(August 21) Robert Benedetto: Wavelets on p-adic fields and related groups — Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = Q_p, the field of p-adic rational numbers (as a group under addition), which has compact open subgroup H = Z_p, the ring of p-adic integers.

Classical wavelet theories, which require a non-trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. We introduce a theory of wavelets on G using coset representatives of the discrete quotient G/H as translating elements. We then construct some wavelet bases for L²(G).

(September 4) Jim Yorke: The Mathematics of Today's Epidemics — The epidemics in question are HIV and flu. I will try to be as non-technical as possible.

(September 11) Jeffrey Bub, Distinguished University Professor: Quantum Games: Einstein and Bohr Meet Alice and Bob — Nonclassical correlations can be exploited to perform computational tasks that are apparently beyond the capacity of a classical computer, and to implement information-theoretic protocols (e.g., sharing a secret key between two parties) that can't be implemented by parties communicating with classical information. In this talk, I consider classical, quantum, and superquantum correlations in terms of two-person communication games in which the players are limited to certain resources. The weird features of quantum mechanics that Einstein called 'spooky action-at-a-distance' are features of the nonclassical correlations associated with games for which there is no winning strategy if the players limited to classical resources, and the debate between Bohr and Einstein about how to understand quantum mechanics can be posed in terms of these games.

(September 25) Andrei Zelevinsky: Quivers with potentials: representations, mutations and applications — A quiver is a finite directed graph, that is, a finite set of vertices some of which are joined by arrows. A quiver representation assigns a finite-dimensional vector space to each vertex, and a linear map between the corresponding spaces to each arrow. A fundamental role in the theory of quiver representations is played by Bernstein-Gelfand-Ponomarev reflection functors associated to every source or sink of a quiver. In a joint work with Harm Derksen and Jerzy Weyman we extend these functors to arbitrary vertices. This construction is based on a framework of quivers with potentials; their representations are quiver representations satisfying relations of a special kind between the linear maps attached to arrows. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras. However no special knowledge will be assumed in any of these subjects, and the exposition aims to be accessible to graduate students.

(October 12) Thomas J. R. Hughes: Isogeometric analysis — Geometry is the foundation of analysis yet modern methods of computational geometry have until recently had very little impact on computational mechanics. The reason may be that the Finite Element Analysis (FEA), as we know it today, was developed in the 1950's and 1960's, before the advent and widespread use of Computer Aided Design (CAD) programs, which occurred in the 1970's and 1980's. Many difficulties encountered with FEA emanate from its approximate, polynomial based geometry, such as, for example, mesh generation, mesh refinement, sliding contact, flows about aerodynamic shapes, buckling of thin shells, etc., and its disconnect with CAD. It would seem that it is time to look at more powerful descriptions of geometry to provide a new basis for computational mechanics.

The purpose of this talk is to describe the new generation of computational mechanics procedures based on modern developments in computational geometry. The emphasis will be on Isogeometric Analysis in which basis functions generated from NURBS (Non-Uniform Rational B-Splines) and T-Splines are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is described. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD description.

In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. Extraordinary accuracy is noted for k-refinement in structural vibrations and wave propagation calculations. Surprising robustness is also noted in fluid and non-linear solid mechanics problems. It is argued that Isogeometric Analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses many advantages. In particular, k-refinement seems to offer a unique combination of attributes, that is, robustness and accuracy, not possessed by classical p-methods, and is applicable to models requiring smoother basis functions, such as, thin bending elements, and strain-gradient and various phase-field theories.

A modelling paradigm for patient-specific simulation of cardiovascular fluid-structure interaction is reviewed, and a précis of the status of current mathematical understanding is presented.

(October 16) Tony Pantev: Geometric Langlands and non-abelian Hodge theory — I will explain a general framework for understanding the geometric Langlands correspondence via non-abelian Hodge theory and Hitchin's abelianization. I will show how this framework can be used to produce Hecke eigensheaves explicitly. If time permits I will illustrate the general strategy on the non-trivial example of the projective line with tame ramification at five points. This is a joint work with R.Donagi and C.Simpson.

(October 23) Yoshi Giga: On the Navier-Stokes Flow with infinity energy and its applications — It has been widely known that the existence of a global-in-time smooth solution for the three-dimensional Navier-Stokes initial value problem with non small initial data is an important open problem. In fact, this is one of seven millennium problems posed by the Clay Institute. To attack this problem various non blow-up criteria have been proposed. My talk today is concerned with non blow-up criteria by developing L^∞ theory of the Navier-Stokes equations. Usually, the solvability of the Navier-Stokes initial value problem is discussed for initial data decaying at space infinity for example of finite kinetic energy. However, it is also important to consider nondecaying initial data so we survey L^∞ theory. We give an application to non blow-up criteria. We show that continuous vorticity alignment implies non blow-up without assuming that the kinetic energy is finite. This provides a different view point for a famous result of Constantin-Fefferman in 1993, where integral estimates play a key role.

(November 6) Oscar Garcia-Prada: Geometry of surface group representations — Given a compact real surface S and a semisimple Lie group G, we consider the moduli space R(S,G) of representations of the fundamental group of S in G (sometimes called the character variety). This moduli space plays a central role in many problems in geometry, topology and physics. By considering a complex structure on the surface S (thus making it a Riemann surface), the moduli space of representations is in bijection with a moduli space of holomorphic objects, known as Higgs bundles. We explain this correspondence and show how to use it to study the topology of R(S,G). We give special attention to the case where G is the isometry group of a non-compact Hermitian symmetric space. In this situation the moduli space has special components that can be regarded in some sense as generalizations of the Teichmueller space of S (which can be identified with a component of the character variety when G=PSL(2,R)).

(November 13) Scott A. Wolpert: The Weil-Petersson geometry of Teichmueller space — Teichmueller space parameterizes the geometry of Riemann surfaces. By the Uniformization Theorem a surface of negative Euler characteristic has a hyperbolic metric. The hyperbolic geometry of individual surfaces leads to the Weil-Petersson geometry on Teichmueller space. We will describe the CAT(0) geometry, including the description of Teichmueller space as an infinite polyhedron and applications to rigidity questions.

(December 4) Philippe G. LeFloch: Einstein spacetimes with bounded curvature — In this lecture, I will present recent results on Einstein spacetimes of general relativity, when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. One such result, in a joint work with B.-L. Chen, concerns the optimal regularity of pointed spacetimes in which, by definition, an ``observer'' has been specified. Under geometric bounds on the curvature and injectivity radius near the observer, there exist a CMC (constant mean curvature) foliation as well as CMC--harmonic coordinates, which are defined in geodesic balls with definite size depending only on the assumed bounds, so that the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.

(February 26) William Goldman: Locally homogeneous geometric manifolds — ABSTRACT. Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modelled on a homogenous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possibly ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S.

In this talk I will survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension.

(March 12) Antoine Chambert-Loir: Algebraicity and rationality of formal power series — Let there be given a power series in, say, one variable. How can one detect whether (or not) it is the Taylor expansion of an algebraic function? Or even of a rational function? And of what use can the answer be? These questions will be addressed in my colloquium talk in which the gentle listener might be pleased to hear the names of historical figures such as Borel, Lindemann, Weierstrass, Dwork, Grothendieck...

(March 26) William P. Byers: The Role of Ambiguity in Mathematics — Mathematics is often taught and discussed as though the only thing that is going on is the logical structure. From the point of view of formal logic, ambiguity is something that must be avoided at all costs. However I shall show that a kind of metaphoric ambiguity is very common in mathematics and, from a mathematical point of view, is often the essential thing that is gong on. Paradoxically even tautologies will be seen to be ambiguous. One of the consequences of this will be a refocusing on what is really important in mathematics, namely, the mathematical ideas.

More generally the talk is an invitation to mathematicians to think about the nature of our subject; in other words, to engage in a kind of informal philosophy of mathematics. It will be based on my recent book, "How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics" (Princeton University Press 2007).

(April 2) Harry Tamvakis: Integer partitions: from Euler to Ramanujan — A partition of a number n is simply a way to write n as a sum of other natural numbers; for example 5 + 5 + 3 + 2 + 2 is a partition of 17. For nearly 300 years people have been exploring the mysteries of integer partitions; they appear in nearly all branches of modern mathematics. In my talk I will focus on the problem of counting the number of partitions satisfying certain properties, and also use the opportunity to discuss the insights of some of the greatest mathematicians of all time.

(April 9) Matthew D. Foreman: The classification problem for ergodic measure preserving transformations — In 1932, von Neumann formulated the problem of classifying ergodic measure preserving transformations of [0,1]. In the nearly 80 years that have followed, there has been considerable positive progress on this problem; notably Ornstein's work using the Sinai-Kolmogorov invariant "entropy" to classify Bernoulli shifts and the theorem of von Neumann and Halmos classifying discrete spectrum transformations.

After reviewing the history, this talk will focus on joint work with Dan Rudolph and Benjamin Weiss that established some sweeping ``anti-classification" results. These theorems show that there are fundamental reasons that any classification is impossible.

(April 16) James Alexander: A historical perspective of the Gibbs phenomenon — The Gibbs phenomenon is a feature of the behavior of Fourier series of a discontinuous function. The story of its elucidation at the turn of the 20th century illustrates that mathematical research is very much a human activity--including such things as misattribution, disputed claims of priority, rhetorical long knives, vague (and not so vague) insults, uncritical parroting, etc. Luminaries in the story besides Gibbs include Albert Michelson, A.E.H. Love, Maxine Bocher (of the AMS Bocher prize), Leopold Fejer, Thomas Groenwall and Henri Poincare. This lecture is largely a historical lecture. We review the history from the late 1800s through the 1910s, with a few pre- and post-echos.

(May 7) Ezra Miller: Unfolding convex polyhedra — Most of us as children saw those paper or cardboard cutouts, which we could call "foldouts", whose edges glue to form (boundaries of) 3-dimensional convex polyhedra. Just how did anyone figure out how to make them? Given a 3-dimensional convex polyhedron, does there always exist a foldout in the plane? What about higher dimensions? These questions have surprising answers, depending on the precise meaning of "foldout". One method is to treat boundaries of polyhedra like Riemannian manifolds. Algorithmic concerns then raise fundamental issues of computational complexity for the combinatorics of geodesics on polyhedra. Ideas from this work spill over into algorithms for doing statistics on spaces of phylogenetic trees and potentially lead to discrete versions of stratified Morse theory.

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