Colloquia, AY 2008-2009
Colloquium, AY 2008-2009
Tea at 3:30 pm in 1425 Stevenson Center
September 4, 2008 |
Topology and Data Abstract: Investigators in all areas of science are aware that data of various kinds are being produced at a rate which appears to be swamping our capacity to analyze it. In this talk, we will present some methods based on topological thinking which permit the qualitative analysis of data in a number of situations, and in addition provide a methodology for extending ideas from statistics into the study of “structural” invariants. I will talk about the theory of these methods, as well as about some examples. |
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September 25, 2008 |
Beyond the geometrization conjecture: models, bounds, and effective rigidity for hyperbolic 3-manifolds Abstract: Though Perelman’s solution of Thurston’s geometrization conjecture raises the possibility of extracting geometric information from a purely topological description of a 3-manifold, it does not directly produce it. In this talk, I will begin with the simple combinatorial data of a certain type of Heegaard splitting of a 3-manifold and extract geometric estimates with constants depending only on the genus of the splitting. This gives rise to a kind of “effective” rigidity theory where one can produce not only the existence of a negatively curved metric but estimates on its shape. |
October 2, 2008 |
Topological stable rank of Banach algebras Abstract: 25 years ago, Rieffel introduced an algebraic invariant for Banach algebras called topological stable rank which generalized the notion of dimension to the non-commutative setting. The topological stable rank has a left and right version, which coincide for C*-algebras and commutative algebras. Moreover, tsr is a Banach algebra variant of the purely algebraic invariant of Bass stable rank for rings-and the left and right versions of Bass stable rank are always equal. So Rieffel asked whether they are always equal? We have calculated the left and right topological stable ranks for the class of nest algebras, and can answer Rieffel’s question negatively. |
October 9, 2008 |
Packing, energy minimization, and exceptional structures Abstract: How should one arrange a collection of charged particles so as to minimize the potential energy between them? This problem arises naturally in physics, but it extends to far more general spaces and potential functions. Almost always, the optimal configuration depends on the potential function, but in certain cases it does not; we call these universal optima. The universal optima highlight some of the remarkable exceptional structures in mathematics (such as E_8, the Leech lattice, and the 27 lines on a cubic surface). This talk will give a survey of this area. I’ll outline what happens in general, what role the universal optima play, how one can prove this, and what else we conjecture. I’ll also briefly discuss the inverse problem, which arises in chemistry and materials science: if we have a desired structure in mind, how can we design a force law under which it will spontaneously self-assemble? |
October 16, 2008 |
On the analysis of Euler’s equations of fluid dynamics with moving boundaries. Abstract: Euler formulated a complete set of equations for the motion of fluids by 1755, and afterwards stated “…it is not the principles of Mechanics which we lack in the pursuit of these researches, but solely Analysis, which is not yet sufficiently cultivated for this purpose.” The past 250 years has seen the development of this Analysis. In this lecture, I will describe some of the major mathematical developments, culminating with recent advances in the understanding of free-boundary and moving interface problems. The latter class spans the classical water-wave problem and the expansion of a gaseous star. |
October 30, 2008 |
A geometric view of random matrices. Abstract: We examine a classical object in random matrix theory – rectangular matrices with random independent entries – from a geometric point of view. How do these matrices act as linear operators in high dimensional spaces? How do they transform convex sets? It often turns out that random matrices are the best transformations one can hope for, with no deterministic constructions known of similar quality. We shall survey a rich history of old and recent developments, focusing on the recent progress on the invertibility problem for random matrices. |
November 6, 2008 |
Complex B-splines, Dirichlet means, and divided differences for multi-dimensions. Abstract: Complex B-splines are a natural extension of classical B-splines. We show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite–Genocchi formula. This formula then allows the definition of a larger class of complex B-splines. The notion of complex B-spline is extended to a multivariate setting by means of ridge functions employing the known geometric relationship between ordinary B-splines and multivariate B-splines. To derive properties of complex B-splines in R^n, 1 < n, the Dirichlet average has to be generalized to include infinite dimensional simplices. Based on this generalization several identities of multivariate complex B-splines are presented. This is joint work with Peter Massopust. Contact person: Akram Aldroubi |
December 4, 2008 |
Derivation of shell theories from 3d nonlinear elasticity. Abstract: A longstanding problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects (such as rods, plates or shells), subject to mechanical deformations, starting from the 3d nonlinear theory. For plates, a recent effort has lead to rigorous justification of a hierarchy of such theories (membrane, Kirchhoff, von Karman). For shells, despite extensive use of their ad-hoc generalizations present in the engineering applications, much less is known from the mathematical point of view. In this talk, I will discuss the limiting behavior (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d mid-surface S. We prove that the minimizers of the 3d elastic energy converge, after suitable rescaling, to minimizers of a hierarchy of shell models. We also discuss convergence of possibly non-minimizing critical points or equilibria. The limiting functionals (which for plates yield respectively the von Karman, linear, or linearized Kirchhoff theories) are intrinsically linked with the geometry of S. They are defined on the space of infinitesimal isometries of S (which replaces the ‘out-of-plane-displacements’ of plates), and the space of finite strains (which replaces strains of the `in-plane-displacements’), thus clarifying the effects of rigidity of S on the derived theories. The different limiting theories correspond to different magnitudes of the applied forces, in terms of the shell thickness. This is joint work with M.G. Mora and R. Pakzad. Tea at 3:30 pm in SC 1425. |
Thursday, January 22, 2008 |
Special Colloquium. |
February 5, 2009 |
Quantum homology and Hamiltonian dynamics. Abstract: This will be a general talk that aims to explain some of the influence of the internal structure of a symplectic manifold (e.g. its pseudoholomorphic spheres) on fundamental dynamical properties of its Hamiltonian diffeomorphisms. I will not assume any knowledge of symplectic geometry. Tea at 3:30 pm in SC 1425. |
February 26, 2009 |
Search for Quantum Symmetry in Topology Abstract: The quantization principle in physics suggests that the quantum world posses more symmetry than the classical world. In this talk, we will demonstrate how to use the quantization principle to discover some new symmetry in mathematics. Tea at 3:30 pm in SC 1425. |
March 12, 2009 |
Shock Reflection-Diffraction Phenomena, von Neumann’s Conjecture, and Nonlinear PDEs of Mixed Type Abstract: We will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the shock reflection-diffraction problems can be formulated into free boundary problems for nonlinear PDEs of mixed-composite hyperbolic-elliptic type and present the von Neumann’s sonic conjecture. The problems involve two types of transonic flow: One is a continuous transition through a pseudo-sonic circle, and the other is a jump transition through the transonic shock as a free boundary. Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including some recent results on the existence, stability, and regularity of global solutions of shock reflection-diffraction by wedges. This talk will be based mainly on joint work with M. Feldman. Tea at 3:30 pm in SC 1425. |
April 2, 2009 |
Subspaces Invariant under Lattice Translations Abstract: In a recent paper, Aldroubi, Cabrelli, Heil, Kornelsen, and Molter answered the following question: If V is a closed subspace of L^2(R) invariant under translation by integers, for which natural numbers n is it true that V is invariant under translation by 1/n? When the real line is replaced by a locally compact abelian group G and V is replaced by a closed subspace of L^2(G) which is invariant under translations by members of a fixed lattice subgroup L of G, the analogous question asks for an explicit description of the family of lattice subgroups containing L whose translations leave V invariant. Hrvoje Sikic and I have obtained an answer to this question. When L’ is a fixed lattice group containing L, our methods provide an algorithm constructing all singly generated L-invariant spaces which are also L’-invariant. |
April 9, 2009 |
Entire Solutions of Completely Coercive Quasilinear Elliptic Equations Abstract: A famous theorem of Sergei Bernstein says that every entire solution u = u(x), x in R², of the minimal surface equation, div{Du(1 + Du²)^(-1/2)} = 0, is a linear function; no conditions whatsoever being placed on the behavior of the solution u. This result however fails to be true in higher dimensions, in fact if x in R^n, with n > 7, there exist entire non-constant solutions (Bombiere, DiGiorgi and Miranda). Our purpose is to consider other quasilinear elliptic equations which do have the following Bernstein–Liouville property, namely that u is constant for any entire solution u, but where no restrictions are placed on the dimension n and no conditions are assumed on the behavior of the solution. Note that the Laplace equation Delta u = 0 does not qualify: while bounded solutions, or even solutions which are bounded either above or below, are constant (the Liouville theorem), one easily finds entire non-constant solutions (of course bounded neither above nor below): e.g., even in two dimensions the solution x² – y² is a case in point. Tea at 3:30 pm in SC 1425. |
April 30, 2009 |
LOCATION: Room 1431 Spectral flow associated to KMS states with periodic KMS group action Abstract: We describe a general framework in which we use KMS states for circle actions on a C*-algebra to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. Tea at 3:30 pm in SC 1425. |
Colloquium Chair (2008-2009): John Ratcliffe
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