local Galois deformation space. Moreover, we define the global eigenvariety for GL(2) over any number field, and give a lower bound of its dimension.

### Jennifer Balakrishnan

Title: Coleman Integration for Hyperelliptic Curves: Algorithms and Applications
Date: Tuesday, April 26, 2011
1:00pm, Room 2-135
Committee: Kiran Kedlaya (thesis advisor), Bjorn Poonen, Abhinav Kumar

#### Abstract

The Coleman integral is a p-adic line integral that can be used to encapsulate several quantities relevant to a study of the arithmetic of varieties. In this thesis, I describe algorithms for computing Coleman integrals on hyperelliptic curves and discuss some immediate applications. I give algorithms to compute single and iterated integrals on odd models of hyperelliptic curves, as well as the necessary modifications to implement these algorithms for even models. Furthermore, I show how these algorithms can be used in various situations. The first application is the method of Chabauty to find rational points on curves of genus greater than 1. The second is Minhyong Kim's recent nonabelian analogue of the Chabauty method for elliptic curves. The last two applications concern p-adic heights on Jacobians of hyperelliptic curves, necessary to formulate a p-adic analogue of the Birch and Swinnerton-Dyer conjecture. I conclude by stating the analogue of the Mazur-Tate-Teitelbaum conjecture in our setting and presenting supporting data.

### Yee Lok Wong

Title: High-performance Computing with PetaBricks and Julia
Date: Tuesday, April 26, 2011
11:00am, Room 1-246
Committee: Alan Edelman (thesis advisor), Gilbert Strang, Saman Amarasinghe (MIT EECS)

#### Abstract

We present two recent parallel programming languages, PetaBricks and Julia, and demonstrate how we can use these two languages to re-examine classic numerical algorithms in new approaches for high-performance computing.

PetaBricks is an implicitly parallel language that allows programmers to naturally express algorithmic choice explicitly at the language level. The PetaBricks compiler and autotuner is not only able to compose a complex program using fine-grained algorithmic choices but also find the right choice for many other parameters including data distribution, parallelization and blocking. We re-examine classic numerical algorithms with PetaBricks, and show that the PetaBricks autotuner produces nontrivial optimal algorithms that are difficult to reproduce otherwise. We also introduce the notion of variable accuracy algorithms, in which accuracy measures and requirements are supplied by the programmer and incorporated by the PetaBricks compiler and autotuner in the search of optimal algorithms. We demonstrate the accuracy/performance trade-offs by benchmark problems, and show how nontrivial algorithmic choice can change with different user accuracy requirements.

Julia is a new high-level programming language that aims at achieving performance comparable to traditional compiled languages, while remaining easy to program and offering flexible parallelism without extensive effort. We describe a problem in large-scale terrain data analysis which motivates the use of Julia. We perform classical filtering techniques to study the terrain profiles and propose a measure based on Singular Value Decomposition (SVD) to quantify terrain surface roughness. We then give a brief tutorial of Julia and present results of our serial blocked SVD algorithm implementation in Julia. We also describe the parallel implementation of our SVD algorithm and discuss how flexible parallelism can be further explored using Julia.

### Linan Chen

Title: Applications of Probability to Partial Differential Equations and Infinite Dimensional Analysis
Date: Friday, April 22, 2011
3:00pm, Room 2-139
Committee: Daniel Stroock (thesis advisor), David Jerison, Scott Sheffield

#### Abstract

This thesis consists of two parts. The first part applies a probabilistic approach to the study of the Wright-Fisher equation, an equation which is used to model demographic evolution in the presence of diffusion. The fundamental solution to the Wright-Fisher equation is carefully analyzed by relating it to the fundamental solution to a model equation which has the same degeneracy at one boundary. Estimates are given for short time behavior of the fundamental solution as well as its derivatives near the boundary. The second part studies the probabilistic extensions of the classical Cauchy functional equation for additive functions both in finite and infinite dimensions. The connection between additivity and linearity is explored under different circumstances, and the techniques developed in the process lead to results about the structure of abstract Wiener spaces. Both parts are joint work with Daniel W. Stroock.

### Lu Wang

Title: Self-shrinkers of Mean Curvature Flow and Harmonic Map Heat Flow with Rough Boundary Data
Date: Tuesday, April 19, 2011
4:00pm, Room 2-142
Committee: Tobias Colding (Thesis Advisor), David Jerison, Daniel Stroock

#### Abstract

In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.

Second, we prove the uniqueness for energy decreasing weak solutions of the harmonic map heat flow from the unit open disk into a closed manifold, given any H 1 initial data and boundary data, which is the restriction of the initial data on the boundary of the disk. Previously, under an additional assumption on boundary regularity, this uniqueness result was obtained by Riviére (when the target manifold is the round sphere and the energy of initial data is small) and Freire (for general target manifolds). The point of our uniqueness result is that no boundary regularity assumption is needed. Also, we prove the exponential convergence of the harmonic map heat flow, assuming that the energy is small at all times.

Third, we prove that smooth self-shrinkers in the Euclidean space, that are entire graphs, are hyperplanes. This generalizes an earlier result by Ecker and Huisken: no polynomial growth assumption at infinity is needed.

### Ben Harris

Title: Fourier Transforms of Nilpotent Orbits, Limit Formulas for Reductive Lie Groups, and Wave Front Cycles of Tempered Representations
Date: Tuesday, April 19, 2011
3:00pm, Room 2-135
Committee: David Vogan (thesis advisor), Sigurdur Helgason, Donald King (Northeastern University)

#### Abstract

In this thesis, the author gives an explicit formula for the Fourier transform of the canonical measure on a nilpotent coadjoint orbit for GL(n,R). If G is a reductive Lie group, and O is a nilpotent coadjoint orbit, a necessary condition is given for O to appear in the wave front cycle of a tempered representation. In addition, the coefficients of the wave front cycle of a tempered representation of G are expressed in terms of volumes of precompact submanifolds of certain affine spaces. In the process of proving these results, we obtain several limit formulas for reductive Lie groups.

### Cathy Lennon

Title: Arithmetic and Analytic Properties of Finite Field Hypergeometric Functions
Date: Tuesday, April 19, 2011
10:00am, Room 2-135
Committee: Benjamin Brubaker (thesis advisor), Bjorn Poonen, Abhinav Kumar

#### Abstract

In this defense we discuss new results concerning the arithmetic and analytic properties of Gaussian (finite field) hypergeometric series. We present two expressions for the number of 𝔽p- points on certain families of varieties as special values of these functions. We also present "hypergeometric trace formulas" for the traces of Hecke operators on spaces of cusp forms of level 3 and 9. These formulas lead to a simple expression for the Fourier coefficients of η(3z)8, the unique normalized cusp form of weight 4 and level 9. We then use this to show that a certain threefold is "modular" in the sense that the number of its 𝔽p-points is expressible in terms of these coefficients. Finally, we discuss congruence relations between Gaussian and truncated classical hypergeometric series, and how these relate to the supercongruence conjectures of Rodriguez-Villegas.

### Ronen Mukamel

Title: Orbifold Points on Teichmueller Curves and Jacobians with Complex Multiplication
Date: Monday, April 18, 2011
2:00pm, Room 2-139
Committee: Curt McMullen (thesis advisor), Bjorn Poonen, Abhinav Kumar, Benedict Gross

#### Abstract

For each integer D > 4 congruent to 0 or 1 mod 4, the Weierstrass curve W_D is the moduli space of Riemann surfaces whose Jacobians have real multiplication by the quadratic order of discriminant D stabilizing a holomorphic one form with double zero up to scale. The curve W_D is important in Teichmuller theory because it is a finite volume hyperbolic Riemann surface and the natural immersion into the moduli space of genus two Riemann surfaces is algebraic and isometric.

The purpose of this thesis is to study the orbifold points on W_D. Our primary goal is to determine the number and type of orbifold points on each component of W_D. Together with previous work of McMullen and Bainbridge, our enumeration of the orbifold points completes the determination of the homeomorphism type of W_D and gives a formula for the genus of its components. We use our formula for give bounds on the genus of W_D and determine the components of Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point of W_D.

### Jose Soto

Title: Contributions on Secretary Problems, Independent Sets of Rectangles and Related Problems
Date: Friday, April 15, 2011
10:00am, Room 2-135
Committee: Michel Goemans (supervisor), Jonathan Kelner, Andreas Schulz

#### Abstract

We study three problems arising from different areas of combinatorial optimization.

1. The random-assignment matroid secretary problem: A hidden collection of weights is randomly assigned to the elements of a matroid. The elements are then revealed in random order. When an element is revealed we decide to accept it or not, while keeping the accepted set independent in the matroid. Our objective is to maximize the total weight of the accepted set. By exploiting the notion of principal partition of a matroid and its decomposition into uniformly dense minors, we give the first constant competitive algorithm for this variant of the secretary problem.

2. The jump number problem: Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that is not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for bipartite posets. For the class of 2-directional orthogonal ray posets, we show that this problem is equivalent to finding a maximum disjoint subset of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable, and improving on the running time of existing algorithms for certain subclasses. This is joint work with C. Telha.

3. Constrained set function minimization: We show an efficient O(n3)-time algorithm to find a nonempty minimizer of a symmetric submodular function over any family of sets closed under inclusion. This algorithm is based on Queyranne's pendant-pair technique for minimizing unconstrained symmetric submodular functions. We extend our algorithm to also report all the inclusion-wise minimal nonempty minimizers under hereditary constraints of functions contained in a slightly larger class.

### James Pascaleff

Title: Floer cohomology in the mirror of the projective plane and a binodal cubic curve
Date: Friday, April 15, 2011
10:00am, Room 2-105
Committee: Denis Auroux (thesis advisor), Tom Mrowka, Paul Seidel

#### Abstract

We construct a family of Lagrangian submanifolds in the Landau-Ginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor. These objects correspond under mirror symmetry to the powers of the twisting sheaf O(1), and hence their Floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. An interesting feature is the presence of a singular torus fibration on the mirror, of which the Lagrangians are sections. This gives rise to a distinguished basis of the Floer cohomology and the homogeneous coordinate ring parameterized by fractional integral points in the singular affine structure on the base of the torus fibration. The algebra structure on the Floer cohomology is computed using the symplectic techniques of Lefschetz fibrations and the TQFT counting sections of such fibrations. We also show that our results agree with the tropical analog proposed by Abouzaid-Gross-Siebert. Extensions to a restricted class of singular affine manifolds and to mirrors of the complements of components of the anticanonical divisor are discussed.

### Steven Sivek

Title: Bordered Legendrian knots and sutured Legendrian invariants
Date: Thursday, April 14, 2011
9:00am, Room 1-150
Committee: Tomasz Mrowka (advisor), Peter Kronheimer (Harvard), Peter Ozsváth

#### Abstract

In this thesis we apply techniques from the bordered and sutured variants of Floer homology to study Legendrian knots. First, given a front diagram for a Legendrian knot K in S3 which has been split into several pieces, we associate a differential graded algebra to each "bordered" piece and prove a van Kampen theorem which recovers the Chekanov-Eliashberg invariant Ch(K) of the knot from the bordered DGAs. This leads to the construction of morphisms Ch(K) → Ch(K') corresponding to certain Legendrian tangle replacements and many related applications. We also examine several examples in detail, including Legendrian Whitehead doubles and the first known knot with maximal Thurston-Bennequin invariant for which Ch(K) vanishes.

Second, we use monopole Floer homology for sutured manifolds to construct new invariants of Legendrian knots. These invariants reside in monopole knot homology and closely resemble Heegaard Floer invariants due to Lisca-Ozsváth-Stipsicz-Szabó, but their construction directly involves the contact topology of the knot complement and so many of their properties are easier to prove in this context. In particular, we show that these new invariants are functorial under Lagrangian concordance.

### David Jordan

Title: Quantized Multiplicative Quiver Varieties
Date: Monday, April 11, 2011
2:30pm, Room 5-217
Committee: Pavel Etingof (thesis advisor), Victor Kac, Valerio Toledano-Laredo (Northeastern University)

#### Abstract

In this thesis, a new class of algebras called quantized multiplicative quiver varieties $A^\xi_d(Q)$, is constructed, depending upon a quiver $Q$, its dimension vector $d$, and a certain "moment map" parameter $\xi$. The algebras $A^\xi_d(Q)$ are obtained via quantum Hamiltonian reduction of another algebra $\mathcal{D}_q(\operatorname{Mat}_d(Q))$ relative to a quantum moment map $\mu^\#_q$, both of which are also constructed herein. The algebras $\mathcal{D}_q(\operatorname{Mat}_d(Q))$ and $A^\xi_d(Q)$ bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural.

When $Q$ consists of a single vertex of dimension $N$ with a single loop, the algebra $\mathcal{D}_q(\operatorname{Mat}_N(Q))$ is isomorphic to the algebra of quantum differential operators on $G = GL_N$. In this case, for any
$n \in \mathbb{Z}_{\ge0}$, we construct a functor from the category of $\mathcal{D}_q$-modules to representations of the type $A$ double affine Hecke algebra of rank $n$. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra $H$, and yields representations of the elliptic braid group of rank $n$.

### Wenxuan Lu

Title: Instanton Correction, Wall Crossing and Mirror Symmetry of Hitchin's Moduli Spaces
Date: Thursday, April 7, 2011
3:00pm, Room 8-205
Committee: Shing-Tung Yau (Harvard), Victor Guillemin, James Mckernan

#### Abstract

The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock- Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We show that the two instanton correction problems are equivalent in an appropriate sense.

### Vedran Sohinger

Title: Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrödinger Equations
Date: Wednesday, April 6, 2011
4:00pm, Room 2-146
Committee: Gigliola Staffilani (Advisor), David Jerison, Richard Melrose

#### Abstract

In this Thesis, we study the growth of Sobolev norms of solutions to Nonlinear Schrödinger Equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate of the low-to-high frequency cascade. We will present a frequency decomposition method which allows us to obtain polynomial bounds in the case of the 1D Hartree equation with sufficiently regular convolution potential, and which allows us to bound the growth of fractional Sobolev norms in the case of the Cubic NLS on the real line. We will also present some 2D and 3D results.

# Winter

### Kaloyan Slavov

Title: The Moduli Space of Hypersurfaces Whose Singular Locus Has High Dimension
Date: Friday, December 17, 2010
1:00pm, Room 2-147
Committee: Bjorn Poonen (thesis advisor), James McKernan, Joe Harris

#### Abstract

Fix integers $n$ and $b$ with $n \ge 3$ and $1 \le b \le n - 1$. Let $k$ be an algebraically closed field. Consider the moduli space $X$ of hypersurfaces in $\mathbb{P}^n_k$ of fixed degree $l$ whose singular locus is at least $b$-dimensional. We prove that for large $l$, $X$ has a unique irreducible component of maximal dimension, consisting of the hypersurfaces singular along a linear $b$-dimensional subspace of $\mathbb{P}^n$.