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Colloquium Seminar Abstracts

University of Maryland
Colloquium Seminar Abstracts

(September 14) Michael Thaddeus: Wonderful compactifications of groups as moduli spaces of principal bundles — I will describe de Concini and Procesi's wonderful compactification of a semisimple group and survey some of its wonderful properties. In the case of PGL(n), I will explain how it can be viewed as parametrizing so-called complete linear maps, that is, sequences of linear maps from the kernel to the cokernel of the previous map. In the general case, as shown in joint work with Johan Martens, it can still be viewed as a moduli space, but of very different objects: principal bundles on chains of lines. This point of view enables us to extend the wonderful compactification to groups with nontrivial center, but at a price: instead of a smooth manifold, it is now an orbifold.

(September 28) Patrick Brosnan: An introduction to essential dimension — Essential dimension is a measure of the complexity of algebraic objects (for instance, field extensions, algebras, quadratic forms or curves of a given genus) introduced by Zinovy Reichstein and Joe Buhler. I'll explain some of the ``classical'' results, generalizations and some of the open problems in the area.

(October 21) David Fisher: Rigidity of group actions, cohomology and compactness — I will begin with a survey of the field of rigidity of large group actions, emphasizing some open questions and some recent progress. The main focus of the talk will be some very recent work on local rigidity of group actions that gives very simple proofs of some old and some new results.

(October 28) Ezra Getzler: A Filtration of Open/Closed Topological Field Theory — A well-known theorem states that there is an equivalence between two-dimensional topological field theories and commutative Frobenius algebras. This theorem is essentially due to Hatcher and Thurston. Using Morse theory, they proved that any pair of pants decompositions of a surface S may be joined by a sequence of moves of two types, respectively associated to embeddings in S of a surface of genus 0 with four holes, and of a surface of genus 1 with one hole.

Moore and Seiberg categorified this result: they gave a presentation of modular functors, that is, two-dimensional field theories taking values in a symmetric monoidal bicategory. They actually proved that on adjoining 2-cells associated to open embeddings in S of a surface of genus 0 with five holes, and of genus 1 with two holes, to Hatcher and Thurston's graph, one obtains a simply connected 2-dimensional cell complex.

In this talk, we show how the study of Teichmueller space shows that these results are the first in a sequence of generalizations, and explain some relations to results in algebraic geometry and other fields.

(November 2) Alex Eskin: Rational billiards and the SL(2,R) action on moduli space — I will discuss ergodic theory over the moduli space of compact Riemann surfaces and its applications to the study of polygonal billiard tables. There is an analogy between this subject and the theory of flows on homogeneous spaces; I will talk about some successes and limitations of this viewpoint. This is joint work with Maryam Mirzakhani.

(November 9) Marie Farge : Why and how do we use wavelets to study turbulence? — We will first define turbulence and explain some open problems. We will then recall what wavelets are and show how we use it to try solving some of those open problems.

(November 16) Tien-Yien Li: Mixed volume computation and solving polynomial systems — In the last few decades, the homotopy continuation method has been established in the U.S. for finding the full set of isolated zeros to a polynomial system numerically. The method involves fi rst solving a trivial system, and then deforming these solutions along smooth paths to the solutions of the target system. Recently, modeling the sparse structure of a polynomial system by its Newton polytopes leads to a major computational breakthrough. Based on an elegant method for computing the mixed volume, the new polyhedral homotopy can find all isolated zeros of a polynomial system much efficiently. The method has been successfully implemented and proved to be very powerful in many occasions, especially when the systems are sparse. We will elaborate the method in this talk.

(December 2) Jinchao Xu: Optimal and Practical Algebraic Solvers for Discretized PDEs — An overview of fast solution techniques (such as multi-grid, two-grid, one-grid and nil-grid methods) will be given in this talk on solving large scale systems of equations that arise from the discretization of partial differential equations (such as Poisson, elasticity, Stokes, Navier-Stokes, Maxwell, MHD, and black-oil models). Mathematical optimality, practical applicability and parallel (CPU/GPU) scalability will be addressed for these algorithms and applications.

(December 7) Sastry G. Pantula: DMS to THRIVE — This talk will discuss the funding and other opportunities at DMS and at NSF.

(February 8) Sergiu Klainerman: On the rigidity of black holes — The rigidity conjecture states that all regular, stationary solutions of the Einstein field equations in vacuum are isometric to the Kerr solution. The simple motivation behind this conjecture is that one expects, due to gravitational radiation, that general, dynamic, solutions of the Einstein field equation settle down, asymptotically, into a stationary regime. A well known result of Carter, Robinson and Hawking has settled the conjecture in the class of real analytic spacetimes. The assumption of real analyticity is however very problematic; there is simply no physical or mathematical justification for it. During the last five years I have developed, in collaboration with A. Ionescu and S. Alaxakis, a strategy to dispense of it. In my lecture I will these results and concentrate on some recent results obtained in collaboration with A. Ionescu.

(February 17) Peter Jones: Product formulas for positive measures and applications — We will discuss a simple product formula for general positive measures on Euclidean space. This formula was introduced (as far as I know) by R. Fefferman, C. Kenig, and J. Pipher) in their work on harmonic measure for certain elliptic PDE's. I will discuss applications to various problems. The first of these (joint work with D. Bassu, L. Ness, V. Rokhlin) is to analysis of volume-like signals that arise in communication networks. We will show how one various coefficients that come from the product formula can be used for classification. We then discuss how an arbitrary measure on the unit circle can be related to a (unique) curve in the plane. Certain classes of random curves arising in SLE processes are intimately related to this procedure (joint work with K. Astala, A. Kupiainen, E. Saksman). Finally we will present a few of the ideas that arise in recent work with M. Csörnyei on a seemingly unrelated problem. An old theorem of Rademacher states that a Lipschitz mapping from one Euclidean space to another is differentiable almost everywhere. We show that, in any dimension, given a set E of Lebesgue measure zero, there is a mapping from Rn to itself that is nowhere differentiable on E. This was previously known when n = 1 (where it is very simple) and n = 2 (which is a difficult recent result due to others). Along the way we need to introduce some new types of objects in the classical Calderon-Zygmund machinery.

(February 22) Michael Hintermüller: Semismooth Newton Methods: Theory, Numerics and Applications — Non-smooth operator equations arise in many practical applications in biomedical or engineering sciences as well as mathematical imaging or finance. In this talk, for the numerical solution of such problems a generalized Newton framework in function space is discussed. Relying on the concept of semismoothness, locally superlinear convergence of the associated Newton iteration is established and its mesh independent convergence behavior upon discretization is shown. The efficiency and wide applicability of the method is highlighted by considering constrained optimal control problems for fluid flow, contact problems with or without adhesion forces, phase separation phenomena relying on non-smooth homogeneous free energy densities and restoration tasks in mathematical image processing.

(March 14) Alexis Vasseur: De Giorgi methods applied to regularity issues in Fluid Mechanics — TBA

(March 30) Hakan Eliasson: Birkhoff Normal Form and a problem of Herman — We shall discuss some questions related to the Birkhoff Normal Form of an analytic Hamiltonian system near a Diophantine equilibrium. This is classical subject but several questions still remain open. We shall in particular report on a recent work (joint with B. Fayad and R. Krikorian) which is related to (but does not solve) a question raised by M. Herman in his ICM lecture in Berlin 1996: is such an equilibrium always accumulated by a set of KAM-tori of positive Lebesgue measure?

(April 25) Albert Fathi: Lyapunov Functions: Towards an Aubry-Mather theory for homeomorphisms? — This is a joint work with Pierre Pageault. For a homeomorphism h of a compact space, a Lyapunov function is a real valued function that is non-increasing along orbits for h. By looking at simple dynamical systems(=homeomorphisms) on the circle, we will see that there are systems which are topologically conjugate and have Lyapunov functions with various regularity. This will lead us to define barriers analogous to the well known Peierls barrier or to the Mań potential in Lagrangian systems. That will produce by analogy to Mather's theory of Lagrangian Systems an Aubry set which is the generalized recurrence set introduced in the 60's by Joe Auslander and a Mań set which is essentially Conley's chain recurrent set. Even if the abstract does not show it, the lecture will be at the level of a first year graduate student. All necessary concepts can be easily explained.

(May 2) Nigel Higson: Contractions of Lie Groups and Representation Theory — The contraction of a Lie group G to a closed subgroup is a Lie group, usually easier to analyze than G itself, that approximates G to first order near the subgroup. The terminology is due to the mathematical physicists, who examined the group of Galilean transformations as a contraction of the group of Lorentz transformations. My focus will be on a related but different class of examples, the prototype of which is the group of isometric motions of Euclidean space, viewed as a contraction of the group of isometric motions of hyperbolic space (the subgroup is the compact group of motions fixing a point). It is natural to expect some sort of approximative relation between representations of G and representations of its contraction. But in the 1970's George Mackey suggested that there is a much more rigid connection between the representations of the two groups. I shall formulate a reasonably precise conjecture that was inspired by subsequent developments in C*-algebra theory and noncommutative geometry, and describe the evidence in support of it, which is by now substantial. However a conceptual explanation of Mackey's rigidity phenomenon remains elusive.

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