Math Colloquia

"Variations and Applications of Carleson's theorem" Abstract: Carleson's theorem of the 1960s states that a Fourier series with square summable coefficients converges almost everywhere. In the past ten years this theorem has been much studied and generalized, and its variants have applications to singular integrals along vector fields and to theorems in ergodic theory. We will present the theorem and elaborate on its variants and applications.

The study of computational complexity presents challenging mathematical problems. In Complexity Theory computational problems are classified into complexity classes, the best known include P, NP and Valiant's class #P for counting problems. A central problem in Valiant's theory is the permanent vs. determinant problem. We will report some latest progress on this problem. Graph homomorphism was introduced by Lovasz over 40 years ago, and it is also called the partition functions in Statistical Physics, and can encode a rich class of counting problems: Given an $m \times m$ symmetric matrix $A$ over the complex numbers, compute the function $Z_A(\cdot)$, where for an arbitrary input graph $G$, \[ Z_A(G) = \sum_{\xi:V(G)\rightarrow [m]} \prod_{(u,v)\in E(G)} A_{\xi(u),\xi(v)}.\] Our foucs is the computational complexity of $Z_A(\cdot)$. With Xi Chen and Pinyan Lu, we have achieved a complete classification theorem for the complexity of $Z_A(\cdot)$. The classification proof is too complicated to present, but we will present the proof of a lemma. It states that in order to be computable in polynomial time, the matrix $A$ must possess a group structure. Another component of the proof uses Gauss sums. (In a subsequent Number Theory Seminar I will present some related work.) No prior knowledge of complexity theory is assumed.

Many classical fractals, such as the Sierpinski gasket, and Julia sets of polynomials can be described through groups generated by finite automata. Automaton groups also provide a rich source of examples (such as Grigorchuk group of intermediate growth) and play an important role in geometric group theory. I will talk about how random walks on fractals can be used to understand their structure.

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, such phenomena has attracted a renewed interest from the engineering community. This talk will cover a survey of the speaker's research and related work in this area ranging from aggregation models in nonlinear PDE to control algorithms and robotic testbed experiments. We conclude with a discussion of some interesting problems for the applied mathematics community.

Mirror symmetry is a phenomenon first discovered by string theorists in 1990. This was a surprising conjectural relationship between different Calabi-Yau manifolds. These are Ricci-flat, three-dimensional complex manifolds. String theory predicted that certain very difficult calculations on one of these manifolds (counting rational curves) was transformed into a manageable problem on the other. Since then, a great deal of effort has been devoted to understanding this phenomenon. In general, however, it is quite a complicated story. I will explain a surprisingly elementary theorem, due to myself, Pandharipande and Siebert, which provides a self-contained example of mirror symmetry. One side of the mirror symmetry picture involves certain commutators in a group of automorphisms of the algebraic torus, while the other side involves counting ratinoal curves on algebraic surfaces.

Effective randomness, also known as algorithmic randomness deals with the infinite binary strings which are intuitively ''random''. I will talk about the different ways in which one can take to calibrate randomness, and discuss some of the recent development in this area. In particular, I will focus on the interactions with computability and logic. This talk is aimed at a general audience.

Niccolo Tartaglia's (1449-1557) solution to solving cubic equations, which renowned mathematician and physician Girolamo Cardano wanted but Tartaglia resisted, led to one of the first intellectual property cases in Western history. Eventually, Tartaglia agreed to give Cardano what he so desired, but only if the latter promised he would not publish it. Cardano promised, and Tartaglia sent him the solution. Wasting little time, however, Cardano published the solution (along with a 'general' solution that he himself developed). Tartaglia was, not surprisingly, furious and began a vicious battle with Cardano's assistant, Ludovico Ferrari (Cardano refused to engage Tartaglia directly). But vitriolic polemics aside, there is something else rather curious about this ordeal: the solution Tartaglia gave Cardano was encrypted in a poem. This talk looks at the motives behind his "poetic solution" and what it says about the close relationship between 'poeisis' and 'mathesis' in this period of mathematics' history.

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Start with n particles at each of k points in the d-dimensional lattice, and let each particle perform simple random walk until it reaches an unoccupied site. The law of the resulting random set of occupied sites does not depend on the order in which the walks are performed, as shown by Diaconis and Fulton. We prove that if the distances between the starting points are suitably scaled, then the set of occupied sites has a deterministic scaling limit. In two dimensions, the boundary of the limiting shape is an algebraic curve of degree 2k. (For k = 1 it is a circle, as proved in 1992 by Lawler, Bramson and Griffeath.) The limiting shape can also be described in terms of a free-boundary problem for the Laplacian and quadrature identities for harmonic functions. I will describe applications to the abelian sandpile, and to Propp's rotor-router model, and show simulations that suggest intriguing (yet unproved) connections with conformal mapping. Joint work with Lionel Levine.

Sometimes the way that magic tricks work is even more amazing than the tricks themselves. I will illustrate with tricks that fool magicians (demonstrations provided). The tricks depend on hidden mathematics; combinatorics and group theory (don't worry, the talk is aimed at a general audience). The Math behind the tricks has applications to secret codes, decoding dna, robot vision and much else. Changing the tricks leads to math problems beyond our current understanding.

When several integers are added, carries occur along the way. This carries process has an amazing transition matrix (Holte). The behavior of the carries is intimately related to the mathematics of ordinary riffle shuffles. I will explain the mathematics of carries, shuffling and their connection. This is joint work with Jason Fulman

A broad overview of recent work on affine geometric properties of convex bodies will be presented. In particular, the shadows of a convex body can be used to construct new convex bodies called projection bodies. We discuss how this construction leads to new affine isoperimetric inequalities that are stronger than the classical Euclidean isoperimetric inequality. Connections to Sobolev inequalities, moment-entropy inequalities, and information theory will also be mentioned.

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The structure of rational solutions to a polynomial equation depends on the structure of corresponding algebraic variety. In case of a curve of genus zero, the problem of finding all solutions can be completely solved using Hasse-Minkowski principle. In case of genus one, the obstruction to the Hasse-Minkowski principle is conjectured to be finite; and the set of rational points is a finitely generated group by the Mordell-Weil theorem if it not empty. In case of genus two or bigger, the set of solutions is finite by Faltings theorem. A major unsolved problem today is the effectivity of solutions for curves of genus one or bigger. For elliptic curves, one has the Birch and Swinnerton-Dyer (BSD) conjecture which relates the Mordell-Weil group and the central values of L-series arising from counting rational points over finite fields. For curves of genus two or bigger, one has the ABC conjecture and its refinements providing some effective bounds for curves. In function field case, these conjectures are consequences of Bogomolov-Miyaoka-Yau.

Gross and Schoen have constructed a cohomologically trivial 1-cycle on the triple product of a curve by a modification of the diagonal cycle. In lecture 2, he will explain his recent formula for the height of this cycle in term of relative dualising sheaf. He will also explain the applications of this formula to ABC conjecture, Bogomolov Conjecture, and Tate's conjecture for variety over finite fields.

Gross and Kudla have conjectured a formula to related the height of Gross--Schoen cycles on Shimura curves and the central derivatives of the triple product L-series for triples of newforms of weight two. This conjecture is proved in his joint work with Xinyi Yuan and Wei Zhang with a great generality. In Lecture 3, Shouwu will explain this formula and its applications to rational points to elliptic curves.

abstract: Dispersive equations, like the Schr\'odinger equation for example, have been used to model several wave phenomena with the distinct property that if no boundary conditions are imposed then in time the wave spreads out spatially. In the last fifteen years this field has seen an incredible amount of new and sophisticated results proved with the aid of mathematics coming from different fields: Fourier analysis, differential and symplectic geometry, analytic number theory, and now also probability and a bit of dynamical systems. In this talk it is my intention to present few simple, but still representative examples in which one can see how these different kinds of mathematics are used in this context.

This talk focuses on some recent developments in the theory of fully nonlinear elliptic equations. We will be particularly concerned with the behavior of solutions of Bellman-Isaacs equations, which arise (for example) in the theory of two-player, zero-sum stochastic differential games. Such elliptic and parabolic equations present a unique challenge, since the usual energy methods-- multiplying by test functions and integrating by parts-- are inapplicable. Instead, one must rely almost entirely on the maximum principle. We will show how to construct fundamental solutions of these equations by exploiting their positive homogeneity and using ideas from principal eigenvalue theory and a deep functional analytic theorem. The shape of the fundamental solution contains useful information about the trajectories of the underlying controlled stochastic process. We will also discover a new perspective on some classical results regarding critical exponents for Laplace's equation.

The usual norm on the complex numbers and its associated analytic geometry (holomorphic functions and differential forms) have been fundamental tools for understanding the geometry and topology of complex algebraic varieties since the beginnings of the subject. Nonarchimedean norms, such as the p-adic norm on the rational numbers, also have an associated analytic geometry which has been used in number theory, but is just beginning to be applied in other areas of mathematics, such as algebraic geometry and dynamical systems. This talk will be an introduction to nonarchimedean geometry with an explanation of its combinatorial manifestation in tropical geometry and relations to intersection theory.

In this lecture I will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. I will begin the lecture by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then I will explain how to generalize the construction to obtain a larger collection of functions that we call The oscillator dictionary . Functions in the oscillator dictionary admit many interesting pseudo-random properties, in particular, I will explain several of these properties which arise in the context of problems of current interest in communication theory. Technical details for my Colloquium lecture will be given in the algebraic geometry seminar at 2:25pm. There I will prove the main properties of the oscillator functions using the new tool that we developed---The Geometric Weil Representation.

The Picard group of an algebraic variety X is the set of complex line bundles over X. In this talk, we will describe the Picard groups of certain finite covers of the moduli space of curves. The methods we use combine ideas from algebraic geometry, finite group theory, and algebraic/geometric topology.

Reaction-diffusion equations are parabolic partial differential equations used in the modeling of phenomena such as propagation of species in an environment or spreading of flames in combustible media. Their general solutions exhibit two basic behaviors, extinction (quenching) and spreading. In this talk we will review recent progress in our understanding of how the motion of the underlying medium, modelled by a fluid flow, affects both the occurence of quenching and the speed of spreading of reaction. The problem turns out to have fruitful connections to questions about mixing effixiency of flows and homogenization of advection-diffusion operators.

At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called skinning maps These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known about their behavior.

In geometric group theory we are interested in studying finitely generated groups as geometric objects. A finitely generated group can be considered as a metric space when endowed with a `word metric'. This word metric depends on the choice of generating set but all such metrics are bilipschitz equivalent. Usually, however, finitely generated groups are studied up to `quasi-isometry'. This is a coarse version of bilipschitz equivalence that allows one to study these groups by studying proper geodesic metric spaces on which they act. I will give examples that show that these two notions are not equivalent. The proof will give a flavor of some of the varied techniques and theorems currently used to study the geometry of finitely generated groups.

The Carleson-Hunt theorem shows that for every p-integrable function f on the circle, 1 < p < infinity, the Fourier series of f converges to f almost everywhere. We give an extension of this theorem which provides quantitative information about about the rate of convergence, and some applications.

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There is a growing need for mathematical methodologies which can provide understanding of high dimensional data sets. These methods also need certain kinds of robustness, so that they should not be too sensitive to changes of scale and to noise, and they should be applicable to various kinds of unstructured data. In these talks we will discuss methods for adapting idealized notions coming from algebraic topology and homotopy theory to the world of point clouds, and show numerous examples of applications of these methods.

There is a growing need for mathematical methodologies which can provide understanding of high dimensional data sets. These methods also need certain kinds of robustness, so that they should not be too sensitive to changes of scale and to noise, and they should be applicable to various kinds of "unstructured data". In these talks we will discuss methods for adapting idealized notions coming from algebraic topology and homotopy theory to the world of "point clouds", and show numerous examples of applications of these methods.

In this talk I want to present several classical and less classical examples to contribute to the understanding of the role of scaling and randomness in the appearance of entropy and irreversibility. The simplest idea is the diffusion limit for the Lorentz equation? Next one should observe that the derivation of the Lorentz equation for system of particles interacting with obstacles has been validated by Galavotti under a randomness hypothesis. Such derivation turns out to be wrong when the obstacles are on a periodic lattice (Golse Bourgain Weinberg). On the other hand in the linear setting the diffusion limit has been obtained (under a finite horizon hypothesis) by Sinai and Bunimowich and I want to present a ``baby model? of such derivation obtained with Golse. Two others observations have to be added. 1 In the Landford derivation of the Boltzmann equation the fact that the problem is genuinely non linear ``helps? in the convergence. 2 To overcome the obstruction in the case of the Golse Bourgain Weinberg example one introduce a ?non markowian closure? and this shares some similarities with what is done (with absolutely no proof ) in turbulence with the so called k-epsilon model.

I will discuss recent work with Felix Schlenk that explains exactly when a 4-dimensional ellipsoid embeds symplectically in a ball. Surprisingly, the answer involves Fibonacci numbers and has connections to lattice counting problems. The talk will not assume any prior knowledge of symplectic geometry.

This talk is about joint work with Ralph Greenberg and Karl Rubin. Given a group C of order 7 with a Galois action (in characteristic not 7), we construct the family of all elliptic curves with a rational subgroup Galois-isomorphic to C. As an application, we show that the images of 7-adic representations of elliptic curves over Q with a rational subgroup of order 7 are as large as they can be, with at most one exception (counted suitably).

In his Disquisitiones Arithmeticae, Gauss published a simple formula for the number of solutions mod p to x^3 - y^3 = 1. In his last diary entry, Gauss gave a similar result for the number of solutions mod p to x^2 + y^2 + x^2y^2 = 1. In modern language, these results can be viewed as part of an extensive history of counting points on elliptic curves over finite fields, which now has applications to cryptography. This talk will discuss some of this history, along with recent joint work with Karl Rubin on counting points on reductions of elliptic curves with complex multiplication.

The relation between complex analysis and algebraic geometry is the subject of a story that began a long time ago, has evolved along a number of interwoven strands, and is still far from fully told. In this talk we will tell a story along one particular strand involving the Laplace equation and equations that evolved from it. We will discuss several major results in the subject, beginning with Kodaira's Embedding Theorem and ending with some more recent results in birational geometry.

I will show where subtle mean value inequalities come into a 2-categorification of the symplectic category, and what the symplectic category is in the first place. I hope to also touch on applications of this abstract framework in low dimensional topology and mirror symmetry.