# Summer 2010

Title: From Manifolds to Invariants of En Algebras
Date: Tuesday August 24, 2010
3:30pm, Room 2-131
Committee: Haynes Miller (thesis advisor), Clark Barwick, Mark Behrens

#### Abstract

The work in this thesis is the first step in an investigation of an interesting class of invariants of $E_n$-algebras which generalize topological Hochschild homology. The main goal of this thesis is to simply give a definition of those invariants.

We define PROPs $E^G_n$, for $G$ a structure group sitting over $GL(n,\mathbb{R})$. Given a manifold with a (tangential) $G$-structure, we define right modules $E^G_n[M]$ constructed out of spaces of $G$-augmented embeddings of disjoint unions of euclidean spaces into $M$. These spaces are modifications to the usual spaces of embeddings of manifolds.

Taking $G=1$, $E1_n$ is equivalent to the $n$-little discs PROP, and $E1_n[M]$ is defined for any parallelized $n$-dimensional manifold $M$.

The invariant we define for a $E^G_n$-algebra $A$ is morally defined by a derived coend $T^G(A;M)$:=$E^G_n[M]\overset{L}{\underset{E^G_n}{\otimes}} A$ for any $n$-manifold $M$ with a $G$-structure.

case $T^1(A;S1)$ recovers the topological Hochschild homology of an associative ring spectrum $A$.

These invariants also appear in the work of Jacob Lurie and Paolo Salvatore, where they are involved in a sort of non-abelian Poincaré duality.

In this thesis defense, I will attempt to explain how these invariants generalize the usual bar construction of associative algebras.

### Peter Buchak

Title: Flow-Induced Oscillation of Flexible Bodies
Date: Wednesday, August 18, 2010
2:00pm - 4:00pm, Room 2-135
Committee: John Bush (thesis advisor), Ruben Rosales, Tom Peacock (MIT Mechanical Engineering)

#### Abstract

We present a combined theoretical and experimental investigation of two systems in which flexible bodies are induced to oscillate by steady flows. The first system we study consists of multiple thin sheets of paper in a steady flow, clamped at the downstream end, which we call the "clapping book". Pages sequentially lift off, accumulating in a stack of paper held up by the wind. When the elasticity and weight of the pages overcome the aerodynamic force, the book "claps" shut; this process then repeats. We investigate this system experimentally and theoretically, using the theory of beams in high Reynolds number flow, and test our predictions of the clapping period.

The second system we discuss is inspired by free-reed musical instruments, which produce sound by the oscillation of reeds, thin strips of metal tuned to specific pitches. Each reed is mounted above a slot on the upstream side of a support plate, a geometry that allows a steady flow to induce finite-amplitude oscillations. We study this system experimentally and propose models, also based on the theory of elastic beams in high Reynolds number flow, offering several possible descriptions of the flow.

### Sawyer Tabony

Title: Deformations of Characters, Metaplectic Whittaker Functions, and the Yang-Baxter Equation
Date: Friday, August 13, 2010
11:00am, Room 2-105

#### Abstract

Recent work has uncovered an unexpected connection between characters of representations and lattice models in statistical mechanics. The bridge was first formed from Kuperberg's solution to the alternating sign matrix (ASM) conjecture. This conjecture enumerates ASMs, which can be used to describe highest weight representations, but Kuperberg utilized a square ice model from statistical mechanics in his proof. Since that work, other results using similar methods have been demonstrated, and this work continues in that vein.

We begin by defining the particular lattice model we study. We then imbue the lattice model with Boltzmann weights suggested by a bijection with a set of symmetric ASMs. These weights define a partition function, whose properties are studied by combinatorial and symmetric function methods. The course of study culminates in the use of the Yang-Baxter equation for our ice model to prove that the partition function factors into a deformation of the Weyl denominator in type B and a generalized character of a highest weight representation.

We will briefly mention work done in the last two sections of the thesis, which deal with two approaches to computing Whittaker coefficients of Eisenstein series and automorphic forms.

### Ben Mares

Title: Some Analytic Aspects of Vafa-Witten Twisted $N=4$ Supersymmetric Yang-Mills Theory
Date: Wednesday, July 7, 2010
1:00pm, Room 2-131
Committee: Tomasz Mrowka (thesis advisor), Peter Kronheimer, Paul Seidel

#### Abstract

Given an oriented Riemannian four-manifold equipped with a principal bundle, we investigate the moduli space ℳ$VW$ of solutions to the Vafa-Witten equations. These equations arise from a twist of $N=4$ supersymmetric Yang-Mills theory. Physicists believe that this theory has a well-defined partition function depending on the coupling parameter. On one hand, the S-duality conjecture predicts that this partition function is a modular form. On the other hand, the Fourier coefficients of the partition function seem to measure the "Euler characteristics" of various moduli spaces ℳ¯$ASD$ of compactified anti-self-dual instantons. For several algebraic surfaces, these Euler characteristics were verified to be modular forms.

Except in certain special cases, it's very unclear how to precisely define the partition function. If there is a mathematically sensible definition of the partition function, we expect it to arise as a gauge-theoretic invariant of the moduli spaces ℳ$VW$. The aim of this thesis is to initiate the analysis necessary to define such invariants. In particular, we establish a variety of estimates for the Vafa-Witten equations, and we use those estimates to give a partial Uhlenbeck compactification of the moduli space.

# Spring 2010 Thesis Defenses

### Jinwoo Shin

Title: Efficient Distributed Medium Access Algorithm
Date: Friday, June 25, 2010
11:00am, Room 32D-677
Committee: Peter Shor, Michel Goemans, Devavrat Shah (thesis advisor)

#### Abstract

Efficient scheduling to resolve contention among multiple entities, also known as Medium Access Control (MAC), is the fundamental algorithmic problem that needs to be resolved for designing a high-performance communication network architecture. MAC algorithms are required to be extremely simple and distributed in order to be implementable while utilizing limited network resources efficiently. Despite a long history starting from the Aloha network in 1970's, a satisfactory simple, distributed MAC algorithm of high performance has remained elusive till recently. The main difficulty comes from the nature of "Implementable simple protocols are hard to analyze, while analytic solutions known in the literature are hard to implement."

In this thesis, we resolve this longstanding open problem of designing an efficient MAC protocol for arbitrary wireless networks. Our solution blends the classical Metropolis Hastings sampling mechanism with insights obtained from analysis of time-varying queuing dynamics or Markov process to obtain a desired protocol. Methodologically, our theoretical framework is applicable to design of efficient distributed scheduling algorithms for a wide class of combinatorial resource allocation problems, including switch scheduling and optical network scheduling.

### Craig Desjardins

Title: Monomization of Power Ideals and Parking Functions
Date: Tueday June 22, 2010
11:00am, Room 2-135
Committee: Alex Postnikov (thesis advisor), Richard Stanley, Daniel Kleitman

#### Abstract

A zonotopal algebra is the quotient of a polynomial ring by an ideal generated by powers of linear forms which are derived from a zonotope, or dually its hyperplane arrangement. In the case that the hyperplane arrangement is of Type A, we can rephrase the definition in terms of graphs. Using the symmetry of these ideals, we can find monomial ideals which preserve much of the structure of the zonotopal algebras while being computationally very efficient, in particular far faster than Grobner basis techniques. We extend this monomization theory from the known case of the central zonotopal algebra to the other two main cases of the external and internal zonotopal algebras.

### Hoda Bidkhori

Title: Classification and Enumeration of Special Classes of Posets and Polytopes
Date: Thursday, May 20, 2010
2:00pm, Room 4-145
Committee: Richard P Stanley (thesis advisor), Alex Postnikov, Henry Cohn (Microsoft Research)

#### Abstract

This thesis concerns combinatorial and enumerative aspects of different classes of posets and polytopes. The first part concerns the finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. Ehrenborg and Readdy gave a complete classification of the factorial functions of infinite Eulerian binomial posets and infinite Eulerian Sheffer posets, where infinite posets are those posets which contain an infinite chain. We answer questions asked by R. Ehrenborg and M. Readdy. We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. This work is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.

In the second topic concerns lattice path matroid polytopes. The theory of matroid polytopes has gained prominence due to its applications in algebraic geometry, combinatorial optimization, Coxeter group theory, and, most recently, tropical geometry. In general matroid polytopes are not well understood. Lattice path matroid polytopes (LPMP) belong to two famous classes of polytopes, sorted closed matroid polytopes and polypositroids. We study several properties of LPMPs and build a new connection between the theories of matroid polytopes and lattice paths. I investigate many properties of LPMPs, including their face structure, decomposition, and triangulations, as well as formulas for calculating their Ehrhart polynomial and volume.

### Michael Manapat

Title: Phase Transitions in Evolutionary Dynamics
Date: Monday, May 17, 2010
2:00pm, Program for Evolutionary Dynamics
Harvard University, One Brattle Square, Ste 6
Committee: Martin Nowak (thesis advisor), Peter Shor, John Bush

#### Abstract

This thesis consists of five essays on evolutionary dynamics, but this talk will only cover one of them: "Irrationality as a public good in the trust game."

In the trust game, two players have a chance to win a sum of money. The "investor" begins with one monetary unit. She gives a fraction of that unit to the "trustee." The amount she gives is multiplied by a factor greater than one. The trustee then returns a fraction of what he receives to the investor. In a non-repeated game, a rational trustee will return nothing. Hence, a rational investor will give nothing. In behavioral experiments, however, humans exhibit significant levels of trust and trustworthiness. Here we show that these predispositions may be the result of evolutionary adaptations. We find that when investors have information about trustees, investors become completely trusting and trustees assume the minimum level of trustworthiness that justifies that trust. "Reputation" leads to efficient outcomes as the two players split all the possible payoff from the game, but the trustee captures most of the gains: "seller" reputation helps "sellers" more than it helps "buyers." Investors can capture more of the surplus if they are collectively irrational: they can demand more from trustees than is rational, or their sensitivity to information about trustees can be dulled. Collective investor irrationality surprisingly leads to higher payoffs for investors, but each investor has an incentive to deviate from this behavior and act more rationally. Eventually investors evolve to be highly rational and lose the gains their collective behavior had earned them: irrationality is thus a public good. We describe online behavioral experiments we performed to validate the model assumptions.

### Leonid Chindelevitch

Title: Extracting Information from Biological Networks
Date: Monday, May 3, 2010
2:00pm, Room 32G-575
Committee: Bonnie Berger (thesis advisor), Aviv Regev, Michel Goemans

#### Abstract

Systems biology, the study of biological systems in a holistic manner, has been catalyzed by the dramatic improvement in experimental techniques, coupled with a constantly increasing availability of biological data. The representation and analysis of this heterogeneous data is facilitated by the powerful abstraction of biological networks. We examine several types of these networks and look in detail at the kind of information their analysis can yield.

We begin by looking at protein interaction networks. We introduce a new algorithm for the pairwise alignment of these networks, and show that these alignments can provide important clues to the function of proteins as well as insights into the evolutionary history of the species under examination. We then examine regulatory networks, and present an approach for validating putative drug targets based on the information contained in these networks. Finally, we analyze metabolic networks, providing new insights into the structure of constraint-based models of cell metabolism and describe a methodology for performing a complete analysis of a metabolic network. This methodology is applied to the metabolic networks of two model organisms, yeast and E. coli, as well as Mycobacterium tuberculosis, the pathogen responsible for almost 2 million deaths around the world every year, providing novel insights.

### Chris Kottke

Title: Index Theorems and Magnetic Monopoles on Asymptotically Conic Manifolds
Date: Friday, April 30, 2010
2:00pm, Room 4-231
Committee: Richard Melrose (thesis advisor), Tom Mrowka, Victor Guillemin

#### Abstract

I will discuss two different extensions of the index theorem originally due to C. Callias and later generalized by N. Anghel and others, concerning operators on a complete Riemannian manifold of the form $D+i$Φ, where $D$ is a Dirac operator and Φ is a family of self-adjoint invertible matrices.

The first result is a pseudodifferential version of this index theorem, in the spirit of of the K-theoretic proof of the Atiyah-Singer index theorem, for an appropriate class of pseudodifferential operators on asymptotically conic (asymptotically locally Euclidean) spaces. The second result is an extension to the case where Φ has constant rank nullspace at infinity.

Finally, I will show how these results can be applied to compute the dimension of the moduli space of magnetic monopoles on asymptotically conic manifolds.

### Fang Wang

Title: Radiation Field for Einstein Vacuum Equations
Date: Friday April 30, 2010
3:30pm- 5:30pm, Room 2-131
Committee: Richard Melrose (thesis advisor), Gigliola Staffilani, Victor Guillemin

#### Abstract

The radiation field introduced by Friedlander provides a direct approach to study the asymptotic expansion of solutions to the wave equation near null infinity. I use this concept to study the asymptotic behavior of solutions to Einstein Vacuum equations, which are close to Minkowski space. By imposing harmonic gauge, the Einstein Vacuum equations reduce to a system of quasilinear wave equations on ℝ$t$,$x1+n$. I show that if the space dimension $n$≥$5$, then the MØller wave operator provides an isomorphism from Cauchy data satisfying the constraint equations to the radiation field satisfying the corresponding constraint conditions on small neighborhoods of suitable weighted b-Sobolev spaces.

#### Emanuel Stoica

Title: Unitary Representation of Rational Cherednik Algebras and Hecke Algebras
Date: Friday, April 30, 2010
10:30am - 12:30pm, Room 2-136
Committee: Pavel Etingof (thesis advisor), Roman Bezrukavnikov, David Vogan

#### Abstract

In this thesis, we begin the study of unitary representations in the lowest weight category of rational Cherednik algebras of complex reflection groups. We also conjecture and partially prove that the KZ functor maps these unitary representations to unitary representations of the corresponding Hecke algebras.

### Ting Xue

Title: Nilpotent Orbits in Characteristic 2 and the Springer Correspondence
Date: Wednesday, April 28, 2010
4:30pm, Room 2-135
Committee: George Lusztig (thesis advisor), Roman Bezrukavnikov, Ju-Lee Kim

#### Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p$, $\mathfrak{g}$ the Lie algebra of $G$ and $\mathfrak{g}$* the dual vector space of $\mathfrak{g}$. When $p$ is large enough, Springer constructs a correspondence (called Springer correspondence) which relates irreducible representations of the Weyl group of $G$ to nilpotent $G$-orbits in $\mathfrak{g}$. For arbitrary $p$, Lusztig constructs generalized Springer correspondence which concerns with unipotent classes in $G$. In this thesis, we study Springer correspondence for $\mathfrak{g}$ (assume $G$ is a classical group) and $\mathfrak{g}$* when $p$ is a bad prime. In particular, we classify nilpotent $G$-orbits in $\mathfrak{g}$ (type $B$, $D$) over finite fields and nilpotent $G$-orbits in $\mathfrak{g}$* over algebraically closed or finite fields.

### Martin Frankland

Title: Quillen Cohomology of Π-Algebras and Application to their Realization
Date: Tuesday April 27, 2010
4:00pm, Room 2-151
Committee: Haynes Miller (thesis advisor), Mark Behrens, Michael Hopkins (Harvard)

#### Abstract

A Π-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object A, one may ask if it can be realized topologically: Is there a space X such that π*X is isomorphic to A as a Π-algebra, and if so, can we classify them?

Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a Π-algebra, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of Π-algebras. What do these groups look like, and can we compute them?

We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated Π-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.

### Christopher Evans

Title: A Strong Maximum Principle for Reaction-Diffusion Systems and a Weak Convergence Scheme for Reflected Stochastic Differential Equations
Date: Friday April 23, 2010
10:30am, Room 4-153
Committee: Dan Stroock (thesis advisor), David Jerison, Scott Sheffield

#### Abstract

This thesis consists of two results.

The first result is a strong maximum principle for certain parabolic systems of equations, which, for illustrative purposes, I consider as reaction-diffusion systems. Using the theory of viscosity solutions, I give a proof which extends the previous theorem to no longer require any regularity assumptions on the boundary of the convex set in which the system takes its values.

The second result is an approximation scheme for reflected stochastic differential equations (RSDE) of the Stratonovich type. This is a joint result with Prof. Stroock. We show that the distribution of the solution to such an RSDE is the weak limit of the distribution of the solutions of the RSDEs one gets by replacing the driving Brownian motion by its N-dyadic linear interpolation. In particular, we can infer geometric properties of the solutions to a Stratonovich RSDE from those of the solutions to the approximating RSDE.

### Brian Lehmann

Title: Numerical Properties of Pseudo-effective Divisors
Date: Friday, April 23rd, 2010
10:00am, Room 2-135
Committee: James McKernan (thesis advisor), Steven Kleiman, Chenyang Xu

#### Abstract

For a big line bundle L, there is a close relationship between certain numerical invariants and the asymptotic behavior of sections of L^m as m increases. We analyze how this relationship extends to divisors on the boundary of the pseudo-effective cone. In the first part, we analyze the notion of numerical dimension, which measures the failure of a line bundle to be big. We show that several natural definitions agree. In the second part, we construct a rational map that approximates the Iitaka fibration for L but depends only on its numerical class. This map is called the pseudo-effective reduction map, and is closely related to work of Eckl.

### Qian Lin

Title: Modules Over De Concini Quantum Group and Modules Over Affine Lie Algebra at Critical Level
Date: Tuesday April 20, 2010
4:00pm, Room 2-135
Committee: Roman Bezrukavnikov (thesis advisor), Pavel Etingof, George Lusztig

#### Abstract

In this thesis, we proved the new t-structure,which introduced by Frenkel-Gaitsgory, on the derived category of coherent sheaves on Steinberg variety is braid positive above the tautological t-structure on the cotangent bundle of flag variety. As an application, we also proved that certain category of modules of De Concini-Kac quantum group at odd root of unity is equivalent to certain category of modules of affine Kac-Moody algebra at critical level.

### Xiaoguang Ma

Title: On Trigonometric and Elliptic Cherednik Algebras
Date: Wednesday, April 14, 2010
4:30pm, Room 2-135
Committee: Pavel Etingof (thesis advisor), David Vogan, Roman Bezrukavnikov

#### Abstract

In this thesis, we study the trigonometric and elliptic Cherednik algebras. In the first part, we give a Lie-theoretic construction of the trigonometric Chered- nik algebras of type $BC_n$. We construct a functor from the category of Harish- Chandra modules of the symmetric pair of type AIII to the category of representations of the degenerate affine and double affine Hecke algebra of type $BC_n$. We also study the images of some D-modules and the principal series modules. In the second part, we define the elliptic Dunkl operators on an abelian variety with a finite group action. Using these elliptic Dunkl operators, we construct a new family of quantum integrable systems.

### Angelica Osorno

Title: An Infinite Loop Space Structure for K-theory of Bimonoidal Categories
Date: Tuesday, April 13, 2010
4:00pm, Room 4-153
Committee: Mark Behrens (thesis advisor), Haynes Miller, Clark Barwick (Harvard)

#### Abstract

In recent work of Baas-Dundas-Richter-Rognes, the authors introduce the notion of the $K$-theory of a bimonoidal category $R$, and show that it is equivalent to the algebraic $K$-theory space of the ring spectrum HR. In this thesis we show that $K(R)$ is the group completion of the classifying space of the 2-category $Mod_R$ of modules over $R$, and show that $Mod_R$ is a symmetric monoidal 2-category. We explain how to use this symmetric monoidal structure to produce a Γ-(2-category), which gives an infinite loop space structure on $K(R)$. We show that the equivalence mentioned above is an equivalence of infinite loop spaces.

### Ricky Liu

Title: Specht Modules and Schubert Varieties for General Diagrams
Date: Tuesday, April 13, 2010
4:00 pm Room 2-135
Committee: Alexander Postnikov (thesis advisor), Richard Stanley, Andrei Zelevinsky (Northeastern)

#### Abstract

The algebra of symmetric functions, the representation theory of the symmetric group, and the geometry of the Grassmannian are related to each other via Schur functions, Specht modules, and Schubert varieties, all of which are indexed by partitions and their Young diagrams. We will generalize these objects to allow for not just Young diagrams but arbitrary collections of boxes or, equally well, bipartite graphs. We will then provide evidence for a conjecture that the relation between the areas described above can be extended to these general diagrams.

In particular, we will prove the conjecture for forests. Along the way, we will use a novel geometric approach to show that the dimension of the Specht module of a forest is the same as the normalized volume of its matching polytope. We will also demonstrate a new Littlewood-Richardson rule and provide combinatorial, algebraic, and geometric interpretations of it.

### Jennifer French

Title: Derived Mapping Spaces as Models for Localizations
Date: Monday, April 12, 2010
4:30pm, Room 2-131
Committee: Mark Behrens (thesis advisor), Mike Hill (University of Virginia), Haynes Miller

#### Abstract

This work focuses on a generalization of the models for rational homotopy theory developed by D. Sullivan and D. Quillen and $p$-adic homotopy developed by M. Mandell to $K(1)$-local homotopy theory. The work is divided into two parts.

The first part is a reflection on M. Mandell's model for $p$-adic homotopy theory. Reformulating M. Mandell's result in terms of an adjunction between -complete, nilpotent spaces of finite type and a subcategory of commutative $H\overline{\mathbb{F}}_p$-algebras, the main theorem shows that the unit of this adjunction induces an isomorphism between the unstable $H\mathbb{F}_p$ Adams spectral sequence and the $H\overline{\mathbb{F}}_p$ Goerss--Hopkins spectral sequence.

The second part generalizes M. Mandell's model for $p$ -adic homotopy theory to give a model for $K(1)$-localization. The main theorem gives a model for the $K(1)$-localization of an infinite loop space as a certain derived mapping space of $K(1)$-local ring spectra. This result is proven by analyzing a more general functor from finite spectra to a mapping space of $K^\wedge_p$-algebras using homotopy calculus, and then taking the continuous homotopy fixed points with respect to the prime to $p$ Adams operations. As an application, we show that the main theorem applies to give a model for the $K(1)$-localization of the odd sphere $S^{2n+1}$ when $n \ge 1$.

### Peter McNamara

Title: Whittaker Functions on Metaplectic Groups
Date: Monday, April 12, 2010
1:00pm, Room 2-147
Committee: Benjamin Brubaker (thesis advisor), Kiran Kedlaya, Solomon Friedberg (Boston College)

#### Abstract

The theory of Whittaker functions is of crucial importance in the classical study of automorphic forms on adele groups. Motivated by the appearance of Whittaker functions for covers of reductive groups in the theory of multiple Dirichlet series, we provide a study of Whittaker functions on metaplectic covers of reductive groups over local fields.

### Amanda Redlich

Title: Unbalanced Allocations
Date: Firday, April 2, 2010
3:00pm, Room 4-153
Committee: Peter Shor (thesis advisor), Joel Spencer (Courant Institute at NYU), Daniel Kleitman

#### Abstract

Recently, there has been much research on processes that are mostly random, but also have a small amount of deterministic choice; e.g., Achlioptas processes on graphs. This thesis builds on the balanced allocation algorithm first described by Azar, Broder, Karlin and Upfal. Their algorithm (and its relatives) uses randomness and some choice to distribute balls into bins in a balanced way. Here is a description of the opposite family of algorithms, with an analysis of exactly how unbalanced the distribution can become.

### Matthew Gelvin

Title: Fusion Action Systems
Date: Tuesday, March 30, 2010
4:00pm, Room 2-151
Committee: Haynes Miller (thesis advisor), Mark Behrens, Bill Dwyer (Notre Dame)

#### Abstract

Fusion systems, an abstraction of the $p$-local structure of finite groups, lie in the intersection of algebraic topology and finite group theory. In this talk I detail a "theory of fusion systems with many objects," generalizing fusion system theory to imitate actions of a finite group on a finite set. This generalization will come in three stages: as a simple condition to be put on a finite $p$-group-set, as algebraic structure added to the fusion system, and finally as an additional layer of structure necessary to construct classifying spaces and do homotopy theory.

### Zhenqi He

Title: Odd Dimensional Symplectic Manifolds
Date: Tuesday, March 30, 2010
10:00am, Room 3-343
Committee: Victor Guillemin (thesis advisor), Tomasz Mrowka, Yi Lin (Georgia Southern University)

#### Abstract

In this thesis, we introduce the odd dimensional symplectic manifolds, and study the Hodge theory on the basic symplectic manifolds. We can define two cohomology theories on them, the standard basic de Rham cohomology gheory and a basic version of the Koszul-Brylinski-Mathieu 'harmonic' symplectic cohomology theory. Among our main results are a collection of examples for which these cohomology theories don't coincide, and, in fact, for which the usual basic cohomology theory is infinite dimensional and the symplectic cohomology theory is finite dimensional. On the other hand, we prove an odd version of the Mathieu theorem and the $d$δ-lemma: the two theories coincide if and only if a basic version of strong Lefschetz property holds.

# Winter 2010 Thesis Defenses

### Karola Mészáros

Title: Root Polytopes, Triangulations, and Subdivision Algebras
Date: Tuesday, February 23, 2010
4:00pm, Room 2-147
Committee: Richard Stanley (thesis advisor), Alexander Postnikov, Ira Gessel (Brandeis)

#### Abstract

In this thesis a geometric way to understand the relations of certain noncommutative quadratic algebras defined by Anatol N. Kirillov is developed. These algebras are closely related to the Fomin-Kirillov algebra, which was introduced in the hopes of unraveling the main outstanding problem of modern Schubert calculus, that of finding a combinatorial interpretation for the structure constants of Schubert polynomials. Using a geometric understanding of the relations of Kirillov's algebras in terms of subdivisions of root polytopes, several conjectures of Kirillov about the reduced forms of monomials in the algebras are proved and generalized. Other than a way of understanding Kirillov's algebras, this polytope approach also yields new results about root polytopes, such as explicit triangulations and formulas for their volumes and Ehrhart polynomials. Using the polytope technique an explicit combinatorial description of the reduced forms of monomials is also given. Inspired by Kirillov's algebras, the relations of which can be interpreted as subdivisions of root polytopes, commutative subdivision algebras are defined, whose relations encode a variety of possible subdivisions, and which provide a systematic way of obtaining subdivisions and triangulations.

### William Lopes

Title: The Seiberg-Witten Equations on a Surface Times a Circle
Date: Friday, January 29, 2010
1:00pm, Room 2-147
Committee: Tomasz Mrowka (thesis advisor), Victor Guillemin, Paul Seidel

#### Abstract

In this thesis I study the Seiberg-Witten equations on the product of a genus $g$ surface and a circle. The circle invariance is exploited to reduce to the vortex equations on the surface and thus completely describe the Seiberg-Witten monopoles.

In the case that the solutions are not Morse-Bott regular an explicit perturbation is found which achieves the Morse Bott property and thus find a candidate for a chain complex computing the Seiberg-Witten Floer homology.