Analysis, Geometry, and Dynamical Systems Seminar   RSS

Past sessions

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02/10/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Axel Grünrock, University of Wuppertal

On well-posedness theory for nonlinear wave equations aside from the Hs -scale

The Cauchy problem for the cubic nonlinear Schrödinger equation iu t+u xx=u 2 u,u(x,0 )=u 0 (x) is considered, with emphasis on the case of one space dimension, i.e., xR. We review the meanwhile classical results of Tsutsumi and Cazenave/Weissler concerning data in the standard Sobolev spaces H s(R n). Two invariance properties of the equation---scaling and Galilean invariance---lead to certain obstructions to (time local) well-posedness. A related counterexample of Kenig, Ponce, and Vega showing ill-posedness is sketched. As a consequence, in the one-dimensional case the restriction to H s(R)-data seems to be inadequate. A generalization of the standard theory due to the author is presented. Finally, we discuss several other canonical dispersive equations, such as KdV, mKdV, and DNLS, for which very similar problems appear.

10/07/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Denis Bonheure, Université de Louvain-la-Neuve

Symmetry breaking in Moser-Trudinger inequalities and a Hénon type problem in dimension two

In this talk, we discuss a Hénon type functional with an exponential nonlinearity in dimension two. To study the symmetry properties of maximizers, we establish a link with Moser-Trudinger inequalities and consider the symmetry properties of the extremal functions for these inequalities.

28/06/2007, 11:00 — 12:00 — Room P3.10, Mathematics Building
Frédéric Paugam, Institut de Mathématiques de Jussieu, Paris

A survey of the geometry of the functional equation of Riemann's zeta function

27/06/2007, 15:00 — 16:00 — Room P4.35, Mathematics Building
Frédéric Paugam, Institut de Mathématiques de Jussieu, Paris

Noncommutative geometry and number theoretical dynamical systems

26/06/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Carleton University, Canada

Spectral computations for self-similar groups

Self-similar groups, also known as automaton groups, have recently been popularized by the Grigorchuk school. They are groups acting on Cantor sets in such a way that their structure mimics the self-similarity of the Cantor set. Many interesting problems have been solved using self-similar groups including Milnor's problem on growth, Day's problem on elementary amenable groups, Hubbard's twisted rabbit problem, to name a few.

Grigorchuk and Zuk used a self-similar action of the lamplighter group to compute the spectral measure for the simple random walk with respect to its self-similar generating set. This led to a counterexample to the strong form of Atiyah's conjecture on L 2 -betti numbers. Dicks and Schick generalized the result to wreath products G wr Z with G a finite group and with "analogous" generators using a different technique. We recover Dicks and Schick's results using the original method of Grigorchuk and Zuk. In the process we clarify the relationship between the Kesten spectral measure and a spectral measure introduced by Grigorchuk and Zuk for self-similar groups. This is joint work with Mark Kambites and Pedro V. Silva.

19/06/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Transversality in Scalar Reaction-Diffusion Equations on a Circle

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction-diffusion equations on a circle \( u_t=u_{xx}+f(x,u,u_x),\ t\in\mathbb{R},\ x\in S^1, \) always intersect transversally. The argument also shows that for a periodic orbit there are no homoclinic connections. The main tool used in the proofs is Matano's zero number theory dealing with the Sturm nodal properties of the solutions.

12/06/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alfonso Sorrentino, Princeton University

On the total disconnectedness of the quotient Aubry set

In Mather's studies on the existence of Arnold diffusion, it turns out that it might be useful to understand certain metric aspects of what is called the quotient Aubry set. In particular, it seems to be interesting to know whether this set has a "small" dimension. We prove that under suitable hypotheses on the Lagrangian, the associated quotient Aubry set, corresponding to a certain cohomology class, is totally disconnected, i.e., every connected component consists of a single point. We will also discuss the relation between this problem and a Morse-Sard-like property for (difference of) critical subsolutions of Hamilton-Jacobi equation.

06/06/2007, 11:00 — 12:00 — Room P3.10, Mathematics Building
, Purdue University

Bundles, Operator Algebras and K-theory

The bundles that occur naturally in functional analysis are not necessarily locally trivial. Dixmier and Douady have shown that a continuous bundle with fiber an infinite dimensional separable Hilbert space H over a compact contractible metric space does not need to be trivial, even though the unitary group of H is contractible. Similar phenomena appear in the theory of operator algebras. The operator algebras with Hausdorff primitive spectrum are known to be isomorphic to algebras of sections in certain continuous bundles which are typically not locally trivial. We plan to give a gentle introduction illustrated by examples to the K-theory methods used in the study of these bundles.

05/06/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, University of Nevada, Reno

On Hausdorff Measures and KMS States

We explore an intriguing correspondence between Hausdorff measures arising from self-similar metrics and KMS states. This correspondence identifies the Hausdorff dimension of the space with the inverse temperature of the KMS state.

This talk is partially based on joint work with Jean Renault.

17/05/2007, 11:00 — 12:00 — Room P3.10, Mathematics Building
Rachid El Harti, University Hassan I, Morocco

Operator Algebras and Amenability

24/04/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Emma D'Aniello, Seconda Universitá degli Studi di Napoli

Chaos, periodic orbits and ω-limit sets

06/03/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Technische Universität München

On a new class of equations arising in the modelling of biofilms

23/02/2007, 11:00 — 12:00 — Room P3.10, Mathematics Building
Jacob Palis, IMPA, Rio de Janeiro

A Global Scenario for Chaotic Systems from Poincaré to Present Time

In simple and reasonably nontechnical language, I shall discuss Poincaré's viewpoint that a key question in dynamics was to describe for most systems their global orbit structure. Poincaré suggested the question more than one hundred years ago and it is still a major question in the area. I will include in the discussion the important topic of homoclinic bifurcations, which curiously played a dramatic role in Poincaré's essay that led him to receive a prize from King Oscar II of Sweden.

13/02/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
David Iglesias Ponte, Instituto de Matemáticas y Física Fundamental, CSIC-Madrid

New geometric techniques in mechanics

We will develop a geometric description of Lagrangian Mechanics on Lie algebroids (generalizing Klein's formalism on the tangent bundle). The main motivation comes from reduction theory of Lagrangian systems. If time permits we will explain how Lie groupoids appear in discrete mechanics and the relation between both theories.

08/02/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Gonzalo Contreras, CIMAT, México

A generic property of families of Lagrangian systems

We prove that a generic Lagrangian has finitely many minimizing measures in all the cohomology classes. In particular, for a generic Lagrangian, the quotient Aubry set is finite for all cohomology classes.

30/01/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Universidade Federal do Rio Grande do Sul, Brazil

Holonomic Probabilities and Ergodic Optimization

23/01/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jan Cholewa, University of Silesia, Katowice, Polónia

Dissipative equations in locally uniform spaces

Semilinear damped wave equations, partly dissipative systems and reaction diffusion equations in N are considered in the locally uniform spaces under the assumptions on the nonlinear term similar to those in bounded domains.

07/12/2006, 14:00 — 15:00 — Room P4.35, Mathematics Building
, Universität Wien

Continuity properties of pressure and entropy for piecewise monotone interval maps

05/12/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Andrey Biryuk, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Analysis of a pressureless dynamical system and an open geometrical problem

We consider the pressureless Euler equation equipped with periodic boundary conditions. The criterion for the global smooth solvability can be given in terms of degeneracy of the initial state: the Jacoby Matrix must be everywhere nilpotent. A simple purely geometrical reformulation of the above condition is possible in the 2D case. The question is open for higher dimensions. An application to turbulence is also discussed.

29/11/2006, 15:00 — 16:00 — Room P4.35, Mathematics Building
Francisco Balibrea, Universidade de Murcia

Geometric unfolding of some difference equations

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