Colloquia, AY 2008-2009

Colloquium, AY 2008-2009

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

September 4, 2008  
Gunnar Carlsson, Stanford University

Topology and Data

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.

Contact person: Guoliang Yu

September 25, 2008  
Jeff Brock, Brown University

Beyond the geometrization conjecture: models, bounds, and effective
rigidity for hyperbolic 3-manifolds

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.

Contact persons: John Ratcliffe and Mark Sapir

October 2, 2008  
Kenneth Davidson, University of Waterloo, Canada

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.

Contact person: Dechao Zheng

October 9, 2008  
Henry Cohn, Microsoft Research, New England

Packing, energy minimization, and exceptional structures

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?

Contact person: Doug Hardin

October 16, 2008  
Steve Shkoller, University of California, Davis

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

Contact person: Gieri Simonett

October 30, 2008  
Roman Vershynin, University of Michigan.

A geometric view of random matrices.

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.

Contact person: Alex Powell

November 6, 2008  
Brigitte Forster-Heinlein, Centre for Mathematical Sciences, Technische Universität München.

Complex B-splines, Dirichlet means, and divided differences for multi-dimensions.

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  
Marta Lewicka, University of Minnesota

Derivation of shell theories from 3d nonlinear elasticity.

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.

Contact person: Gieri Simonett

Thursday, January 22, 2008 

Special Colloquium.

February 5, 2009  
Dusa McDuff, Columbia University

Quantum homology and Hamiltonian dynamics.

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.

Contact person: Guoliang Yu

February 26, 2009  
Yongbin Ruan, Department of Mathematics, University of Michigan.

Search for Quantum Symmetry in Topology

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.

Contact person: Guoliang Yu

March 12, 2009  
Gui-Qiang Chen, Northwestern University

Shock Reflection-Diffraction Phenomena, von Neumann’s Conjecture, and Nonlinear PDEs of Mixed Type

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.

Contact person:  

April 2, 2009  
Edward N. Wilson, Washington University, St. Louis

Subspaces Invariant under Lattice Translations

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.

Contact person: Akram Aldroubi

April 9, 2009  
James Serrin, University of Minnesota

Entire Solutions of Completely Coercive Quasilinear Elliptic Equations

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.

Contact person:  

April 30, 2009  
Ryszard Nest, University of Copenhagen

LOCATION: Room 1431

Spectral flow associated to KMS states with periodic KMS group action

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.

Contact person: Guoliang Yu


Colloquium Chair (2008-2009): John Ratcliffe

Back Home