September 27, 2007
Speaker: Mark Watkins, Bristol
Title of talk: Searching for points near curves using lattice reduction (apres Elkies)
Abstract: Given a plane curve $f(x,y)=0$, can we find rational numbers $x$ and $y$
with small denominator such that $(x,y)$ is close to the graph of $f$? A efficient method for this was introduced by Elkies about a decade ago. We can view the method as breaking the curve into many pieces, and then taking a linear approximation to $f$ on each. Using the famed lattice reduction algorithm of Lenstra, Lenstra, and Lovasz, we can find points close to each linear approximation. One can generalise to nonplanar curves, and to varieties of higher dimension. It turns out that embedding our curve into a higher-dimensional projective space is often wise, as then we can take a higher-order Taylor series as an approximation. The conditions on the curve are not very restrictive, only involving suitable smoothness; for instance, we could find rational points close to $x^{\pi}+y^{\pi}=1$ if desired.

It was noted by Mazur that finding points close to a curve includes the special case of finding points on it. Here we can introduce $p$-adic methods, which are numerically more robust. These have been used by Tom Fisher to find points on a genus 1 curve represented by 54 quadrics in $P^{11}$ which correspond to points of height more than 600 on a elliptic curve.

January 17, 2008

Speaker: Howie Weiss, Georgia Tech
Title of talk: Can You Hear the Shape of a Potential?
Abstract: Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction on a lattice determines a lattice spin system with associated potential. The pressure and free energy of the potential are fundamental characteristics of the system. However, even for the simplest systems, the information about the (microscopic) potential that the (macroscopic) free energy captures is subtle and poorly understood.

We study whether, or to what extent, potentials are determined by their free energy. In particular, we show that for a one- dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy.

We show (and exploit) that this rigidity problem has striking analogies to inverse problems in spectral geometry that Kac summarized ``Can you hear the shape of a drum?", and to inverse problems in algebraic number theory.

No physics background will be needed to understand the talk.

January 24, 2008

Speaker: Daniel Krashen, University of Pennsylvania
Title of talk: The u-invariant of fields
Abstract:  The u-invariant of a field is defined to be the maximal
dimension of a quadratic form which has no nontrivial zeros. Although there are some expectations for what  u-invariants should be of most "naturally occuring" fields, these  invariants are unknown in a great number of situations. For example,  if $F$ is a nonreal number field, it is known that $u(F) = 4$, and it  is expected that the u-invariant of the rational function field $F(t)$  should be $8$. At this point, however, there is no known bound for  $u(F(t))$ (and no proof it is even finite).

Important progress on this type of problem was obtained by Parimala  and Suresh late last year, who showed that the u-invariant of a  rational function field $F(t)$ is $8$ when $F$ is $p$-adic ($p$ odd).  In this talk I  will describe joint work with David Harbater and Julia Hartmann  in which we give an independent proof and a generalization of this  result using the method of "field patching."

January 25, 2008 - note this is a Friday.

Speaker: Angela Gibney, University of Pennsylvania
Title of talk:  Fulton's conjecture and upper and lower bounds for Mori cones
of algebraic varieties
Abstract:  The Mori cone is a fundamental, often illusive, invariant  of an algebraic variety and is the central object of study in higher  dimensional algebraic geometry.  In this talk I will explain Fulton's conjecture, which predicts a very simple description of the Mori cone of the moduli space of curves.  I'll show how one can naturally obtain  upper and lower bounds for the Mori cone of a large class of  varieties.  In the case of the moduli space of curves, the upper bound is the cone described by Fulton's conjecture.  In particular, this gives a new possibility for the Mori cone and a new perspective on Fulton's conjecture.

January 29, 2008 - note this is a Tuesday.

Speaker: Xiangdong Xie, Georgia Southern University
Title of talk: Negative curvature and quasiconformal analysis
Abstract: In the original proof of Mostow's rigidity theorem for hyperbolic manifolds,  quasiconformal analysis on   Euclidean space played an important role. To study rigidity property of general negatively curved spaces, one needs to do  analysis on more  general metric spaces. For example, for complex hyperbolic spaces, one has to do analysis on Heisenberg groups. In this talk I shall explain the connection between negative curvature and quasiconformal analysis.  I shall particularly discuss the quasiisometric rigidity question for Hadamard manifolds:  Does every quasiisometry between two Hadamard manifolds lie at a finite distance from a bilipschitz map?

January 30, 2008 - note this is a Wednesday

Speaker: Ken Baker, Georgia Tech
Title of talk:  Genera of knots in 3-manifolds
Abstract:  A knot is an embedding of a circle into a 3-manifold.  Knots in the 3-sphere bound compact, orientable surfaces commonly known as Seifert surfaces.  The genus of a knot is defined to be the minimum genus among all its Seifert surfaces.  What enables a knot to have a Seifert surface is that, when oriented, it represents the trivial element of the 1st homology of the ambient 3-manifold.  We extend this notion of genus to non-null homologous knots in (closed, orientable) 3-manifolds and investigate the set of genera of knots within a given 1st homology class. These sets for non-trivial homology classes have curious structures and are related to outstanding problems in Dehn surgery.

January 31, 2008

Speaker: Xuhua He, Stony Brook University, NY
Title of talk: Introduction to G-stable pieces
Abstract: The notion of G-stable pieces was introduced by Lusztig in the study of parabolic character sheaves. It has a rich structure, combining combinatorics, group theory, geometric representation theory and other fields. In this talk, we will discuss the G-stable pieces from the point of view of combinatorics of Weyl groups. We will also talk about some connection with algebraic geometry and Poisson geometry.

February 1, 2008 - note this is a Friday.

Speaker: Stefan Wenger, NYU
Title of talk: Isoperimetric inequalities and the large scale geometry of metric spaces
Abstract: Isoperimetric problems have been widely studied in various contexts and appear in areas such as analysis, geometry and geometric group theory. A suitable framework for their study is given by the theory of currents (in metric spaces), which provides powerful tools for attacking area-minimization problems.

Isoperimetric inequalities can in particular be used to study the large scale geometry of Riemannian manifolds and, more generally, singular metric spaces. This will be the focus of the talk, in which we will survey both old and recent results . Among other things, we will present an optimal theorem which states that from the point of view of isoperimetric inequalities, nothing exists between Euclidean space and spaces of strictly negative curvature on a large scale, or more generally Gromov hyperbolic spaces. This generalizes and strengthens previous theorems of Gromov and others.

February 5, 2008 - note this is a Tuesday

Speaker: Simon Foucart, Vanderbilt University
Title of talk: From Approximation Theory to Compressive Sampling via Banach Spaces Geometry: a Computational Tour
Abstract: Starting with an issue on computation stability, I will introduce the notion of condition number for a system spanning a normed space $V$. I will show how optimization techniques can be used to calculate the minimum of these condition numbers. The latter is an intrinsic constant of the space $V$, and I will examine its connections with the projection constant of $V$. In particular, I will raise a question --- formulated only in terms of projections --- related to the $P_\lambda$-problem. The arguments will lead me to the new and exciting field of Compressive Sampling. The paradigm that only few information on a signal is necessary for its reconstruction will be illustrated by some striking yet simple results, including  a proof of Kashin's theorem on widths as a byproduct. All along, an eye will be kept on the computational aspect of the theory. 

February 7, 2008

Speaker: Michael Ching, Johns Hopkins University
Title of talk: Derivatives of the identity and chain rules for calculus of functors
Abstract: Goodwillie's calculus of functors is a way to apply some of the concepts
of ordinary real-variable calculus to study homotopy theory. The main
tool is a "Taylor series" that consists of a sequence of approximations to a functor of topological spaces. The analogy goes deeper than you might expect with, for example, sensible (and useful) notions of the derivatives of a functor.

In this talk, I'll address the existence of "chain rules" in this context and point out the similarities with, and differences to, ordinary calculus. Underlying such chain rules are the derivatives of the identity functor (which are more interesting than you might expect). I'll try to explain how these derivatives explain to some extent the appearance of Lie algebras in, for example, rational homotopy theory.

February 12, 2008 - note this is a Tuesday

Speaker: Adrian Butscher, Stanford University
Title of talk: Gluing Constructions for Constant Mean Curvature Surfaces.
Abstract: I will review the now classical Kapouleas gluing construction for CMC surfaces in Euclidean space and present some results and work in progress concerning the extensions of this theory to general ambient manifolds.  An important feature which emerges is that the ambient Riemannian curvature seems to play a significant role in the existence of such surfaces; and exploiting this, it seems possible to construct examples of CMC surfaces having properties very different from their Euclidean analogues.

February 21, 2008

Speaker: Christopher Hacon,University of Utah
Title: Finite generation of canonical rings
Abstract: Let X be a smooth projective algebraic variety. A pluri-canonical form is an section of a power of the canonical linear bundle on X. The canonical ring formed by the pluri-canonical forms is an invariant that is of fundamental importance in the study of the birational geometry of X.

In dimension 2, its properties were understood by the Italian school of Algebraic Geometry at the beginning of the 20th century. The 3 dimensional case was understood in the 1980's by celebrated work of Mori and others. In this talk I will discuss joint work with Birkar, Cascini and McKernan towards understanding the geometry of algebraic varieties of arbitrary dimension. In particular I will discuss the following theorem:

Theorem. The canonical ring of any smooth projective algebraic variety X is finitely generated.

(Note that this Theorem was independently proven by Siu.)

February 27, 2008 - note this is a Wednesday

Speaker: Carl Pomerance, Dartmouth
Title of talk: TBA
Abstract: TBA

March 6, 2008
Speaker: Marty Golubitsky, University of Houston
Title: Symmetry breaking and synchrony breaking
Abstract: A coupled cell system is a network of interacting dynamical systems.  Coupled cell models assume that the output from  each cell is important and that signals from two or more  cells can be compared so that patterns of synchrony can  emerge.  We ask: Which part of the qualitative dynamics  observed in coupled cells is the product of network  architecture and which part depends on the specific equations?

In our theory, local network symmetries replace symmetry as  a way of organizing network dynamics, and synchrony breaking  replaces symmetry breaking as a basic way in which transitions  to complicated dynamics occur. Background on symmetry breakingand some of the more interesting examples will be presented.

April 17, 2008
Speaker: Professor Yang Wang, Michigan State University
Title: The Golden Ratio Encoder for Analog-to-Digital Conversion
Abstract: The advent of computer and digital information technologies and their development have greatly changed the world we live in. Today digital technologies are everywhere in our lives. A key step that makes all those technologies possible is to convert analog data into digital ones, a process known analog-to-digital conversion, or A/D conversion. With demand for higher precision and more cutting-edge technologies, the mathematics of A/D conversion algorithms plays an important role in this quest. In this talk we review algorithms for A/D conversions and the associated problems concerning stability and robustness. We discuss a new algorithm for A/D conversion using the golden ratio expansion for real numbers and the Fibonacci sequence. We show that this new A/D encoder have many advantages over the traditional A/D encoders.

August 28, 2008
Speaker: Andrei Zelevinsky (Northeastern University)
Title of talk: Cluster algebras