The graduate student seminar is a forum for graduate students to present interesting mathematics to other graduate students. It is not recommended that you speak on your own work, as that is in most cases not accessible enough. We encourage self contained talks which are accessible and appeal to a wide audience of graduate students.

If you would like to give a talk please email <gradseminar@> or talk with Andy Manion or Ila Varma.

Fall 2011 schedule: The seminar will be held in Fine 314 from 12:15pm on Tuesdays.

You may be interested in other seminars at the Mathematics Department and Princeton University.

## Calendar: Fall 2011

September 27, 2011 The polynomial method for Harmonic analysis Kevin Hughes

September 20, 2011 Cohen-Lenstra Heuristics Kevin Wilson

## Archive

### Spring 2010

February 23, 2010 Infinite Ergodic Theory and Continued Fractions Francesco Cellarosi

March 2, 2010 Stationary phases and Spherical averages Po Lam Yung

March 4, 2010 Magma Mark Watkins

March 8, 2010 Fixed point theorems and applications in PDEs Shiwu Yang

March 23, 2010 The Conquest of Topology by Algebra, or, Topological Spaces as Relational Beta-Modules Owen Biesel

March 30, 2010 Iwasawa Theory Xin Wan

April 6, 2010 Representation varieties in 3-manifold topology Will Cavendish

April 13, 2010 Representation varieties in 3-manifold topology Eleftherios Tsiokos

April 27, 2010 Transcendence Boris Alexeev

April 20, 2010 Smooth Poincare conjecture: is there an exotic four-sphere? Zhong Tao Wu

May 4, 2010 Regularity of the Hardy Littlewood function Kevin Hughes

### Fall 2009

December 3, 2009 Bigger is better Kevin Hughes

November 19, 2009 Extend your function now! Results guaranteed or your money back Arie Israel

November 12, 2009 The Reflector Antenna Problem, and its Connection to Mass Transport June Kitagawa

October 29, 2009 The Congruence Subgroup Problem for SL(n,Z) Daniel Shenfeld

October 22, 2009 Curve enumeration from matrix integrals Michael McBreen

October 15, 2009 Harmonic measure Phil Sosoe

October 8, 2009 Strong multiplicity one and l-adic Galois representations Stefan Patrikis

Oct 1, 2009 ODEs and solutions from actions of Lie Groups Mohammad Farajzadeh Tehrani

September 24, 2009 Nonlinear Waves Jonathan Luk

### Spring 2009

May 14, 2009 Deligne-Lusztig Andrew Snowden

April 30, 2009 Some Relationships Between Moduli Spaces of Curves Wei Ho

April 23, 2009 TBA Kevin Hughes

April 16, 2009 Equivariant Cohomology Iman Setayesh

April 9, 2009 Mathai-Quillen's Thom form and Atiyah-Hirzebruch's Riemann-Roch theorem Guangbo Xu

April 2, 2009 Poisson summation formula Ali Altug

March 26, 2009 Sum-product estimates via combinatorial geometry Po-Shen Loh

March 12, 2009 Morse theory Michael McBreen

March 5, 2009 Lie Groups: Decomposition and Exponentiation Kevin Wilson

February 26, 2009 Models and Fields: A delicate Passage to Characteristic p Jacob Tsimerman

February 19, 2009 Bend and Break Bhargav Bhatt

February 12, 2009 Tamagawa numbers Arul Shankar

February 5, 2009 The Nielsen Realization Problem Will Cavendish

### Fall 2008

December 11, 2008 Ratner's Theorem Ilya Vinogradov

December 4, 2008 Fibered Knots Margaret Doig

November 20, 2008 An Overview of the l-adic and p-adic Monodromy Theorems Stefan Patrikis

November 13, 2008 Algebraic Curves and their Realizations Yaim Cooper

November 6, 2008 Embedded Surfaces in 4-Manifolds Josh Green

October 23, 2008 Differential Galois Theory Vivek Shende

October 16, 2008 Ergodic Theory Sam Ruth

October 9, 2008 GRH and polynomial-time primality testing Jacob Tsimerman

October 2, 2008 The Composition Problem in Measure Spaces Philip J Isett

September 25, 2008 Grushko's Theorem Sucharit Sarkar

September 18, 2008 Random Walks on Groups Simon Marshall

### Spring 2008

May 1, 2008 The Six-Sphere Yanir Rubinstein

April 24, 2008 Secant Varieties and Applications Wei Ho

April 17, 2008 Matriod Theory Melody Chan

April 10, 2008 Undecidability and Hilbert's 10'th Problem Jacob Tsimerman

April 3, 2008 The Moment Map and Delzant Polytopes Mohammad F.Tehrani

March 27, 2008 Counting Circles and Other Things Elena Fuchs

March 13, 2008 Counting Curves Iman Setayesh

March 6, 2008 Prime Splitting Laws Stefan Patrikis

February 28, 2008 Hyperbolic 3-Manifolds and Arithmetic Simon Marshall

February 21, 2008 Differential Equations and Arithmetic Ali Altug

February 14, 2008 An Outline of the h-Cobordism Theorem Sucharit Sarkar

February 7, 2008 The Probabilistic Method and A Classical Result of Erdos Sasha Fradkin

### Fall 2007

November 29, 2007 The ABC Conjecture and Other Things I Can't Prove Jacob Tsimerman

November 15, 2007 Ergodic Theory and Equidistribution Simon Marshall

November 8, 2007 The Borel-Weil-Bott Theorem Vivek Shende

October 25, 2007 The Adams Conjecture Bhargav Bhatt

October 18, 2007 In P, or not in P? Sucharit Sarkar

October 11, 2007 Sidon Sets and Digit Reversal Boris Bukh

October 4, 2007 Getting to Know Stacks Melanie Wood

September 27, 2007 4-Manifolds Josh Greene

September 20, 2007 Artin on Real Closed Fields Balin Fleming

### Spring 2007

May 22, 2007 Counting Lattice Points Po Lam Yung

May 8, 2007 Picard Groups of Moduli Problems Bhargav Bhatt

May 1, 2007 Interesting Characters Erik Carlsson

April 24, 2007 Probabilistic Methods in Ramsey Theory Jacob Fox

April 17, 2007 Knot Homology Theories Josh Greene

April 10, 2007 Sieves in Number Theory Gihan Marasingha (Oxford University)

April 3, 2007 The Banach-Tarski Paradox Lillian Pierce

March 13, 2007 What Everyone Should Know About Topology (but I had to look up) Margaret Doig

March 6, 2007 Constructing Quotients in Algebraic Geometry Wei Ho

February 27, 2007 Kakeya Problem or Why Taming Hedgehogs Is So Much Fun Boris Bukh

February 20, 2007 Representations of Nilpotent Groups Brian Street

February 6, 2007 Harmonic Analysis and Group Representations Simon Marshall

January 30, 2007 Hanna Neumann Theorem Sucharit Sarkar

January 23, 2007 Representations of Quivers Andrew Snowden

### Fall 2006

December 12, 2006 Good versus Evil - a not so epic battle Blair D. Sullivan

December 5, 2006 All about the Riemann Hypothesis (or at least some) Elena Fuchs

November 28, 2006 Adeles and the Zeta Function Simon Marshall

November 21, 2006 Intersection Patterns of Geometric Objects Jacob Fox

November 14, 2006 Random Phenomena in Number Theory Melanie Matchett Wood

November 7, 2006 The Development of Morse Theory (or, On the Morse-Thom-Bott-Smale-Milnor-Witten Complex) Josh Greene

October 24, 2006 The Jones Polynomial (and other cool facts about Knot Theory) Margaret Doig

October 17, 2006 Hypersurfaces in Toric Varieties Alan Stapledon (Institute for Advanced Study)

October 10, 2006 Small Forbidden Configurations Balin Fleming

October 3, 2006 All about 1089 Lara Pudwell (Rutgers University)

September 26, 2006 How to paint the plane? Boris Bukh

### Spring 2006

April 25, 2006 From polygonal billiards to Teichmuller dynamics Corinna Ulcigrai

April 18, 2006 Discriminant Varieties Lanie Wood

April 11, 2006 The Alexander Polynomial of a Knot Yi Ni

April 4, 2006 Random Matrices and Universality Louis-Pierre Arguin

March 28, 2006 A Counterexample to the Poincar\'e Conjecture? Josh Greene

March 14, 2006 When Adding Things Up Gets You Back Where You Started Blair Dowling Sullivan

March 7, 2006 Very Even Numbers of Odd Partitions Chris Mihelich

February 28, 2006 Toplogical Dynamics and van der Waerden's Theorem Shimon Brooks

February 21, 2006 Hypoellipticity Brian Street

February 14, 2006 Braid Groups Margaret Doig

February 7, 2006 Towards a degree two Atkin-Lehner theory. Michael Volpato

### Fall 2005

December 14, 2005 Tell Me Your Values and I Will Tell You What You Are! Boris Bukh

December 7, 2005 The Prime Number Theorem on the Nose Alex Kontorovich (Columbia University)

November 30, 2005 The Face Numbers of Simplicial Polytopes Balin Fleming

No talk November 23 -- Thanksgiving

November 16, 2005 Szemeredi's Regularity Lemma Po-Shen Loh

November 9, 2005 You call that a 7-sphere?! Chris Mihelich

No talk November 2 -- Fall Break

October 26, 2005 Regular Homotopy Classes of Singular Maps Andras Juhasz

October 19, 2005 Quaternionic Analysis Andrew Snowden

October 12, 2005 A Polynomial with a Big Galois Group Elena Fuchs

October 5, 2005 Rational (!) Cubic Reciprocity Paul Pollack

September 28, 2005 11-torsion on Elliptic Curves Wei Ho

September 21, 2005 A Minor Theorem Blair Dowling

### Spring 2005

May 18, 2005 The Margulis Measure Alexander Bufetov

May 11, 2005 A Milnor Theorem Josh Greene

April 20, 2005 Linear invariants of points in the complex plane Andrew Snowden

April 13, 2005 Nonstandard Analysis Brian Street

March 30, 2005 A curious pair of integers Michael Volpato

March 23, 2005 The Absolute Galois Group and Dessins d'Enfants Melanie Wood

February 25, 2005 Dealing with Complexity: The Statistical Physics Approach Louis-Pierre Arguin

Special Schedule!

February 16, 2005 Finding Arithmetic Flavor in Quantum Chaos Lior Silberman

### Spring 2004

May 14, 2004 Rank One Transformations: Ornstein's example Alexander I. Bufetov

April 30, 2004 A stability approach to hypergraph Tur and agrave;n problems Peter Keevash

April 16, 2004 Dynamics of Quasi-Periodic Schroedinger Equations Kristian Bjerklöv

February 18, 2004 Why rational curves? Carolina Araujo

February 4, 2004 Pseudoholomorphic Curves and Periodic Orbits in Hamiltonian Dynamics Jake Solomon (Massachusetts Institute of Technology)

### Fall 2003

December 11, 2003 Ricci Flow and Geometrization of 3-manifolds Baris Coskunuzer

December 5, 2003 Singular Sets in Conformal Geometry: from PDE to Topology Maria del Mar Gonzalez

Special Schedule!!

October 13, 2003 Arithmetic Quantum Chaos: Semiclassical Limits of Automorphic Forms Lior Silberman

### Spring 2003

April 18, 2003 An introduction to extremal graph theory Peter Keevash

April 4, 2003 On Bowen's Construction of Gibbs measures Alexander I. Bufetov

March 31, 2003 Prospective Graduate Student Open House Maria Chudnovsky, Chris Hall, Philip Gressman, Peter Milley

Special Open House Seminar

February 28, 2003 Null Yang–Mills fields on $S^3\times S^1$ Jonathan Holland

cancelled

### Fall 2002

December 13, 2002 Using Torsion to Bound the Rank of an Elliptic Curve Chris Hall

December 6, 2002 Tube Volumes and Small Hyperbolic Three-Manifolds Peter Milley

November 22, 2002 Ultraproducts, Scaling limits and Fixed points: from Model Theory to discrete groups Lior Silberman

November 8, 2002 Elliptic curves, Root numbers, ultrametric analysis, pliable functions, sieve theory and Del Pezzo surfaces Harald Helfgott

This and future seminars in Fall 2002 will be @ Fine 1201.

October 11, 2002 Knots with Large Volume and Lens Space Surgeries Ken Baker (University of Texas at Austin)

### Spring 2002

May 17, 2002 A survey of 3 dimensional geometry and topology Baris Coskunuzer

May 10, 2002 Restriction of the Fourier Transform to Quadratic Submanifolds Adrian Banner

April 26, 2002 Weakly symmetric spaces Oksana Yakimova (Independent University of Moscow)

April 12, 2002 Uniform Approximate Functional Equation for Principal L-functions Gergely Harcos

April 5, 2002 Explicit Construction of Expanding Graphs and Kazhdan's Property T Lior Silberman

March 29, 2002 Introduction to Rationally Connected Varieties Carolina Araujo

March 15, 2002 Fractional-moment methods for continuum Schrödinger operators Jeff Schenker

March 8, 2002 Interval exchange transformations and the Teichmueller flow Alexander Bufetov

March 1, 2002 Cryptologic Properties of GammaPi Class Seth Patinkin

### Fall 2001

October 26, 2001 Regulators on Milnor K-theory Matt Kerr

October 19, 2001 The Problem of Parity in Elliptic Curves and Analytic Number Theory Harald Helfgott

### Spring 2001

May 11, 2001 The Nilpotent Variety in its Natural Setting Julianna Tymoczko

May 4, 2001 Multiplier operators in stratified groups Ricardo Saenz

April 20, 2001 The wonderful world of CR geometry Andreea Nicoara

March 30, 2001 Operator Valued Holomorphic Functions with Positive Real Part Jeff Schenker

February 23, 2001 KKM on trees Eli Berger

February 16, 2001 Combinatorial Nullstellensatz Peter Keevash

February 9, 2001 Radon Transform, inversion formulas, and a glimpse of singularity Hadi Jorati

January 19, 2001 Harmonic Tori: An Algebro-Geometric Perspective Emma Carberry

### Fall 2000

December 1 & 8, 2000 Structures of Quantum Field Theory Slava Rychkov

November 11, 2000 Sidon Sequences Ben Green

October 27, 2000 Triangulated Spheres and Systems of Disjoint Representatives for Families of Hypergraphs Maria Chudnovsky

October 13, 2000 Minuscule Representations of Lie Algebras Jacob A. Lurie

October 6, 2000 The Creation of Spectral Gaps by Graph Decoration Jeffrey H. Schenker

### Spring 2000

May 5, 2000 Zeros of Statistical Mechanical Partition Functions Logan Kleinwaks

April 21, 2000 The Convex Minorant of Random Walks and Brownian Motion Toufic Suidan

April 14, 2000 Can One Hear the Shape of a Bell? Jade Vinson

April 7, 2000 A geometric proof of existence of Whitney's stratification Vadim Kaloshin

March 24, 2000 Covering lattice points by subspaces Gergely Harcos

March 10, 2000 Hodge Theory and Algebraic Cycles Matthew Kerr

March 3, 2000 A simple description of a loop group David Nadler

February 25, 2000 Properties of projections of fractal sets from infinite into finite dimensional spaces Vadim Kaloshin

February 18, 2000 On Branched Covers of the Torus by Surfaces of Genus 2 David Goldberg

February 4, 2000 Physics and Mathematics of Oscillatory Integrals Slava Rychkov

### Fall 1999

December 10, 1999 Volume Minimizing Submanifolds via Calibrated Geometry Dan Grossman

November 12, 1999 Counting Orbits of Branched Covers of an Elliptic Curve David Goldberg

October 29, 1999 Elliptic Integrals and Intersectin Theory Matt Kerr

October 7, 1999 An Introduction to Anderson Localization Jeff Schenker

### Spring 1999

April 15, 1999 Semi-Classical Limits of Eigenvalues and Witten's Proof of the Morse Inequalities Jeff Schenker

April 8, 1999 Feigenbaum Universality in Period Doubling Bifurcations Michael Tehranchi

March 25, 1999 On Carleson's Theorem Slava Rytchkov

March 11, 1999 Braid Groups and Branched Coverings Beyond the Spherical Case Dave Goldberg

March 4, 1999 The Four Color Theorem Thor Johnson

February 25, 1999 Existence and Uniqueness for 2-D and 3-D Navier Stokes Equations Vadim Y. Kaloshin

February 18, 1999 Boundaries of symmetric spaces Adrian Banner

February 11, 1999 Thomas Hales' Proof of the Kepler Conjecture Jade Vinson

## Abstracts

### September 27, 2011

The polynomial method in Harmonic analysis
Kevin Hughes
12:15 @ Fine 314

We'll discuss incidence problems in harmonic analysis. I'll focus on recent progress which interestingly comes from algebra and topology. In particular, we'll examine Dvir's proof of the Kakeya conjecture over finite fields, Guth's endpoint estimate for the multilinear Kakeya function which used tools from algebraic topology and the joints problem.

### September 20, 2011

Cohen-Lenstra heuristics
Kevin Wilson
12:15 @ Fine 314

The class group of a number field, in colloquial terms, measures the failure of unique factorization in its ring of integers. Despite its central role in number theory, information about its structure, and even its size, mostly remain mysteries. In this talk, we will discuss conjectures on the asymptotics of the structure of class groups (the Cohen-Lenstra heuristics) as well as a more general notion of "categorical randomness" which has led to several other conjectures.

## Abstracts Archive

### May 4, 2010

Regularity of the Hardy Littlewood function
Kevin Hughes
12:30–1:30 @ Fine 314

Lp boundedness of the Hardy-Littlewood function is a classic result in harmonic analysis. But not much is understood about the regularity of it. For instance, if your function f is in the Sobolev W1,1 space, is its maximal function in L1? The answer to this question is unknown, but I will discuss our partial understanding.

### April 20, 2010

Smooth Poincare conjecture: is there an exotic four-sphere?
Zhong Tao Wu
12:30–1:30 @ Fine 314

After briefly reviewing the history of (smooth) Poincare conjecture, we shall focus on the case of four-dimensional sphere. We would like to discuss a construction due to Gluck that provides many candidates of exotic four-spheres, i.e., manifolds homeomorphic but not diffeomorphic to the standard four-sphere. We will prove that Gluck's manifolds are indeed homeomorphic to four-sphere and then sketch an argument by Freedman et al. that could potentially lead to the discovery of exotic four-spheres.

### April 27, 2010

Transcendence
Boris Alexeev
12:30–1:30 @ Fine 314

We all know that e and π are transcendental. How about numbers like e+π, e^π, or sqrt(2)^[sqrt(2)] If 2n, 3n, and 5n are all integers, must n be an integer as well? What if only 2n and 3n are integers? In this talk, I will talk about these and related questions. In particular, I hope to prove the well-known fact above: e and π are transcendental. I will also mention the applications of transcendence theory to other subjects, like logic and integration.

### April 13, 2010

Representation varieties in 3-manifold topology
Eleftherios Tsiokos
12:30–1:30 @ Fine 314

I will start with Dirichlet's Theorem in arithmetic progressions and then I will talk about Chebotarev Density Theorem (which is a generalization) and how it is related with class field theory.

### April 6, 2010

Representation varieties in 3-manifold topology
Will Cavendish
12:30–1:30 @ Fine 314

An important technique in 3-manifold topology is to study representations of the fundamental group of a 3-manifold into a Lie group. Under appropriate circumstances, a collection of such representations can be used to cut out an algebraic variety that is well defined up to birational equivalence. This is useful not only because it produces algebro-geometric invariants of 3-manifolds, but also because it allows one to reinterpret certain topological questions as (hopefully easier) questions in algebraic geometry. In this talk I'll describe the basic ingredients of celebrated work of Culler and Shalen that uses these techniques together with Bass-Serre theory to find special surfaces in 3-manifolds. I'll also tell you a bit about the significance of these surfaces, and why 3-manifold topologists are willing to go to such lengths to find them.

### March 30, 2010

Iwasawa Theory
Xin Wan
12:30–1:30 @ Fine 314

I'll talk about Iwasawa theory. I'll start with the structures of the class groups of number fields, and see how these generalize to various main conjectures in Iwasawa theory. If there's time, I'll also sketch the main idea of the proof and review what's known so far, including the recent result of Skinner-Urban.

### March 23, 2010

The Conquest of Topology by Algebra, or, Topological Spaces as Relational Beta-Modules
Owen Biesel
12:30–1:30 @ Fine 314

We explore the relationship between topological spaces and modules of the ultrafilter monad, Beta. Several basic properties of topological spaces X are rephrased as properties of "convergence" relations BX--->X.

### March 8, 2010

Fixed point theorems and applications in PDEs
Shiwu Yang
12:30–1:30 @ Fine 314

I'll begin with Banach's fixed point theorem in which strict contraction is required and then give examples to show how this simple theorem implies the local and global existence to vary kinds of evolution equations. I'll also introduce Schauder and Schaefer's fixed point theorem which would be of importance in elliptic theory and verify this by several examples if time permitted.

### March 4, 2010

Magma
Mark Watkins
12:30–1:30 @ Fine 314

We give a quick tour of some features of the Magma computer algebra system. These will include: modular forms, algebraic geometry (sheaf cohomology and Groebner bases), computing with L-functions, machinery for function fields, lattices, and some group/representation theory. No experience with Magma will be assumed.

### March 2, 2010

Stationary phases and Spherical averages
Po-Lam Yung
12:30–1:30 @ Fine 314

In this talk we will give an expository account of the following theorem of Stein about spherical averages, which asserts that if f is a function in Lp on Rn, with n≥3 and p>n/(n-1), then for almost every x in Rn, the average of f over a sphere of radius r centered at x is well-defined, and converges to f(x) as r tends to 0. Along the way we will see some beautiful ideas in harmonic analysis, and their connections to other subjects.

### February 23, 2010

Infinite Ergodic Theory and Continued Fractions
Francesco Cellarosi
12:30–1:30 @ Fine 314

I will compare (classical) Ergodic Theory and Infinite Ergodic Theory, i.e. when the space has infinite measure. In particular I will describe how to modify the Birkhoff Ergodic Theorem in the infinite setting. As examples, I will discuss the (classical) Euclidean continued fractions and the (less classical) continued fractions with even partial quotients. Time permitting, I will show the connection of such continued fraction expansions with theta sums.

### December 3, 2009

Bigger is better
Kevin Hughes
12:30–1:30 @ Fine 314

Originally, Hardy and Littlewood developed their "circle method" to study Waring's problem on the representation of numbers as the sums of k^th-powers. In the circle method, one decomposes the circle into "major" and "minor" arcs. Some rough estimates on the minor arcs give a power saving, and the work is then to study the major arcs. The guiding principle is "Bigger is better", i.e. the best estimates arise from making the major arcs as large as possible. Recently, the circle method has been applied to discrete analogues in harmonic analysis. I will discuss the classical circle method, the spherical maximal function and higher degree analogues, and then discuss how these combine to give a discrete spherical maximal function.

### Nov 19, 2009

Arie Israel
12:30–1:30 @ Fine 314

A question that is often asked in Extension Theory is the following: Given $E \subset \mathbb{R}^n$, and $f:E \rightarrow \mathbb{R}$. Is it possible to extend $f$ to a function lying in the space $X(\mathbb{R}^n)$ ?. This question has been answered in the case when $X$ is the space $C^m$ of functions continuously differentiable through order $m$. I will prove the relevant theorem in the special case $m=2$ for finite sets $E$, and time permitting discuss some other interesting variants.

### November 12, 2009

The Reflector Antenna Problem, and its Connection to Mass Transport
June Kitagawa
12:30–1:30 @ Fine 314

The reflector antenna problem is the problem of constructing a reflective surface which directs a specified energy distribution on the sphere to another specified energy distribution on the so-called far-field sphere. I will discuss some of the basic analytic and geometric facts of this problem, and a connection that has been discovered in recent years to Monge's optimal mass transportation problem.

### October 29, 2009

The Congruence Subgroup Problem for SL(n,Z)
Daniel Shenfeld
12:30–1:30 @ Fine 314

The group SL(n, k) (k a field) is simple modulo its center, but for a ring A, SL(n, A) is not: the kernels of the maps SL(n,A)-> SL(n,A/I) (I

### October 22, 2009

The Curve enumeration from matrix integrals
Michael McBreen
12:30–1:30 @ Fine 314

I will explain the connection between matrix integrals and Riemann surface counting, which has caused a big stir in mathematics over the last few decades.

### October 15, 2009

Harmonic measure
Phil Sosoe
12:30–1:30 @ Fine 314

Harmonic measure is a measure on the boundary of domains in the complex plane whose associated integral generates the solution of the Dirichlet problem. I will explain Nevanlinna's "principle of harmonic measure", and give a number of examples of its use. Time allowing, I will discuss recent geometric results about harmonic measure, and its relation to other conformal invariants such as Beurling's extremal length. The talk will be completely elementary!

### October 8, 2009

"> Strong multiplicity one and l-adic Galois representations
Stefan Patrikis
12:30–1:30 @ Fine 314

The strong multiplicity one theorem and its refinements amount to a local-to-global principle in the theory of automorphic representations of GL(N). I will discuss a Galois-theoretic analogue with a surprisingly elementary proof, along with some related questions about the images of l-adic Galois representations.

### October 1, 2009

"> ODEs and solutions from actions of Lie Groups
12:30–1:30 @ Fine 314

We consider ODE's arised from the action of Lie groups on manifolds and discuss how solvability of the corresponding Lie algebra helps to find explicit solutions.

### September 24, 2009

Nonlinear Waves
Jonathan Luk
12:30–1:30 @ Fine 314

Given a nonlinear wave equation whose nonlinearity contains derivatives of the unknown function: $-\partial_t^2\phi+\sum_{i=1}^ 3 \partial_{x_i}^2\phi=N(\partial\phi)$ in $3+1$ dimensions, what can we say about the global existence of solutions? What if the initial conditions are small and compactly supported? If time permits, I will discuss the case of a wave map.

### May 14, 2009

Deligne-Lusztig
Andrew Snowden
12:30–1:30 @ Fine 314

Let G be a finite group of Lie type, for instance GL(n, Z/pZ). It is not difficult to find representations of G on characteristic p vector spaces -- for instance, GL(n, Z/pZ) acts on $(Z/pZ)^n$ in the obvious way -- but it is not so clear how to construct the characteristic zero representations of G. The point of Deligne-Lusztig theory is to provide a uniform construction of these representations. The main bridge from characteristic p (where G lives) to characteristic zero (where we want to be) is provided by etale cohomology. I will assume some familiarity with complex Lie groups but not much from algebraic geometry.

### April 30, 2009

Some Relationships Between Moduli Spaces of Curves
Wei Ho
12:30–1:30 @ Fine 314

We present several constructions relating various moduli spaces of decorated curves. To motivate the study of moduli spaces, we will describe examples with applications to algebraic geometry, number theory, topology, and even analysis. We will then use representation theory to find explicit connections between certain moduli spaces of curves, and we will discuss constructions of Recillas and Vakil that provide other perspectives on this link.

### April 23, 2009

TBA
Kevin Hughes
12:30–1:30 @ Fine 314

What's the smallest area a subset of the plane containing a unit line segment in every direction can have? Besicovitch showed that you can get 0 measure and Fefferman used the existence of this set to provide a solution to the ball multiplier problem. Kakeya sets have been important in understanding other phenomena like the Bochner-Riesz conjecture as the Hausdorff/Minkowski dimension of Kakeya sets relate to the two. Building on Fefferman's work, Bourgain improved our understanding introducing an ingenious bush argument . I will use the analytic story to motivate the corresponding problems for varieties over finite fields and recent work done there.

### April 16, 2009

Equivariant Cohomology
Iman Setayesh
12:30–1:30 @ Fine 314

A cohomology theory is a way of associating a complex of abelian groups to a topological space. Study of such complexes gives geometric information about the structure of the space. Moreover if we have an action of a suitable group on a space we can associate another such complex to that space which carries more information. I will sketch main ideas of this construction and provide some examples.

### April 9, 2009

Mathai-Quillen's Thom form and Atiyah-Hirzebruch's Riemann-Roch theorem
Guangbo Xu
12:30–1:30 @ Fine 314

After Hirzebruch's generalization of the classical Riemann-Roch formula, Grothendieck extended this result to a relative version. Then Atiyah and Hirzebruch gave a "differentiable analogue" of Grothendieck's theorem, which is called Atiyah-Hirzebruch's Riemann-Roch theorem. In spite of the original topological proof, I will present another one, in a more differential-geometric flavor, using Mathai-Quillen's construction of the Thom form.

### April 2, 2009

Poisson summation formula
Ali Altug
12:30–1:30 @ Fine 314

I will first talk about the classical Poisson summation formula and then about a vast generalization of it, namely the trace formula.

### March 26, 2009

Sum-product estimates via combinatorial geometry
Po-Shen Loh
12:30–1:30 @ Fine 314

Every two-dimensional drawing of any graph with V vertices and E ≥ 4V edges necessarily has at least E3/V2 pairs of crossing edges. Also, for every set A of real numbers, one of A+A (the set of all pairwise sums of elements of A) or A·A (the set of all pairwise products) has size at least |A|5/4. What could these two theorems possibly have in common, besides the fact that Endre Szemerédi co-authored both? Surprisingly, quite a lot. We will see the proof of the first result, followed by a series of fascinating consequences which culminate in the second result. Of course, the Probablistic Method will make a crucial appearance.

### March 12, 2009

Morse theory
Michael McBreen
12:30–1:30 @ Fine 314

Morse theory gives the cell structure of a manifold in terms of the critical points of a 'random' real-valued function on this manifold. Besides being clever and pleasing to the eye, it has given us Bott periodicity, counts of geodesics, periodic orbits of dynamical systems, Heegard Floer homology, the foundations of Mirror Symmetry and many many more riches. I will explain the construction in detail, then sketch the applications.

### March 5, 2009

Lie Groups: Decomposition and Exponentiation
Kevin Wilson
12:30–1:30 @ Fine 314

A manifold with a smooth group structure is called a Lie group. Most of the information about Lie groups is captured by the tangent space at the identity and its Poisson bracket. The map relating these two structures is the exponential map (which in the compact case is the same as the geodesic exponential map). In this talk I'll start from first principles, give a brief overview of some decompositions of Lie groups based on this algebraic/analytic interplay, and use these facts to prove some interesting theorems about the image of $\exp$, viz., 1) If $\exp$ is a homeomorphism, then the group is solvable; and 2) while $\exp$ need not be surjective, the connected component of the identity is the square of the image of $\exp$.

### February 26, 2009

Models and Fields: A delicate Passage to Characteristic p
Jacob Tsimerman
12:30–1:30 @ Fine 314

If a polynomial map from $C^n$ to itself is injective, then it is also surjective. The first proof of this elegant result actually used a passage to characteristic p. I'll introduce the relevant notions from model theory and explain the proof, which will naturally yield the following much stronger result: Every statement in the first order logic of rings is true in the algebraic closure of $F_p$ for almost all p iff it is true in C.

### February 19, 2009

Bend and Break
Bhargav Bhatt
12:30–1:30 @ Fine 314

Our goal is to explain the following theorem due to Mori: Given a compact complex manifold X whose tangent bundle has lots of (holomorphic) determinantal sections (a Fano manifold), any pair of points on it lie in the image of a holomorphic map from the Riemann sphere $P^1$. Despite being a complex geometric statement, the only known proof of this result is by reduction to characteristic p. In this talk, we'll discuss what Fano manifolds are, explain the techniques that go into Mori's proof (reducing mod p, deformation theory of maps from curves), and the proof itself.

### February 12, 2009

Tamagawa numbers
Arul Shankar
12:30–1:30 @ Fine 314

I'll explain what Tamagawa numbers are, perhaps motivate their importance a little and explain how to compute them for some groups.

### February 5, 2009

The Nielsen Realization Problem
Will Cavendish
12:30–1:30 @ Fine 314

The purpose of this talk will be to describe some problems in topology whose solutions rely on geometric techniques. I will introduce the mapping class group of a space X, which is the group of homeomorphisms of X up to isotopy. I will then outline Steve Kerckhoff's solution to the Nielsen realization problem, which shows that any finite subgroup of $Mod_g$, the mapping class group of a genus g surface, can be realized as the automorphism group of a Riemann surface of genus g. The proof relies heavily on an understanding of the geometry of Teichm\"uller space $T_g$, and ties in to many natural geometric questions about this space which I hope to describe.

### December 11, 2008

Ratner's Theorem
12:30–1:30 @ Fine 314

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### December 4, 2008

Fibered Knots
Margaret Doig
12:30–1:30 @ Fine 314

A fibered knot is a knot whose complement can be filled "nicely" by copies of an oriented surface bounded by the disk, i.e., is filled by $S^1$ copies of $D^2$ (in fact, this fibration is globally trivial: $S^3-K \conj S^1 \times D^2$). By the time the pizza is all eaten, we should even be able to understand Milnor's construction of a fibration of the $(p,q)$ torus knot by surfaces of genus $(p-1)(q-1)/2$. You may care about fibered knots if you have ever been or will ever be interested in any of the following: \begin{itemize} \item hyperbolic structures \item algebraic knots and links \item unbranched cyclic covers \item open book decompositions \end{itemize}

### November 20, 2008

Stefan Patrikis
12:30–1:30 @ Fine 314

I will introduce(focusing on the case of elliptic curves)two essential theorems of arithmetic geometry that bind the geometry of an algebraic variety over a local field(or number field) to its arithmetic, Galois-theoretic properties. The l-adic mondromy theorem is in fact entirely elementary (one need not even know what 'monodromy' means !) but will motivate the subtler p-adic theorem. Together these results will help us make sense of Fontaine-Mazur conjecture,one of the fundamental open problems in arithmetic geometry.

### November 13, 2008

Algebraic Curves and their Realizations
Yaim Cooper
12:30–1:30 @ Fine 314

Given an abstract curve, how can it be embedded in P^n? What should we mean by "how"? Come learn about one of the first results in this rich field.Along the way, we will use some of the basic machinary used to study curves, and will try to develop intuition for how this machinary works(for example Riemann-Roch).

### November 6, 2008

Embedded Surfaces in 4-Manifolds
Josh Green
12:30–1:30 @ Fine 314

Given a 2-dimensional homology class in a closed 4-manifold, you can try to represent it by an embedded, closed, orientable surface. What is the minimum genus of such a surface? Come find out.

### October 23, 2008

Differential Galois Theory
Vivek Shende
12:30–1:30 @ Fine 314

Algebraic groups, differential equations, and Galois theory: who isn't scared of at least one of those? It being the last GSS before Halloween, I'll do my best to mention all three.The ostensible purpose of doing so will be to explain why certain indefinite integrals cannot be written in terms of elementary functions. No knowledgs of elementary functions will be presupposed.

### October 16, 2008

Ergodic Theory
Sam Ruth
12:30–1:30 @ Fine 314

I will be discussing two basic invariants of Ergodic Theory, entropy and ergodicity, through the problems they were invented to solve. This talk will be heavy on examples and low on rigour.

### October 9, 2008

GRH and polynomial-time primality testing
Jacob Tsimerman
12:30–1:30 @ Fine 314

For a long time, a famous open problem was to figure out wether a number was prime quickly (in polynomial time). It's interesting to see how under the generalized Riemann hypothesis, the problem becomes completely straightforward.I will introduce the relevant concept and present the simple proof. During the second half, I will present a provably polynomial time test, without reliance on GRH. If time permits, I will say some more about L-functions and their application to computing.

### October 2, 2008

The Composition Problem in Measure Spaces
Philip J Isett
12:30–1:30 @ Fine 314

Let X,Y be measurable spaces and \eta : X \to Y be a measurable function. Under what conditions on \eta is the composition with f : Y \to C a well defined operation when f is only specified almost everywhere? Does composition with \eta induce a map between L^p spaces ? I will show how one generally would answer these questions , give an algebraic prespective on the problem(and on measure spaces in general), and give a complete solution when the map\eta is multipilication on the p-adic integers.

### September 25, 2008

Grushko's Theorem
Sucharit Sarkar
12:30–1:30 @ Fine 314

Grushko's Theorem states that rank(G*H)=rank(G)+rank(H) where rank is the minimum number of generators for a group, and "*" denotes the free product. We will present Stallings' (topological) proof of Grushko's Theorem.

### September 18, 2008

Random Walks on Groups
Simon Marshall
12:30–1:30 @ Fine 314

I will talk about the large scale structure of groups,presenting some results on word growth and random walks. I will also describe the link between random walks and spectral theory, and explain an elegant trick from complex analysis which can be used to give exact answers to questions about walks on free products of groups.

### May 1, 2008

The Six-Sphere
Yanir Rubinstein
12:30–1:30 @ Fine 214

Kirchhoff's problem (1947) asks whether $S^6$ is a complex manifold. We will review basic notions in complex and almost-complex geometry and discuss a bit of what is known regarding this problem, making contact with the octonions, $G_2$, Lie brackets, and curvature.

### April 24, 2008

Secant Varieties and Applications
Wei Ho
12:30–1:30 @ Fine 214

The notion of secant varieties in algebraic geometry is classical, but not much is known about these objects, even for many simple cases. Surprisingly, it is possible to translate certain questions in fields as disparate as complexity theory, statistics, and biology into straightforward (but often unsolved) problems about secant varieties. I will introduce secant varieties and present some examples of applications to other fields and to algebraic geometry itself. No knowledge of algebraic geometry, or of these other fields, will be assumed.

### April 17, 2008

Matriod Theory
Melody Chan
12:30–1:30 @ Fine 214

Every maximal acyclic subgraph of a given graph has the same number of edges. Every maximal independent subset of a given set of vectors in $\mathbb{R}^n$ has the same size. The common generalization of these lies in matroid theory, which, roughly speaking, is an abstraction of the notion of linear independence. Despite the fact that matroids "forget" quite a lot of structure, they are immensely useful to graph theorists. In this talk, I'll give an introduction to the theory of matroids and point to some of its highlights, keeping graph theory in mind as our primary motivation and source of examples.

### April 10, 2008

Undecidability and Hilbert's 10'th Problem
Jacob Tsimerman
12:30–1:30 @ Fine 214

Hilbert's famous 10'th problem asks whether there is a general algorithm to decide whether a polynomial in many variables has a solution. I will explain how Robinson, Davis, Putnam, and Matijasevic answered the question in the negative by proving a much more interesting theorem: that any listable set can be listed using a polynomial equation. I'll also give some awesome mind-blowing consequences of this theorem, such as prime-producing polynomials! No background in anything will be required.

### April 3, 2008

The Moment Map and Delzant Polytopes
12:30–1:30 @ Fine 214

In symplectic geometry, the moment map (or momentum map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic quotients, symplectic cuts and sums. During the lecture I will introduce this concept and I will show you how to construct symplectic manifolds having some specific image for the moment map. Also I'll say something about the relation between combinatorial data in the image of the moment map and geometric data on the manifold

### March 27, 2008

Counting Circles and Other Things
Elena Fuchs
12:30–1:30 @ Fine 214

Take a circle of radius one and inscribe in it two circles that are tangent to each other. Now add another circle into the original one so that it's tangent to all three. If we repeat this process over and over, we get an old picture known as the Apollonian circle packing. What radii will we get from one such packing? There are many ways to approach this simple question, and I'll tell you about some in this talk.

### March 13, 2008

Counting Curves
Iman Setayesh
12:30–1:30 @ Fine 214

A basic question in enumerative geometry in its most general form is to find the number of geometric objects satisfying a given collection of properties. In this talk we shall first introduce some tools from geometry and then explain how one can use them to solve a problem of this type. The talk will be elementary, and requires no special background (even in "counting").

### March 6, 2008

Prime Splitting Laws
Stefan Patrikis
12:30–1:30 @ Fine 214

Algebraic number theory seeks to understand and somehow classify all Galois extensions of a number field K. One perspective on this basic classification problem emerges from the Cebotarev density theorem, which implies that such extensions L/K are determined by the primes of K that split completely in L. When L/K is abelian, associating such a 'splitting law' is classical, and after defining the relevant terms I will illustrate this theory with some representative examples. The corresponding question for non-abelian extensions remains a great mystery, however, but I hope to use this to motivate a very concrete introduction to some of the modern machinery of number theory and the Langlands program. No knowledge of number theory will be assumed.

### February 28, 2008

Hyperbolic 3-Manifolds and Arithmetic
Simon Marshall
12:30–1:30 @ Fine 214

Thurston's geometrisation program tells us that we can understand all closed 3-manifolds if we can understand those with a finite volume hyperbolic structure. This is not easy, however, and it is not even clear at the outset that there exist more than a few such manifolds. I will decribe number theoretic methods which construct a rich class of finite volume hyperbolic 3-manifolds, and beautiful converse results which show that all such manifolds have useful arithmetic information associated with them.

### February 21, 2008

Differential Equations and Arithmetic
Ali Altug
12:30–1:30 @ Fine 214

One of the questions in number theory is to represent integers by quadratic forms. To be more precise, the question asks for a characterization of the integers which can be represented by a given form. In this talk we will deal with representing integers as sums of two squares and if time permits as sums of four squares. The talk will be completely elementary. (If I were to deliver the talk in the 19th century it wouldn't be called so!!)

### February 14, 2008

An Outline of the h-Cobordism Theorem
Sucharit Sarkar
12:30–1:30 @ Fine 214

In the first half of the talk, we shall go through the definitions of manifold, cobordism, h-cobordism, Morse functions and gradient-like flows, and demonstrate many of their properties. In the second half of the talk we shall give an outline of the proof. In the process, we shall prove lemmas about commutation and cancellation of critical points, and see the Whitney trick in action. Time permitting, we shall show how the h-cobordism theorem proves the topological Poincare conjecture in high dimensions.

### February 7, 2008

The Probabilistic Method and A Classical Result of Erdos
12:30–1:30 @ Fine 214

The basic idea behind the 'probabilistic method' in combinatorics is as follows: in order to prove the existence of an object with a certain property, one defines an appropriate (non-empty) probability space and shows that a randomly chosen object from this space has the desired property with positive probability. This simple idea (with a few variations) has been applied to prove many beautiful and sometimes surprising results in combinatorics. I will mostly focus on just one such classical result, namely the theorem of Erdos showing that there exist graphs with arbitrarily large girth and arbitrarily large chromatic number. No previous knowledge of graph theory will be assumed.

### November 29, 2007

The ABC Conjecture and Other Things I Can't Prove
Jacob Tsimerman
12:30–1:30 @ Fine 214

The ABC conjecture was made by Oesterle and Masser in 1985. While simple to state, it is likely difficult to prove as it would provide a unified method of solving a lot of very difficult problems. I'll give a small sampling of its power by showing some neat corollaries, and then try to explore some of its analogues and relations with geometry. Maybe we'll even get to do some analytic number theory while we're at it. Who knows? I do.

### November 15, 2007

Ergodic Theory and Equidistribution
Simon Marshall
12:30–1:30 @ Fine 214

Ergodic theory has many uses in the proof of equidistribution results, from buttering your toast in the morning to fighting baldness. However, some situations require a more thorough application of ergodic methods. This week, I will describe some applications in Numbers theory! I will warm up by talking about measures on the circle, before hopefully describing Margulis' proof of the Oppenheim conjecture and recent work of Lindenstrauss on Littlewood's conjecture.

### November 8, 2007

The Borel-Weil-Bott Theorem
Vivek Shende
12:30–1:30 @ Fine 214

What are all the irreducible representations of a complex semisimple Lie group G? One can use the one-dimensional representations of a Borel subgroup B to produce line bundles on the flag manifold G/B; some of these bundles have (a finite dimensional vector space of) holomorphic global sections. These vector spaces carry irreducible G-actions, and in fact account for all the irreducible representations. Even the bundles without global sections have irreducible representations hiding somewhere in their higher cohomology. That may sound like quite a mouthful, but it's not so bad for, say, SL(2): G/B is just the Riemann sphere, whose line bundles can be written down quite explicitly. I'll explain what's going on in this case, then discuss SL(3), and then say a few more words about things in general.

### October 25, 2007

Bhargav Bhatt
12:30–1:30 @ Fine 214

One can associate a sphere to a complex vector space by identifying the boundary of its unit ball with a point. Globalising, for a topological space X, one can define a natural map J:K(X) -> SF(X) where the complex K-theory group K(X) classifies complex vector bundles over X, while the cohomotopy group SF(X) classifies sphere fibrations over X. The Adams conjecture describes the kernel of this homomorphism in terms of certain explicit linear algebraic operations on K(X). In particular, since sphere fibrations are closely related to homotopy theory, the Adams conjecture is usually understood as describing the part of homotopy theory (hard) that comes from complex K-theory (easy). Concretely, it describes the image of a map from a known sequence of groups, the complex K-theory groups of a point, to the stable homotopy groups of spheres (which we don't know in general). We will introduce the principle players in the Adams conjecture, and give an overview of Quillen's mod-p-algebraic-geometry-based "proof" of it.

### October 18, 2007

In P, or not in P?
Sucharit Sarkar
12:30–1:30 @ Fine 214

I shall define P, NP, L, SL and all those other fancy complexity classes. I shall show why certain problems are NP-complete and mention that L=SL. No prior knowledge is required, the speaker having none himself. There will be pizza. There might be games.

### October 11, 2007

Sidon Sets and Digit Reversal
Boris Bukh
12:30–1:30 @ Fine 214

When knowledge of addition, multiplication and even of division does not get us where we want, it is time to turn to the operations of lower repute. In this talk we will see how nice-looking and useless numbers can be turned into ugly-looking and useful numbers by reversing their digits. We will encounter some technical difficulties on the way, but the happy ending is guaranteed!

### October 4, 2007

Getting to Know Stacks
Melanie Wood
12:30–1:30 @ Fine 110

A stack is a type of space that comes up naturally in several contexts, but is often hidden behind a lot of technical defi

### October 18, 2007

In P, or not in P?
Sucharit Sarkar
12:30–1:30 @ Fine 214

I shall define P, NP, L, SL and all those other fancy complexity classes. I shall show why certain problems are NP-complete and mention that L=SL. No prior knowledge is required, the speaker having none himself. There will be pizza. There might be games.

nitions. We'll talk about where and how you might encounter stacks, and see how to recognize "stacky" behavior. We'll see some examples of stacks that are friendly, and give advice for how to be less afraid when you meet a stack that isn't.

### September 27, 2007

4-Manifolds
Josh Greene
12:30–1:30 @ Fine 214

I will survey some of the amazing properties of $4$-manifolds in this action-packed lecture. Sample topics may include: Michael Freedman's classification of simply-connected closed topological $4$-manifolds in terms of unimodular symmetric bilinear forms; Simon Donaldson's diagonalizability theorem for the intersection pairing of a smooth closed definite $4$-manifold; exotic smooth structures on $4$-manifolds; and the genus of embedded surfaces in $\mathbb{C} P^2$. A first pass at algebraic topology should be enough to understand the talk.

### September 20, 2007

Artin on Real Closed Fields
Balin Fleming
12:30–1:30 @ Fine 214

If the algebraic closure $\overline{K}$ of a field $K$ is a finite extension of $K$, what kind of extension can it be? We know for example that $\mathbb{C} = \mathbb{R}(i)$ is a degree two extension formed by adjoining a square root of -1 to $\mathbb{R}$. In 1927 Artin and Schreier showed that this is the only possible behaviour in the following sense. We call $K$ (formally) {\it real} if -1 is not a sum of squares of elements of $K$, and {\it real closed} if $K$ is real and no algebraic extension of $K$ is real. If $\Omega$ is any algebraically closed field which is a nontrivial finite extension of a subfield $K$ then $K$ is real closed and $\Omega = K(i)$, where $i$ is a root of $x^2 + 1$. (In particular, $\Omega$ therefore has characteristic zero). This we'll show.

### May 22, 2007

Counting Lattice Points
Po Lam Yung
12:30–1:30 @ Fine 110

How many integer points are there in a ball of radius $\lambda$ in $\mathbb{R}^n$? For geometers, this is the number of eigenvalues of the standard Laplacian on the flat torus that are smaller than $\lambda$, and for number theorists, this is basically $\sum_{0 \leq m \leq \lambda} r_n(m)$ if $r_n(m)$ is the number of ways to write the integer $m$ as the sum of $n$ squares. Along the way we will come across some simple and yet beautiful applications of Fourier analysis. The talk will be accessible to a general audience.

### May 8, 2007

Picard Groups of Moduli Problems
Bhargav Bhatt
12:30–1:30 @ Fine 110

I'll ostensibly calculate the Picard group (isomorphism classes of holomorphic line bundles under tensor product) of the moduli (classifying) space of elliptic curves. The main goal is to demonstrate the utility of the functorial approach to geometry in making calculations via a concrete example.

### May 1, 2007

Interesting Characters
Erik Carlsson
1:00–2:00 @ Fine 110

In representation theory of simple lie algebras, characters are polynomial functions which determine a representation. The polynomials which can come from characters of a lie algebra are just those which are symmetric under that lie algebra's Weyl group. I'll write down an infinite lie algebra, \hat{sl_2}(C), and show that Weyl group symmetry forces its characters to satisfy the more interesting properties of a theta function. Then I'll write down an explicit representation, and verify those properties directly.

### April 24, 2007

Probabilistic Methods in Ramsey Theory
Jacob Fox
12:30–1:30 @ Fine 110

Ramsey theory consists of a large body of deep results in mathematics that deal with the existence of unavoidable regularity in large structures. The Probabilistic Method is a powerful tool in tackling many problems in this area. In fact, many of the strongest results in Ramsey theory were obtained using probabilistic arguments. In this talk, I will describe several classical results and show how a simple yet powerful probabilistic technique was used recently to make progress on several longstanding problems in Ramsey theory.

### April 17, 2007

Knot Homology Theories
Josh Greene
12:30–1:30 @ Fine 110

We will begin by talking about some basic properties of knots and the use of polynomial invariants (like the Jones and Alexander polynomials) in understanding them. Then we will enter the 21st century and discuss what knot homology theories are, how they "categorify" the polynomial invariants, and what news they bring about these basic properties. Proofs will be assigned as exercises, and I will probably not talk about everything I had planned to.

### April 10, 2007

Sieves in Number Theory
Gihan Marasingha (Oxford University)
12:30–1:30 @ Fine 110

The modern theory of sieves originated in attempts by Brun and Selberg to gain a handle on problems such as the Goldbach conjecture and the twin primes conjecture. I'll introduce some of Selberg's ideas and discuss the parity problem, a fundamental obstacle in traditional sieve methods which we are now beginning to overcome.

### April 3, 2007

Lillian Pierce
12:30–1:30 @ Fine 110

A 1924 theorem of Banach and Tarski states that given A and B any two bounded subsets of three dimensional Euclidean space, each having non- empty interior, then each can be expressed as a disjoint finite union of sets $\{A_j\}$ and $\{B_j\}$, $j=1,...,n$, such that there exist rigid motions $r_j$ with $r_j(A_j) = B_j$ for each $j=1,...,n$. That this is a paradox can be seen in the more colloquial phrasing of this theorem: you can take an orange, cut it into finitely many pieces, and reassemble it into two oranges the same size as the original one, or into a ball the size of the Jupiter, or into a (very Princetonian) zebra. We will give an elementary proof of this intriguing theorem, meeting the Hausdorff Paradox along the way.

### March 13, 2007

What Everyone Should Know About Topology (but I had to look up)
Margaret Doig
12:30–1:30 @ Fine 110

An assortment of short, interesting facts about topology.

### March 6, 2007

Constructing Quotients in Algebraic Geometry
Wei Ho
12:30–1:30 @ Fine 110

Given a group acting on a variety, how do we construct a "nice" quotient? This sort of question often arises when trying to form moduli spaces, for example. Geometric invariant theory, developed by Mumford, gives a concrete and geometric method for describing these quotients. I will sketch the main ideas in GIT and provide many examples.

### February 27, 2007

Kakeya Problem or Why Taming Hedgehogs Is So Much Fun
Boris Bukh
12:30–1:30 @ Fine 110

Kakeya problem asks for the smallest set in $R^d$ which contains a unit line segment in every direction. Amazingly there are sets of measure zero with this property. I will show the construction, and explain why every such set in the plane is two-dimensional.

### February 20, 2007

Representations of Nilpotent Groups
Brian Street
12:30–1:30 @ Fine 110

In the 60s, Kirillov gave a beautiful characterization of the irreducible unitary representations of nilpotent Lie groups. I will discuss this characterization and mention some of its applications to harmonic analysis.

### February 6, 2007

Harmonic Analysis and Group Representations
Simon Marshall
12:30–1:30 @ Fine 110

I will talk about some assorted topics related to representation theory and Lie groups. Hopefully this will include how to do Fourier analysis on symmetric spaces such as the upper half plane, and how group representations give you harmonic functions and vice versa. Some of Harish-Chandra's work on the unitary representations of noncompact Lie groups may appear as well, in particular to describe how these representations, despite being infinite dimensional, may still have a trace defined for them.

### January 30, 2007

Hanna Neumann Theorem
Sucharit Sarkar
12:30–1:30 @ Fine 214

Abstract: Let $H$,$K$ and $H\cap K$ be subgroups of a free group, with ranks $m$, $n$ and $r$ respectively. Hanna Neumann showed that if $mn\neq 0$, then $r-1\leq 2(m-1)(n-1)$ (which in particular proves that, $m$ and $n$ are finite implies $r$ is finite), and conjectured that the factor $2$ can be removed. We shall present her original (topological) proof.

### January 23, 2007

Representations of Quivers
Andrew Snowden
12:30–1:30 @ Fine 214

Homological algebra is usually presented as an abstract subject -- in a typical treatment, one does not even get a feeling for the structure of its most basic objects, abelian categories. However, one can give a very concrete way to picture (at least some) abelian categories, using objects called quivers. Various constructions such as derived functors become mechanical, to the point that one can do interesting examples by hand. I will give an introduction as to how all this works.

### December 12, 2006

Good versus Evil - a not so epic battle
Blair D. Sullivan
12:30–1:30 @ Fine 214

A gentle introduction to Conway's epic battle between celestial beings and their demonic counterparts. I plan to prove that good will triumph, even if severely handicapped, as long as it moves quickly (say, with speed at least 5). The methods used are based upon recent work of Mathe and Bowditch, and the talk will conclude with open problems including knights, time-bombs, and higher dimensions.

### December 5, 2006

All about the Riemann Hypothesis (or at least some)
Elena Fuchs
12:30–1:30 @ Fine 214

Motivated by the previous talk about the Riemann-zeta function, I will talk about the Riemann hypothesis. I'll discuss several implications of the hypothesis, particularly the distribution of prime numbers, as well as some of its history -- don't worry, I won't talk too much about bogus proofs and only a little about its appearance in pop culture.

### November 28, 2006

Simon Marshall
12:30–1:30 @ Fine 214

In his 1950 Princeton thesis, John Tate gave a proof of the functional equations of a class of L-functions using Fourier analysis on rings called Adeles, which can be associated to any number field. I will describe the Adele ring of the rational numbers and the various things you can do on it, and apply Tate's methods to prove the functional equation for the zeta function.

### November 21, 2006

Intersection Patterns of Geometric Objects
Jacob Fox
12:30–1:30 @ Fine 214

Suppose we have a collection of n Jordan arcs in the plane. How large a subcollection can we find such that either every pair of arcs in the subcollection intersect or every pair of arcs in the subcollection are disjoint? The qualitative answer is, "quite large." The problem is closely related to a famous conjecture of Erdos and Hajnal in graph Ramsey theory. I will survey related results and conjectures concerning intersection patterns of geometric objects in the plane, general surfaces, and in higher dimensions. We will discuss how to employ algebraic, combinatorial, geometric, and topological methods to tackle these problems. The talk will be accessible to a general audience.

### November 14, 2006

Random Phenomena in Number Theory
Melanie Matchett Wood
12:30–1:30 @ Fine 214

Motivated by Dirichlet's theorem that there are just as many primes congruent to 1 mod 3 as to 2 mod 3, we'll explore some of the possible senses of "just as many" (including some for which the theorem is surprisingly false!). This raises the issues of how we can rigorously encapsulate the idea that some determinate mathematical phenomenon can be "random" (such as the reductions of the primes mod 3), and what kind of random behavior we expect from "natural" mathematical phenomena. As an example, we'll talk about the Cohen-Lenstra-Martinet heuristics that predict a certain kind of "random" behavior of the class groups of number fields.

### November 7, 2006

The Development of Morse Theory (or, On the Morse-Thom-Bott-Smale-Milnor-Witten Complex)
Josh Greene
12:30–1:30 @ Fine 214

Morse Theory is a foundational tool in topology and has played a key role in some of the field's most striking results, such as in Bott's proof of his periodicity theorem, Milnor's construction of exotic 7-spheres, and in Smale's work on the generalized Poincar\'e conjecture. The plan of the talk is to give a broad overview of this subject with an emphasis on these applications. Time permitting I will give an indication of the role Morse theory has played in the development of Floer homology, which may be the title of a GSS talk next semester. A portion of the talk will be given by Marston Morse himself.

### October 24, 2006

The Jones Polynomial (and other cool facts about Knot Theory)
Margaret Doig
12:30–1:30 @ Fine 214

We'll be talking about knots and knot invariants in the hopes of understanding a little bit about the mysterious but useful Jones polynomial, which allows us to prove, among other things, that alternating knots can be drawn nicely - one of those results that you really want to be true but which is so nice you don't expect to be able to prove it. (NB: There will be colored chalk and possible audience participation.)

### October 17, 2006

Hypersurfaces in Toric Varieties
Alan Stapledon (Institute for Advanced Study)
12:30–1:30 @ Fine 214

The complicated sounding title is really a cover. The true goal of the talk is to explain how to cheat on the algebraic geometry general exam. All the words of the title will be mentioned, probably even in the same sentence, but only to maintain the shroud of secrecy.

### October 10, 2006

Small Forbidden Configurations
Balin Fleming
12:30–1:30 @ Fine 214

We call a (0,1)-matrix A simple if it has no repeated columns. If F (the "forbidden configuration") is another (0,1)-matrix, we say that A has no configuration F if it has no submatrix which is a row and column permutation of F. Write forb(m, F) for the maximum number of columns of a simple matrix with m rows and no F. The problem is to find this number, exactly or (more usually) asymptotically. We'll look at some simple cases, talk about the basic bounds of Sauer and Fu\"redi (which imply in particular that forb(m, F) is at most polynomial in m), and describe other recent results and some results we hope might soon become recent.

### October 3, 2006

Lara Pudwell (Rutgers University)
12:30–1:30 @ Fine 214

I will tell you why 1089 is cool without any scary (insert most feared math field of choice) techniques whatsoever. This talk is perfect for anyone who has a short attention span mid-week as I'll be changing gears every 10-15 minutes. Along the way I'll teach you a fun game (to be translated as nifty trick to baffle all your non-mathematical friends), I'll explain an early paper of a famous mathematician whose work you've probably never read, and I'll tell you a bit about British experimental mathematics of the late 1990s. All thanks to 1089.

### September 26, 2006

How to paint the plane?
Boris Bukh
12:30–1:30 @ Fine 214

How large can a set in a plane be without containing points which are distance one or one zillion apart? For sufficiently large zillion, we know the answer. Surprisingly, the solution involves some Fourier analysis. I will explain the solution, talk about similar coloring questions as well as some other problems involving forbidden distances. No background in harmonic analysis is required for eating the pizza.

### April 25, 2006

From polygonal billiards to Teichmuller dynamics
Corinna Ulcigrai
12:30–1:30 @ Fine 224

A polygonal billiard is a dynamical system given by the motion of a free particle in a polygon, with elastic reflections at the sides. The study of their ergodic properties exploits a fruitful and beautiful connection with geodesic flows on flat surfaces and Teichmuller dynamics. In this talk we will first introduce billiards and describe the construction of Zemlyakov and Katok, which associates to a rational polygonal billiard (i.e. with commensurate angles) a flow on a flat surface. We will then try to explain how dynamical properties of a single billiard relate to questions about dynamics in the spaces of all such flat surfaces. In particular, will describe some results about minimality (Zemlyakov and Katok), unique ergodicity (Masur's criterium), and, time permitting, Veech dynamical dichotomy (Veech-Smillie-McMullen). The talk will be introductory and self-contained.

### April 18, 2006

Discriminant Varieties
Lanie Wood
12:30–1:30 @ Fine 224

Given two curves in the plane, for example, $ax^2y+bx^3y^8 +cx^7=0$ and $exy+fy^9+gx^7y^7=0$ (where $a,b,c,e,f,g$ are coefficients), when do they have a multiple common intersection? This is a condition on $a,b,c,e,f,g$, which defines some variety in 6-dimensional space. What does this variety look like? Another question: Given a continuously varying family of varieties, we might wish to understand which varieties in the family are singular. If, for example, the varieties are defined by the parameters $a, b$, and $c$, we ask for the condition on $a, b$, and $c$ for them to define a singular variety. This condition is a variety (called an $A$-discriminant) in the parameter space--what does it look like? Varieties which are $A$-discriminants (for some family of varieties) turn up in all sorts of places, and include all the classical discriminants and resultants, as well as the first example of this abstract. We can understand $A$-discriminants well in certain nice ("smooth") cases, but some in some natural and interesting cases (like the first one given above!) it seems hard to say much at all about the variety! I will present a simple and combinatorial conjecture for the degree of the variety in the first case given above.

### April 11, 2006

The Alexander Polynomial of a Knot
Yi Ni
12:30–1:30 @ Fine 224

The Alexander polynomial is a simple but powerful tool used to study knots. I will start with tricolorability", which is perhaps the simplest knot invariant, and then show how this invariant is generalized to a polynomial knot invariant: the Alexander polynomial. Several definitions of the Alexander polynomial will be given. Finally, I will discuss various generalizations of the Alexander polynomial.

### April 4, 2006

Random Matrices and Universality
Louis-Pierre Arguin
12:30–1:30 @ Fine 224

One of the aims of probability theory is to characterize probability measures that naturally emerge from well-defined classes of problems with randomness. The canonical example for such "universal" measures is the Gaussian distribution. In the last 15 years, other "universal" probability measures have arisen from the field of Random Matrix Theory (RMT) and appear to be as fundamental from a mathematical point of view. RMT was popularized in the 50's by Wigner, a physicist, who studied the spectrum of random Hermitian matrices. Since then, it has grown as a mathematically rich theory that produced striking results in a variety of fields including combinatorics and number theory. In this talk, I will leisurely introduce the basic features of a particular distribution of random matrices known as the Gaussian Unitary Ensemble (GUE). We will see that GUE features surprisingly emerge in the study of the zeros of the Riemann zeta function, in the distribution of random permutations, and in the chaotic bus system of Cuernevaca, Mexico (if time permits).

### March 28, 2006

A Counterexample to the Poincar\'e Conjecture?
Josh Greene
12:30–1:30 @ Fine 224

Around the turn of the twentieth century, Henri Poincar\'e laid the groundwork for algebraic topology; amongst his early contributions to the field were the introduction of singular homology theory and the fundamental group. Both of these invariants are strong enough to distinguish the homeomorphism types of closed $2$-manifolds, and led Poincar\'e in 1900 to phrase the claim that a closed, orientable $3$-manifold with the same homology groups as the sphere $S^3$ must be homeomorphic to $S^3$. As stated this claim is untrue, and we will discuss a number of approaches to a counterexample to it. The talk should be fun and accessible to anyone unintimidated by the abstract.

### March 14, 2006

When Adding Things Up Gets You Back Where You Started
Blair Dowling Sullivan
12:30–1:30 @ Fine 224

The Caccetta-Haggkvist conjecture is an open problem in graph theory relating minimum outdegree in directed graphs to the lengths of their directed cycles. This conjecture (from 1978) has spawned many related problems, almost none of which have been solved. I will talk a little bit about what's known and what some of the interesting variations are, and then give a neat proof of the conjecture for Cayley graphs using additive number theory.

### March 7, 2006

Very Even Numbers of Odd Partitions
Chris Mihelich
12:30–1:30 @ Fine 224

Many partition functions such as $p(n)$, the number of ways to write $n$ as an unordered sum of positive integers, and $p_2(n)$, the number of those sums in which the summands are odd, have generating functions that are (essentially) modular forms. Such partition functions therefore have remarkable divisibility properties such as Ramanujan's famous congruence $p(5n + 4) \equiv 0 \pmod{5}$, the establishment of which was one of the leading applications of modular forms before they were known to have something to do with Fermat's conjecture. A modern theorem of Serre can be used to show that every power of 2, however large, divides almost all values of the odd-partition function $p_2$. We shall prove this striking fact constructively using resolutely classical methods. No background in modular forms or partition arithmetic will be assumed.

### February 28, 2006

Toplogical Dynamics and van der Waerden's Theorem
Shimon Brooks
12:30–1:30 @ Fine 224

van der Waerden's Theorem states that, given a finite coloring of the integers, you can find (arbitrarily) long arithmetic progressions of one color. The original proof was combinatorial and very long (which, according to your taste, is either "good-but-not-great" or "bad-to-worse"). I'll explain a proof of Furstenberg and Weiss which is dynamical, "but" (or "and-even-better") quite short and elementary. This approach is also easier to generalize to stronger theorems, and time permitting I will hand-wave some of these ideas as well.

### February 21, 2006

Hypoellipticity
Brian Street
12:30–1:30 @ Fine 224

We call an operator $L$ acting on distributions ("generalized functions") $C^\infty$-{\it hypoelliptic} if whenever $Lu$ is $C^\infty$ on a neighborhood of a point $p$, then $u$ is also $C^\infty$ on a neighborhood of $p$. A classical example of a hypoelliptic operator is the laplacian on $\mathbb{R}^n$. I will present a brief introduction to the study of hypoellipticity via outlines of the proofs of two of the main theorems in the area. In the process, I will have occasion to mention such cross cultural phrases as "characteristic variety" and Kohn's "multiplier ideals"(though only in the context of analysis).

### February 14, 2006

Braid Groups
Margaret Doig
12:30–1:30 @ Fine 224

We'll define braid groups, both in a traditional geometric fashion first introduced by Artin and via a more general topological method, by which the $n$-strand braid group of a space $X$ is the fundamental group of its $n$-point configuration space, that is, the fundamental group of the space of all size $n$ subsets of $X$. An element of the braid group can also be thought of as a motion of $n$ distinct particles inside $X$. We'll also discuss some basic results on the braid groups of surfaces and see a few applications to knot theory, for instance, that the $n$-string braid group of the plane is the fundamental group of the complement of a link in $S^3$.

### February 7, 2006

Towards a degree two Atkin-Lehner theory.
Michael Volpato
12:30–1:30 @ Fine 224

After a brief review of classical Atkin-Lehner theory for genus one modular forms, we describe the theory of Siegel modular forms of genus two and a geometric analogue of some recent work of Ralf Schimdt on the decomposition of the space of Siegel cusp forms of genus two into new and old subspaces.

### December 14, 2005

Tell Me Your Values and I Will Tell You What You Are!
Boris Bukh
12:30–1:30 @ Fine 224

Suppose you know the values a rational function; can you tell what the rational function is? When the ground field is the the rational numbers, the answer is almost yes. I will prove this and other cheerful facts about images of polynomials and rational functions.

### December 7, 2005

The Prime Number Theorem on the Nose
Alex Kontorovich (Columbia University)
12:30–1:30 @ Fine 224

When Gauss was a little boy, he did many wonderful things. One of them was conjecture the asymptotic formula for the number of primes less than some fixed bound, which of course implies an asymptotic formula for the size of the n-th prime. About a century later, his conjecture was answered affirmatively and is now known as the Prime Number Theorem. We will describe (read: handwave) a recent result on how many primes are exactly equal to their average value (we call these "primes on the nose"). No prior knowledge is assumed.

### November 30, 2005

The Face Numbers of Simplicial Polytopes
Balin Fleming
12:30–1:30 @ Fine 224

A polytope $K$ is the convex hull of a finite collection of points in $R^n$. Let $f_i$ be the number of i-dimensional faces of $K$. We ask: what vectors $(f_0, ..., f_{n-1})$ arise as the face numbers of some $n$-dimensional $K$? For simplicial polytopes McMullen proposed a conjecture. This was proved by Billera and Lee (sufficiency) and Stanley (necessity). Stanley's proof uses deep results in algebraic geometry by tying the $f_i$ to the cohomology of an associated toric variety, but later McMullen gave another, more combinatorial proof. We'll discuss a variation on McMullen's argument, by passing from K to its normal fan, combinatorially constructing the cohomology as the ring of continuous conewise polynomial functions, and investigating the structure of this ring.

### November 16, 2005

Szemeredi's Regularity Lemma
Po-Shen Loh
12:30–1:30 @ Fine 224

A famous result of Roth states that for any fixed density $\delta > 0$, for sufficiently large $n$ every $S \subset \{1, 2, \ldots, n\}$ with $|S| \geq \delta n$ contains a 3-term arithmetic progression. While this theorem may seem to have little to do with graph theory, it turns out to have an elegant proof via Szemer\'{e}di's Regularity Lemma---a powerful tool in extremal combinatorics! The Lemma states (roughly speaking) that very large graphs exhibit characteristics found in random graphs, for which many results are known. In this talk, we will introduce the Regularity Lemma, catch a glimpse of the main ideas of its proof, and use it to prove Roth's Theorem. No prior knowledge of graph theory or combinatorics will be required.

### November 9, 2005

You call that a 7-sphere?!
Chris Mihelich
12:30–1:30 @ Fine 224

After a good course in multivariable calculus, we understand the notion of a smooth function on a 7-sphere. Or do we? It turns out that there are exactly 28 inequivalent structures of a differentiable manifold on the 7-dimensional sphere. This is so counterintuitive that when John Milnor first constructed a few of the 27 exotic 7-spheres, at first he thought he had found counterexamples to the Poincar\'e conjecture (roughly, that if the algebraic topologist's usual toolbox of algebraic invariants can't tell a space from a sphere, then it is indeed a sphere). In this talk I shall present Milnor's construction of exotic spheres after assembling a few necessary tools from algebraic topology. No prior knowledge of algebraic topology will be assumed.

### October 26, 2005

Regular Homotopy Classes of Singular Maps
Andras Juhasz
12:30–1:30 @ Fine 224

Two locally generic maps $f,g : M^n \to R^{2n-1}$ are called regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if $n$ is not equal to $3$ and $M^n$ is a closed n-manifold then the regular homotopy class of every locally generic map $f : M^n \to R^{2n-1}$ is completely determined by the number of its singular points provided that $f$ is singular (i.e., $f$ is not an immersion). In the case $n=3$ a geometric classification is given for immersions of orientable $3$-manifolds into $5$-space up to regular homotopy.

### October 19, 2005

Quaternionic Analysis
Andrew Snowden
12:30–1:30 @ Fine 224

Of the complex valued functions of a complex variable, the holomorphic functions stand out as those posessing the most remarkable properties. It is natural then to wonder if there is any analagous class of quaternionic valued functions of a quaternionic variable. It seems, though, that it was almost a hundred years after the invention of the quaternions by Hamilton that Fueter was the first to seriously consider the question. He gave a definition for a "regular" quaternionic function and showed that these functions satisfy analogues of the standard propositions of complex analysis: Cauchy's theorem, existence of Laurent expansions, etc. Later, in 1978, Sudbery expanded upon Fueter's ideas. Particularly interesting in Sudbery's paper is the role of harmonic analysis on $S^3$ in the theory, in a manner analagous to the role of harmonic analysis on $S^1$ in complex analysis. In my talk I'll sketch some of the theory of quaternionic analysis and discuss some of its shortcomings.

### October 12, 2005

A Polynomial with a Big Galois Group
Elena Fuchs
12:30–1:30 @ Fine 224

Given an irreducible polynomial $f$ over $\mathbb Q$, we can represent its Galois group as a group of permutations on the $n$ roots, where $n$ is the degree of $f$. A natural question to ask is when the Galois group is all of ${\bf S}_n$ -- i.e., when is it very big", and thus useful for all kinds of stuff? There are several examples of such polynomials, and a particularly nice one is the truncated exponential polynomial, $$f_n(x) = 1+x +\frac{x^2} {2!} + \cdots + \frac{x^n}{n!},$$ which Schur proved to have Galois group ${\bf S}_n$ or ${\bf A}_n$. In this talk, we give a sketch of a recent and much simpler proof due to R. Coleman, involving Newton Polygons, which are amazing on their own.

### October 5, 2005

Rational (!) Cubic Reciprocity
Paul Pollack
12:30–1:30 @ Fine 224

The high point of a first course in elementary number theory is the proof of Gauss's "Theorema Aureum," the law of quadratic reciprocity, which characterizes the primes p for which a given prime q is a square. In this talk we take up the corresponding problem for cubes, and describe (without proof) an early cubic reciprocity law found by Jacobi (ca. 1827), as recently reformulated by Z.-H. Sun. Unlike the usual "cubic reciprocity law" found in textbooks, Jacobi's law is entirely "rational," and doesn't require any algebraic number theory for its formulation. Anyone who has understood this abstract (and/or seen a little elementary number theory) should have no problem following the talk.

### September 28, 2005

11-torsion on Elliptic Curves
Wei Ho
12:30–1:20 @ Fine 224

The rational points of an elliptic curve form a finitely generated abelian group, and Mazur proved that there are only a few choices for the torsion part. In this talk, based on lectures by Tom Weston, we'll "prove" a little bit of Mazur's theorem, namely that there cannot be any 11-torsion. We'll find a space that somewhat classifies possible elliptic curves with 11-torsion, and it will turn out to be an elliptic curve itself! This very special curve doesn't have many rational points, which will help us finish the proof. We'll start from scratch, so no knowledge of elliptic curves is assumed.

### September 21, 2005

A Minor Theorem
Blair Dowling
12:30–1:20 @ Fine 224

A well-known graph theory fact (aka Kuratowski's theorem) is that you can draw a graph in the plane without any crossings if and only if it does not contain one of two special graphs ($K_{3,3}$ and $K_5$) as a minor. It seems only natural to ask whether such a simple characterization exists for drawing graphs on other surfaces. The answers to these questions and more are given by a result "which dwarfs any other result in graph theory" (Diestel) - Robertson and Seymour's Graph Minors Theorem, which states that in any infinite collection of (finite) graphs, one contains another as a minor. I will give an overview of this theorem, its immediate consequences, and some of the most important ideas and lemmas in the proof. Little to no background knowledge of graph theory will be assumed.

### May 18, 2005

The Margulis Measure
Alexander Bufetov
12:30–1:30 @ Fine 1201

The talk will be devoted to some of the results of Margulis's thesis (1970).

Consider the geodesic flow on a compact manifold of negative curvature. The flow preserves the foliations of the phase space into the stable and the unstable horospheres.

Margulis constructs an invariant measure for the geodesic flow satisfying the "uniform expansion" property: namely, conditional measures on the unstable horospheres are expanded uniformly under the action of the flow [note that in the case of variable curvature the Liouville measure does not have this property]. The expansion constant is the topological entropy of the flow.

The Margulis measure is the unique measure of maximal entropy for the geodesic flow.

Periodic orbits of the flow are equidistributed with respect to the Margulis measure.

The Margulis measure allows to obtain the multiplicative asymptotics for the number of periodic orbits of the flow [a logarithmic asymptotics had been earlier established by Sinai].

The talk will be an introduction to these classical results of Margulis. No special background will be assumed, all definitions will be given.

### May 11, 2005

A Milnor Theorem
Josh Greene
12:30–1:30 @ Fine 1201

When John Milnor was a freshman at Princeton, he proved that the total curvature of a non-trivial knot in 3-space is at least $4 \pi$. I will present a recent and very charming proof of this fact due to Ari Turner, which he also discovered while an undergraduate here. Since all results presented are the work of undergraduates, the talk will be accessible to everyone.

### April 20, 2005

Linear invariants of points in the complex plane
Andrew Snowden
12:30–1:30 @ Fine 1201

The general linear group $GL(2, C)$ acts on the complex plane (together with infinity) via linear fractional transformations. One of the classical problems of invariant theory is to describe the algebraic expressions in n points which remain invariant under the group action. In 1894 Kempe gave a beautiful and elementary answer to one aspect of this problem, which I will present. I will also mention the relationship of this problem to that of finding linear invariants of polynomials (such as the discriminant) and some recent work I have been involved with on this problem.

### April 13, 2005

Nonstandard Analysis
Brian Street
12:30–1:30 @ Fine 1201

I will give a brief introduction to Abraham Robinson's nonstandard analysis. Nonstandard analysis can be applied to every area of mathematics, not just analysis. When applied to the real numbers it gives a theory of infinitely large and small real numbers, and these concepts can be generalized to any topological space. Among other things, this gives a nice characterization of compactness. In this talk, I will discuss the basic concepts of nonstandard analysis, and show how it may be applied to standard areas of mathematics. At the end, I will make some remarks on Nelson's internal set theory, and how nonstandard analysis can be used to develop mathematical theories that are interesting in their own right. The talk will be elementary, and require no special background.

### March 30, 2005

A curious pair of integers
Michael Volpato
12:30–1:30 @ Fine 1201

In the Gentleman's Diary of 1795, mathematician J. Saul observed a pair of rather fascinating integers, namely the pair (345, 184). They are interesting because their sum, sum of squares and sum of cubes are all simultaneously perfect squares. Using the theory of elliptic curves we will show that the set of pairs of integers whose sum, sum of squares, and sum of cubes are all simultaneously perfect squares is infinite, and we give a recursion relation which generates all such pairs. [No knowledge of elliptic curves will be assumed.]

### March 23, 2005

The Absolute Galois Group and Dessins d'Enfants
Melanie Wood
12:30–1:30 @ Fine 1201

The Galois group of the algebraic numbers over the rationals ("absolute Galois group") is a fundamental object in number theory, yet it remains very mysterious. In fact, the only element of this infinite group we know how to write down is the identity! We describe an action of the absolute Galois group on simple, combinatorial structures called dessins d'enfants (children's drawings). This action is faithful, and we can try to understand the absolute Galois group better by trying to determine the orbits of this action. We'll discuss some invariants of this action and see examples of the current limits of our knowledge. [Only Galois theory and a bit of complex analysis will be assumed for this talk.]

### February 25, 2005

Dealing with Complexity: The Statistical Physics Approach
Louis-Pierre Arguin
12:00–1:00 @ Fine 1201

Statistical physics is roughly the study of structures with many degrees of freedom.

The first part of the talk will be an introduction to mathematical statistical physics and its basic tools. These tools have mathematical interests of their own, though they have been developped at first by physicists.

In the second part, we will look at the so-called spin glass systems where randomness is added in the structure. These structures exhibit a rich behavior which seems to be fundamental in nature, yet lacks a complete mathematical understanding. Spin Glass Theory finds applications in Optimization and Neural Networks.

This talk is meant to be an elementary introduction to the subject and will use only basic probability and analysis.

### February 16, 2005

Finding Arithmetic Flavor in Quantum Chaos
Lior Silberman
12:30–1:30 @ Fine 1201

The motion of a mechanical system can be described from the points of view of both Classical and Quantum Mechanics. Observing that classical, Newtonian, mechanics offers an excellent approximation to the motion of everyday objects, we expect Quantum Mechanics to do the same. I will describe attempts to formalize this expectation, known as the "correspondence principle", and study its implications. In particular, sufficiently complicated (chaotic) behaviour of the classical dynamics should be visible in the quantum mechanical description, at least in limit of sufficiently high energy.

I will first give a general introduction to this problem, known as the "Quantum Chaos" problem. This is a hard problem in the intersection of dynamical systems and harmonic analysis. Most investigation has been numerical, but some analytical results are known.

Secondly, I will explain why an analytic number theorist like me would care about this general framework: natural questions about the analytical properties of functions known as "automorphic forms" can be though of as very special cases of this problem. In the language above I will describe a case where the system exhibits additional symmetries associated with the additional arithmetic structure.

Finally, I will give a few ideas from the groundbreaking results of Prof. Lindenstrauss on the "arithmetic" case, leading up to his solution of the problem on compact hyperbolic surfaces. Time permitting I will also mention more general results obtained by A. Venkatesh and myself.

### May 14, 2004

Rank One Transformations: Ornstein's example
Alexander I. Bufetov
4pm–5pm @ Fine 314

A measure-preserving transformation is said to be of rank one if it is very well approximable by circle rotations (precisely, if its Rohlin towers generate the full sigma-algebra).

Donald Ornstein (1967) constructed a family of examples showing that such transformations can be strongly mixing.

Jean Bourgain (1993) proved that the spectrum of Ornstein's transformations is almost surely singular. El Houcein El Abdalaoui (1999) developed Bourgain's methods to show that Ornstein's transformations are almost surely disjoint in the sense of Furstenberg.

In the talk, I shall explain Ornstein's construction and attempt to give an outline of Bourgain's proof. Since we shall be considering one precise family of examples, the talk will be very elementary and will not require any background at all.

### April 30, 2004

A stability approach to hypergraph Tur and agrave;n problems
Peter Keevash
4pm–5pm @ Fine 322

I will discuss the stability method for solving extremal problems, as illustrated by some results from my thesis. The method consists of the following two-step process. In order to show that a given configuration is a unique optimum for an extremal problem, we first prove an approximate structure theorem for all constructions whose value is close to the optimum and then use this theorem to show that any imperfection in the structure must lead to a suboptimal configuration. Benny Sudakov and I recently used this approach to resolve two conjectures of Sos and Frankl.

### April 16, 2004

Dynamics of Quasi-Periodic Schroedinger Equations
Kristian Bjerklöv
4pm–5pm @ Fine 214

In this talk we consider the discrete Schroedinger equation $$-(u_{n+1}+u_{n-1})+\lambda V(\theta+n\omega)u_n=Eu_n,$$ where $V$ is a real $C^1$-function defined on the one-dimensional torus and $\omega$ is irrational. Besides a general introduction, I will present a dynamical method for analysing this equation in the large coupling regime ($|\lambda|$ large). This analysis gives (non-trivial) lower estimates of the Lyapunov exponent for a large (in measure) set of energies $E$, all lying in the spectrum of the associated Schroedinger operator (and hence giving a lower estimate of the measure of the spectrum). It also shows the existence of exponentially decaying eigenfunctions. Moreover, the detailed analyse enables us to prove that the projective flow on the two-dimensional torus induced by the Schroedinger equation can be minimal and non-ergodic.

### February 18, 2004

Why rational curves?
Carolina Araujo
11am–12pm @ Fine 314

In this talk I will explain some of the reasons why it is important to study rational curves on complex algebraic varieties. The talk will have two parts.

First I will briefly introduce the Minimal Model Program, which aims at classification of projective algebraic varieties. As I shall explain, rational curves play a very important role in this theory.

In the second part of the talk, I will concentrate on varieties that contain many rational curves. I will explain how we can recover important properties of such varieties by studying the geometry of certain families of rational curves on them.

This will be an introductory talk, and I will not assume any background in algebraic geometry.

### February 4, 2004

Pseudoholomorphic Curves and Periodic Orbits in Hamiltonian Dynamics
Jake Solomon (Massachusetts Institute of Technology)
11am–12pm @ Fine 314

Gromov's discovery of the theory of pseudo-holomorphic curves in symplectic manifolds, combining techniques of elliptic PDE with intuition from complex algebraic geometry, has led to many striking results in classical mechanics. In this talk, I will define symplectic and contact manifolds as a vehicle for abstracting classical Hamiltonian mechanics and properly formulating Weinstein's conjecture about existence of periodic orbits in fixed energy hypersurfaces. Then, I hope to explain intuitively how one can apply pseudo-holomorphic curves to prove certain instances of Weinstein's conjecture.

### December 11, 2003

Ricci Flow and Geometrization of 3-manifolds
Baris Coskunuzer
3:45pm–4:45pm @ Fine 314

I'll try to give an elementary sketch of Perelman's proof of Geometrization Conjecture and Poincaré Conjecture without going into analytical details too much.

### December 5, 2003

Singular Sets in Conformal Geometry: from PDE to Topology
Maria del Mar Gonzalez
12:30pm–1:30pm @ Fine 1201

I am interested in the relations between fully non-linear equations and conformal geometry. For this, consider the $\sigma_k$ curvatures of a locally conformally flat manifold: they constitute a natural generalization of the scalar curvature, and contain a lot of topological information. From the analysis point of view, we have a fully non-linear elliptic equation.

I will try to make this talk self contained. A bit of standard Riemannian geometry and PDE will be useful, but I will skip all the technicalities and look at the general ideas.

### October 13, 2003

Arithmetic Quantum Chaos: Semiclassical Limits of Automorphic Forms
Lior Silberman
4pm–5pm @ Fine 322

I will attempt to explain the title of the talk in three stages.

The first part will introduce the "Quantum Unique Ergodicity" problem, a question of Harmonic analysis on a dynamical system. After a brief introduction to the ergodic (chaotic) behaviour of a classical system, we will define the quantum mechanical version of that system (a process known as "quantization"). At high energies ("semi-classical" domain) we will expect the motion of the quantum particle to reflect the behaviour of the classical particle. There has been some (limited) success attacking the problem in this generality.

At the second part we will see how a related version of this problem arises in analytic number theory ("Arithmetic QUE") as a question about analytic properties of automorphic forms. In the terminology of the first part, these systems have multiple "time directions" as well as additional "symmetries", and we are concerned with quantum states which respect these.

Time permitting, I will conclude with a brief discussion of the recent proof due to Lindenstrauss of Arithmetic QUE on the upper half-plane.

### April 18, 2003

An introduction to extremal graph theory
Peter Keevash
12:30pm–1:30pm @ Fine 1201

The birth of extremal graph theory can be dated to the seminal work of Turán in the 1940's, although this was foreshadowed by sporadic results going back to the early 20th century. Since then it has been a flourishing branch of mathematics, popularised in particular by Erdös, and its scope has been greatly expanded from its original formulation as the 'forbidden subgraph' problem. A particularly fruitful source of related problems stems from the connection with Ramsey theory, another branch of combinatorics, that gives precision to the notion that "complete disorder is impossible".

I will present the original theorems of Turán and Ramsey, and then discuss a related result of Goodman. Following this thread of the theory leads to a couple of related conjectures of Erdös, on which I can report some progress (joint work with Benny Sudakov). This talk is intended to be introductory, so I will not assume any background in graph theory.

### April 4, 2003

On Bowen's Construction of Gibbs measures
Alexander I. Bufetov
12pm–1pm @ Fine 1201

A Gibbs measure for a stationary process is a natural generalization of a Markov measure. Roughly speaking, it is a measure for which distant past of the process has little influence over its future. For such measures, one can obtain limit theorems for the process and give explicit rates of convergence of time averages to the space average. In the talk, I shall review Bowen's classical construction of Gibbs measures. The talk would assume no background.

### March 31, 2003

Maria Chudnovsky, Chris Hall, Philip Gressman, Peter Milley
10:30am–11:45am @ Fine PL

Maria ChudnovskyThe Strong Perfect Graph Theorem: A graph is called perfect if for every induced subgraph the size of its largest clique equals the minimum number of colors needed to color its vertices. In 1960's Claude Berge made a conjecture that has become one of the most well-known open problems in graph theory: any graph that contains no induced odd cycles of length greater than three or their complements is perfect. This conjecture is know as the Strong Perfect Graph Conjecture. In 2002, in joint work with Neil Robertson, Paul Seymour and Robin Thomas we proved this conjecture.

Chris HallTwin Primes and Elliptic Curves: Let $f$ be a polynomial in $F_q[x]$. We say $(f,f-1)$ is a twin-prime pair if $f$ and $f-1$ are irreducible in $F_q[x]$. I will show that for $q\neq 3$ there are infinitely many twin prime pairs. Given a twin-prime pair I will construct an elliptic curve $E/F_q(t)$ such that $E(F_q(t)) = Z/2+Z/2$.

Philip GressmanThe Kakeya Needle Problem: Suppose that you're given a unit line segment (needle) in the plane and asked to move it continuously in such a way that when you've finished, the needle has pointed in every possible direction. You could, for example, rotate the needle through 180 degrees, but what should the strategy be to minimize the area that the needle "sweeps out"? This talk will show just how much space is needed and explain how this 80-year-old problem relates to some modern mathematics.

Peter MilleyWhy Three-Manifolds Vex Me: Come watch the speaker make a valiant attempt to explain, in only ten minutes, why three-manifold topology fascinates him so much. This talk will not focus on any one particular problem but will instead try to present a handful of those aspects of the subject which make it so rich, interesting, and at times downright perplexing.

### February 28, 2003

Null Yang–Mills fields on $S^3\times S^1$
Jonathan Holland
12:30pm–1:30pm @ Fine 1201
(This seminar was cancelled)

We present all local solutions of the linear Yang–Mills equations on Lorentzian $S^3\times S^1$ with the prescribed geometrical behavior that they be null and twisting. We explore the intimate connection between such solutions and the CR geometry of some associated twistor spaces, as well as some embedding problems which arise.

### December 13, 2002

Using Torsion to Bound the Rank of an Elliptic Curve
Chris Hall
12:30pm–1:30pm @ Fine 1201

Given an elliptic curve $E$ over a global field $K$ the basic form of the Birch–Swinnerton-Dyer (BSD) conjecture asserts that the analytic rank and algebraic rank of the curve are equal. When $K=F_q(C)$ is a function field a theorem of Tate implies that the analytic rank bounds the algebraic rank from above. Using an elementary argument I will show how one can use $l$-torsion to deduce the mod-$l$ reduction of the associated L-function $L(T,E/K)$. We will apply this to infinite families of curves to show that the analytic rank and hence algebraic rank are at most one. Using an unpublished result of D. Ulmer one may deduce that BSD holds for these families.

Most of the talk will focus on one concrete case, and I will define things as they are needed. Those familiar with elliptic curves and L-functions will certainly be more comfortable, but I will try to explain things so that others will get something out of the talk as well.

### December 6, 2002

Tube Volumes and Small Hyperbolic Three-Manifolds
Peter Milley
12:30pm–1:30pm @ Fine 1201

In two dimensions, the Gauss–Bonnet theorem says that the area of a complete hyperbolic surface is just a constant multiple of its Euler characteristic. In three dimensions, however, the situation is much more complicated; in particular, the question of which hyperbolic three-manifolds have the smallest volume is still open. In this talk I will give a brief history of this problem and my own efforts to tackle it using, among other things, a rigourous computer-aided computation.

### November 22, 2002

Ultraproducts, Scaling limits and Fixed points: from Model Theory to discrete groups
Lior Silberman
12:30pm–1:30pm @ Fine 1201

An ultrafilter on a set $I$ is a notion of "majority" for subsets of $I$. The existence of ultrafilters is a useful technical device in several fields. For example, they can be used to assign a unique limit to any sequence of real numbers, where this limit satisfies many of the usual properties of limits. Ultrafilters are also used in mathematical logic to form Ultraproducts of models, a useful construction. A combination of the two ideas leads to a definition of the limit of a sequence of metric spaces. I will show how M. Gromov uses this construction in the context of the representation theory of discrete groups and Kazhdan property (T).

### November 8, 2002

Elliptic curves, Root numbers, ultrametric analysis, pliable functions, sieve theory and Del Pezzo surfaces
Harald Helfgott
12:30pm–1:30pm @ Fine 1201

What is a root number? What are its statistical properties? Why do they matter? How do you get at them? What does the seventh word in the title mean? What is hard about the integers? What is hard and what is not? What is no longer hard? Why should a non-number-theorist care? Are you going to explain what you talked about last spring?

### October 11, 2002

Knots with Large Volume and Lens Space Surgeries
Ken Baker (University of Texas at Austin)
12:30pm–1:30pm @ Fine 801

John Berge gave a conjecturally complete list of knots with lens space surgeries. In this talk I will exhibit a subset of these knots which have arbitrarily large hyperbolic volume and discuss a consequence of this result.

### May 17, 2002

A survey of 3 dimensional geometry and topology
Baris Coskunuzer

Known results about classification of 3-manifolds, and the developments in geometrization conjecture.

### May 10, 2002

Restriction of the Fourier Transform to Quadratic Submanifolds

In this talk I shall describe the problem of $L^p$-$L^2$ restriction of the Fourier transform to a model class of submanifolds of Euclidean space, namely quadratic submanifolds. The cases of codimension 1 and 2 have been understood for some time, whereas little is known for codimension greater than 3. I will show how an aspect of the theory of resolution of singularities can be used to get new results in this case.

### April 26, 2002

Weakly symmetric spaces
Oksana Yakimova (Independent University of Moscow)

Weakly symmetric spaces were introduced by Selberg in his paper on the trace formula. They have a number of intriguing properties: for instance, any weakly symmetric space is a real form of a complex spherical homogeneous space; then, the algebra of invariant differential operators of a weakly symmetric spaces is always commutative. The classification of weakly symmetric spaces is an open problem. In the talk, I shall speak about some approaches to this problem.

### April 12, 2002

Uniform Approximate Functional Equation for Principal L-functions
Gergely Harcos

I shall describe an approximate functional equation for central values of L-functions attached to cusp forms (with unitary central character) on GL_m over number fields. The formula generalizes a classical theorem of Hardy and Littlewood (for the Riemann zeta function) and is useful for establishing subconvexity or nonvanishing results in certain families. It is somewhat unexpected that the best known bounds for the Ramanujan-Selberg conjectures (due to Luo, Rudnick and Sarnak) are utilized in the proof.

### April 5, 2002

Explicit Construction of Expanding Graphs and Kazhdan's Property T
Lior Silberman

An "expander" is a highly connected sparse graph, which is a useful tool for many problem in theoretical computer science. While the existence of expanders follows from counting arguments (a.k.a. the probabilistic method), for many applications an explicit construction is needed. This was first achieved by Margulis using a result on the representation theory of lattices in Lie groups, known as Kazhdan's "property T".

I will first define expanders and discuss some of their properties. Turning to representation theory, I will explain the definition of property T, and show Margulis' construction, assuming Kazhdan's result.

In future talks this semester (not at the GSS) I will discuss recent papers by Zuk and Gromov detailing randomized constructions of discrete groups with property T using expanders.

### March 29, 2002

Introduction to Rationally Connected Varieties
Carolina Araujo

I'll will give an introduction to the concept of Rationally Connected Varieties. These should be thought of as the higher dimensonal analogs of rational curves and rational surfaces. I will emphasize the geometric ideas involved in handling rational curves on varieties, rather than the technical details. I will assume little knowledge of Algebraic Geometry.

### March 15, 2002

Fractional-moment methods for continuum Schrödinger operators
Jeff Schenker

I will describe the phenomena of Anderson localization and its relevance to physics and present a new technique for analyzing localization phenomena which occur with continuum Shrödinger operators. The technique is an extension of the Aizenman-Molchanov method for discrete operators.

### March 8, 2002

Interval exchange transformations and the Teichmueller flow
Alexander Bufetov

Assume we take an interval, cut it into a finite number of subintervals, perhaps of different lengths, then permute these subintervals. We obtain in this fashion a map of the original interval into itself, called an interval exchange transformation; it clearly preserves the Lebesgue measure; for example, an exchange of two intervals is a circle rotation. First results on ergodic behaviour of interval exchange transformations were obtained by V. I. Oseledets (finite multiplicity of the spectrum - 1966). In 1982 H. Masur and W. Veech, proved (independently and simultaneously) that almost all interval exchange nsformations admit of no invariant probability measure other than Lebesgue (unique ergodicity; conjectured by Keane in 1975). In that study, H. Masur discovered a beautiful relationship between the behaviour of interval exchanges and the Teichmueller flow, that is, the geodesic flow on the moduli space of closed surfaces. In the talk, I shall give the main ideas of the papers of Masur and Veech.

More recently, A. Zorich, A. Zoriuch and M. Kontsevich, and G. Forni have studied the Lyapunov spectrum of the Teichmueller flow and its relationship to the decay of correaltions for interval exchanges; if time permits, I shall speak about these results as well. The talk would strive to be elementary, self-contained and accessible to undergraduate students; it would presuppose no knowledge of either ergodic or Teichmueller theory.

### March 1, 2002

Cryptologic Properties of GammaPi Class
Seth Patinkin

I will present a new cryptologic primitive which maybe used in a variety of contexts, both symmetric and asymmetric. This method is based on the construction of a class of mappings from the space of real-entry matrices to the space of characteristic functions defined on bounded finite-dimensional geometric regions. Novelty, computational infeasibility, efficiency, and known attacks shall be addressed.

### October 26, 2001

Regulators on Milnor K-theory
Matt Kerr

We'll discuss how a generalization of Liouville's theorem leads directly into arithmetic phenomena related to the Beilinson conjectures (e.g. polylogarithms and special values of L-functions), via elementary complex algebraic geometry and hodge theory.

### October 19, 2001

The Problem of Parity in Elliptic Curves and Analytic Number Theory
Harald Helfgott

One of the most important things in the life of an elliptic curve is the sign of its functional equation, also called the "root number". It gives you the kind of symmetry of the L-function and the modular function associated to the curve, and, if you believe the BSD conjecture, it tells you the parity of the algebraic rank as well. Usually you hope or expect root number 1 and root number -1 to be equally common in whatever family of elliptic curves you are working with. However, proofs (and counterexamples) have been given only in a few special cases so far. We will see that the general case is intimately bound with a basic problem in analytic number theory. Specifically, if you have an elliptic surface E over Q with non-constant j-invariant and at least one place of multiplicative reduction over Q(t), the average root number of the fibers will be zero if the Moebius function averages to zero over the integers represented by a homogeneous polynomial P_E(x,y). This is a classical problem in analytic number theory which until now has been open for deg(P)>2. I will show how to solve it for deg(P)=3, P reducible, using the Selberg sieve and the large sieve in various devious ways. I will also sketch how one should be able to take advantage of the work of Friedlander, Iwaniec and Heath-Brown if P is irreducible.

It goes without saying that all of the above refers to elliptic curves over Q and that you know perfectly well whom to see if you want to know about elliptic curves over function fields.

### May 11, 2001

The Nilpotent Variety in its Natural Setting
Julianna Tymoczko

The nilpotent variety lies at the crossroads of many different mathematical paths, including geometry, representation theory, combinatorics, even numerical analysis.  It is defined as follows: first, a flag in an n-dimensional complex vector space V is a nested collection of vector subspaces $V_1 \subseteq V_2 \subseteq \ldots\subseteq V_n = V$ — namely, a line contained in a plane contained in a three-dimensional subspace contained in a ... Given a nilpotent operator N, the collection of flags whose subspaces are each preserved by the action of N, so that $N V_i \subseteq V_i$ for each i, is called the nilpotent variety. We will provide an introduction to this and certain closely related varieties, placing them in their context and, if time permits, discussing some recent results.

### May 4, 2001

Multiplier operators in stratified groups
Ricardo Saenz

We will discuss the boundedness of the operator m(L), where L is a sublaplacian of the stratified group G, and m is a bounded function. By the spectral theorem, m(L) is bounded in L^2(G). However, if m satisfies a regularity condition of order Q/2, where Q is the homogeneous dimension of G, then m(L) is also bounded in L^p(G), for 1<p<\infty. We will also show that, if G is the Heisenberg group, or in general is an H-type group, then a regularity condition of order D/2 is suficient, where D is the real dimension of G (in general D < G). These results are analogous to the Hörmander-Mihilin multiplier theorem on R^n.

### April 20, 2001

The wonderful world of CR geometry
Andreea Nicoara

This will be a very introductory lecture on CR (Cauchy-Riemann) geometry. I will define CR manifolds and discuss the kinds of problems  that arise in this field along with some of the methods used to attack them. I might even touch upon Penrose's application of CR geometry to relativity which is the only non-mathematical application, as far as I know.

### March 30, 2001

Operator Valued Holomorphic Functions with Positive Real Part
Jeff Schenker

The classical theory of holomorphic maps of the disk with positive real part is quite elegant.  I will show how this theory can be translated to describe operator valued holomorphic functions.  The arguments require only basic Hilbert space and measure theory.

### February 23, 2001

KKM on trees
Eli Berger

The Knaster-Kuratowski-Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this talk a version of KKM theorem for trees is introduced and used to prove several combinatorial theorems. A 2-trees hypergraph is a family of nonempty subsets of the union of two trees T and R, each of which has a connected intersection with T and with R. A homogeneous 2-trees hypergraph is a family of subsets of T each of which is the union of two connected sets.  For each such hypergraph H we denote by $\tau(H)$ the minimal cardinality of a set intersecting all sets in the hypergraph and by $\nu(H)$ the maximal number of disjoint sets in it. In this talk I prove that in a 2-trees hypegraph $\tau(H) \leq 2 \nu(H)$ and in a homogeneous 2-tree hypergraph $\tau(H) \leq 3 \nu(H)$. This improves the result of Alon, that $\tau(H) \leq 8 \nu(H)$ in both cases.

### February 16, 2001

Combinatorial Nullstellensatz
Peter Keevash

A Sidon Sequence is a sequence of integers containing no solutions to the equation a + b = c + d apart from the obvious ones. I shall attempt to do justice to an extremely nice paper of Imre Ruzsa in which progress is made towards an old conjecture of Erdos. The conjecture is that there are Sidon sequences which grow about as quickly as certain reasonably obvious upper bounds allow. Ruzsa's remarkable and implausible construction involves rearranging the binary expansions of the numbers clog(p) for p a prime, where c is a random real number.

I then hope to present a small selection of other interesting open problems in the area. For the benefit of any harmonic analysts that might attend I may briefly mention the (different) objects that they call Sidon Sequences. Confusingly enough the Sidon sequences of interest to me have a role to play in certain problems connected with the other type of Sidon sequence, as was discovered by Rudin.

### February 9, 2001

Radon Transform, inversion formulas, and a glimpse of singularity

The so called Radon transform, introduced early this century by Funk and Radon, has applications in analysis and tomography. After going around the subject for a little while, we will begin looking for generalizations and specially wonder what happens if we introduce a singular kernel to the problem and when the singularity goes on a manifold. If time permits I will sketch a proof by Christ, Nagel, Stein and Wainger about L^p boundedness of singular Radon transform. It should be accessible to a wide range of audience.

### January 19, 2001

Harmonic Tori: An Algebro-Geometric Perspective
Emma Carberry

Harmonic maps from a Riemann surface to a compact Lie group or symmetric space are of both physical and geometric interest. The harmonic map equations are then a reduction of the self-dual Yang-Mills equations, and thus physicists study harmonic maps of $\mathbb R^2$ and $\mathbb R^{1,1}$. A surface has constant mean curvature precisely when its Gauss map is harmonic, and Willmore surfaces and surfaces of constant negative Gauss curvature also possess characterizations in terms of such harmonic maps.  We focus on a simple case of particular interest, harmonic maps $f$ from a 2-torus (with any conformal structure $\tau$) to the 3-sphere (with standard metric). In 1990 Hitchin showed that the data $(f,\tau)$ is in one-to-one correspondence with certain algebro-geometric data, namely a hyperelliptic curve (called the spectral curve), a pair of meromorphic differentials on this curve, and a line bundle, all satisfying certain conditions. He proved (using ad hoc methods) that for $g\leq 3$, there are curves of genus g that support the required data, and hence describe harmonic maps $f:(T^2,\tau)\rightarrow S^3$. One is particularly interested in conformal harmonic maps as their images are minimal surfaces. We show that for each $g\geq 0$, there is a conformal harmonic map $f:(T^2, \tau)\rightarrow S^3$ whose spectral curve has genus $g$.

### December 1 & 8, 2000

Structures of Quantum Field Theory
Slava Rychkov