Math Graduate Student Seminar

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 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

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

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

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

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

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

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

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

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

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

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 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

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

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

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

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 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

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December 4, 2008

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

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

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

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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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 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 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 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

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

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

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

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

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

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.

September 27, 2007

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

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

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

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

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

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

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

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

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

An assortment of short, interesting facts about topology.

March 6, 2007

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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 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

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

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

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

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 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

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

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

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

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

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

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

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

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

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

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

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

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 — The 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 Hall — Twin 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 Gressman — The 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 Milley — Why 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

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

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

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

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

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

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

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

May 10, 2002

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 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

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

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

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

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

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

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

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

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 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

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

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

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

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

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

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 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

In 2 lectures (12/1/00 and 12/8/00) I will try to give an exposition of basic objects and ideas of Quantum Field Theory. The list of topics will include: Feynman path integrals, Wightman axiomatics, Feynman diagram techniques for perturbation theory, renormalization group, and conformal field theories. If time permits, we will discuss asymptotic freedom and mass gap in gauge field theories, which is the subject of one of the $1,000,000 Clay Institute Problems. No prior knowledge of QFT will be assumed.

November 11, 2000

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.

October 27, 2000

A hypergraph is a collection of subsets, called edges, of a given vertex set. It is a generalization of the notion of a graph, where all the edges are of  size 2.   We prove a  generalization of Hall's theorem to families of hypergraphs, namely give sufficient conditions for a family of hypergraphs to  have a system of  disjoint representatives (a choice of edges from each  hypergraph, such that two edges chosen from different hypergraphs are disjoint). In the course of the proof we show that  a triangulation of S^n-1  can be extended it to a triangulation of B^n  by adding points in the interior of B^n with some restrictions on the degrees of these points.

October 13, 2000

I will present a nice way of constructing simply-laced Lie algebras and some of their representations. If time permits, I'll give some applications which can help to understand the exceptional group E_6.

October 6, 2000

We present a mechanism for the creation of gaps in the spectra of self-adjoint operators defined over a Hilbert  space of functions on a graph. The mechanism is based on the process of "graph decoration" which we define. The resulting operators can be viewed as associated with discrete models exhibiting a repeated local structure and a certain bottleneck in hopping amplitudes.

May 5, 2000

The existence and properties of phase transitions in a statistical mechanical model can be determined from the distribution of the zeros of its partition function in the complex parameter space. Where the zeros impinge on the physical parameter space with non-zero density, a first-order phase transition will be observed; the discontinuity in the order parameter can be computed from the density of the zeros. The earliest result in this area dates from 1952, and is due to Lee and Yang, who showed that the zeros of the partition function for the ferromagnetic Ising model lie on the unit circle in the complex fugacity plane. The topic has attracted attention ever since, but few results have been proven for other models. Recently, the distribution of partition function zeros for a wide class of models exhibiting first-order phase transitions, including the q-state Potts model (with q large enough) and the Blume-Capel model, has been described using Pirogov-Sinai Theory. These results and supporting mathematical ideas, such as contour representations of lattice spin systems, cluster expansions, and metastable free energies, will be discussed.

(Joint work with M. Biskup, C. Borgs, J.T. Chayes, and R. Kotecky. To appear in Physical Review Letters.)

April 21, 2000

The convex minorant of symmetric random walk processes turns out to be a key tool in solving several interesting physics problems. We briefly describe one such problem and proceed to compute several useful statistical properties of the convex minorant. The computations will be elementary but not standard. We relate these calculations to Brownian motion.

April 14, 2000

On a planar domain or a Riemannian surface, the eigenvalues of the Laplacian carry a lot of geometric information: the area, the perimeter, the genus, and the lengths of some (sometimes all) closed geodesics. Do the eigenvalues carry enough information to recover the domain or surface (the inverse spectral problem)? In at least some cases, no: we'll see two planar domains with the same spectrum. In some cases, yes or almost.

We now consider the inverse spectral problem for surfaces of revolution, and review known results by Bruning/Heintze and Zelditch. The speaker then presents a reconstruction algorithm for surfaces of revolution and conjectures that it almost always succeeds. The almost-sure success is proven for a toy problem.

April 7, 2000

A stratification of a set, e.g. an analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together "regularly". A "regular" partition is usually called a Whitney stratification. The stratification theory was originated by Thom and Whitney for the algebraic and analytic sets. It was one of key ingredients for Mather's proof of topological stability theorem. During the talk we present an elementary geometric proof of existence of Whitney stratification based on Rolle's lemma.

March 24, 2000

I shall find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C, symmetric about the origin 0. This problem arised in connection with the higher dimensional Littlewood-Offord problem. Apart from some auxiliary results in number theory the pigeon-hole principle will be the only tool utilized in the talk. Joint work with I. Barany, J. Pach and G. Tardos.

March 10, 2000

We demonstrate the dependence of rational equivalence upon the field of definition of a subvariety with a very concrete, cute example. This talk should be of special interest to anyone who would like to see geometric motivation for the Beilinson conjectures, though I think it will be accessible and fun for anyone who knows Abel's theorem.

March 3, 2000

The space of maps from a circle to a compact Lie group G is itself a group - it is often called the loop group of G. Although it is generally infinite dimensional, its topology and group structure are in some ways simpler than that of the original group G.

The aim of this talk is to give an introduction to some of the beautiful topology of loop groups. Specifically, I would like to describe a simple model for the loop group of SO(3,R). To understand the model one only needs to know what a group is and what a topology is. (In fact what SO(3,R) is is not important!) But with the model, one can see remarkable connections - first observed by Lusztig - between the topology of the loop group and representation theory. I hope to give an elementary presentation of this, and to briefly describe the model for the loop group of an arbitrary compact group if there is some time at the end.

February 25, 2000

(joint with B.Hunt)

Consider a compact set $X$ in $\bf R^N$ of Hausdorff dimension $D$. The classical result of Marstrand and Mattila says that for an almost every linear projection of $X$ into a subspace of dimension at least $D$ Hausdorff dimension is preserved, i.e. Hausdorff dimension of $X$ and its image are the same.

Now consider a compact set $X$ in a Hilbert space and its image under typical linear projections $\pi$ into ${\bf R}^N$. For any positive number $D$ we give an example of a set $X$ with Hausdorff dimension $D$ in the real Hilbert space $l^2$ such that for {\it all} projections $\pi$ into ${\bf R}^N$, no matter how large $N$ is, the Hausdorff dimension of $\pi(X)$ is less than $2$ no matter how large $d$ is. This nonpreservation phenomenon is unexpected and will be discussed with more details.

February 18, 2000

There is a natural topology on the set of branched covers of the torus by surfaces of genus 2 which allows us to construct a parameter space for such covers. How many components will the parameter space have in each degree? What other invariants will distinguish the components? Is it possible to tell when two maps are in the same component? Answers will be forthcoming.

February 4, 2000

We will have fun with a number of things related to short-wave asymptotics of oscillatory integrals. Probably the easiest physical interpretation of the theory is the intensity of light near caustics. The method of calculating the asymptotics is based on resolving the singularities of the phase function by pulling it back to a toric variety. The construction is governed by the Newton polyhedron of the phase function.

December 10, 1999

November 12, 1999

I will talk about branched covers of Riemann Surfaces, how to describe them, and how they behave under the action of the Braid Group. I will use these methods to derive some results about branched covers of the torus. A given cover induces a map on homology which can be represented as a 2g x 2 integral matrix. Which such matricies can be induced by a cover? How many disjoint braid-orbits of covers are associated to each matrix? Is there an analogue to the Luroth Theorem for simply branched covers of the sphere?

October 29, 1999

This will be an elementary, expository talk on the evolution of Abel's theorem — in particular, where it comes from, beginning essentially with calculus. Proofs of the easier results will be sketched. Time permitting (if we get past the 19th century), I'll discuss generalizations for singular/relative varieties and higher (co)dimensions, and perhaps indicate where Chow groups, polylogarithms and K-theory come in.

October 7, 1999

Certain disordered quantum systems are known to satisfy a ``localization'' condition which is most simply described as the absence of so-called ``extended states.'' I will describe a model system, define the notion of localization, and describe what results are expected and which of these are known to be true. Time permitting, I will present a short proof of localization by the Aizenman - Mochanov technique.

April 15, 1999

A proof, due to Witten, of the Morse inequalities relies upon the "semi-classical" limit of certain Schroedinger type operators. I will sketch Witten's proof and describe in more detail the proof of the existence of the "semi-classical" limit.

April 8, 1999

I plan to talk about some quantitatively universal features that emerge in the period doubling route to chaos. In particular, I will outline the heuristic argument that Feigenbaum gave in the 70's to explain some numerical observations about the dynamics of one-dimension maps of an interval onto itself. I will also sketch Lanford's computer aided proof of Feigenbaum's conjectures. And time permitting, I'll talk about why people think this sort of stuff is cool.

April 1, 1999

March 25, 1999

In 1966, Lennart Carleson proved his famous theorem: the Fourier series of any square integrable function f converges to f almost everywhere.

This talk will be an introduction to Carleson's original proof of the theorem. I will apply Carleson's method in its simplest form to prove that the Fourier partial sums do not grow faster than loglog(n) almost everywhere. This is already much better than the classical Kolmogorov-Seliverstov-Plessner bound (log n)^{1/2}. Then I will indicate modifications needed to get the full strength of Carleson's result.

March 11, 1999

March 4, 1999

This talk will be an introduction to the Four Color Theorem (4CT), the proof, and some equivalent formulations and generalizations. We will begin with the necessary background graph theory and a brief history of the 4CT. We will then begin to motivate the key ideas used in the current proof and possibly explain why one should expect these ideas to yield a proof. Finally, we will present some theorems equivalent to the 4CT and say a few words regarding some extensions, both open and resolved.

February 25, 1999

We shall present Mattingly-Sinai's proof of a global existence and uniqueness theorem for the 2-D Navier-Stokes equation. In order to make the talk a bit different from the one Sinai gave last week, we also describe how this proof can be extended to prove local existence and uniqueness for the 3-D Navier-Stokes equation.

February 18, 1999

The hyperbolic plane has an ideal boundary, which we can identify with the circle or the union of the real axis and the point at infinity, depending on whether we use a disk or upper half-plane model. The behavior of certain functions on the hyperbolic plane is essentially determined by their extensions to the boundary. In particular, the Cayley transform which takes the disk model to the upper half-plane model extends to a conformal map of the boundary. We use so-called groups of Heisenberg type to generalize to all symmetric spaces of rank one of noncompact type.

Note: this talk will be substantially different from the one I gave at the student seminar in Fall '97.

February 11, 1999

The Kepler Conjecture (first stated in 1611) asserts that there is no sphere packing more dense than the face-centered cubic. Last Fall Thomas Hales announced a proof. He presented this proof over the course of a week-long workshop at IAS in January. NOTE: The proof has yet to be checked carefully — this was the motivation for the workshop.

Hales' proof relies on computers for several steps. A score is assigned to the decomposition star around each sphere in the packing, in such a way that an upper bound of 8pt for the score implies the Kepler Conjecture. The most critical uses of computers are:

1. to enumerate the graphs associated to decomposition stars,
2. using interval arithmetic to majorize the score by a linear function,
3. using linear programming to derive upper bounds for the score, which falls short of 8pt for most of the 5000 cases in part 1.