Analysis, Geometry, and Dynamical Systems Seminar   RSS

Past sessions

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07/05/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Hildebrando Rodrigues, Universidade de São Paulo (São Carlos)

Synchronization and Applications

23/04/2014, 15:00 — 16:00 — Room P3.10, Mathematics Building
Diogo Oliveira e Silva, Hausdorff Center for Mathematics da Universidade de Bona

On extremizers for Fourier restriction inequalities

This talk will focus on extremizers for a family of Fourier restriction inequalities on planar curves. It turns out that, depending on whether or not a certain geometric condition related to the curvature is satisfied, extremizing sequences of nonnegative functions may or may not have a subsequence which converges to an extremizer. We hope to describe the method of proof, which is of concentration compactness flavor, in some detail. Tools include bilinear estimates, a variational calculation, a modification of the usual method of stationary phase and several explicit computations.

08/04/2014, 14:00 — 15:00 — Abreu Faro Amphitheatre
Lloyd Demetrius, Harvard University and Max Planck Institute of Molecular Biology, Berlin

An entropic selection principle of evolutionary theory

The outcome of competition between populations of replicating entities is predicted by evolutionary entropy, a nonequilibrium analogue of the Gibbs-Boltzmann entropy in statistical thermodynamics. This talk will outline the mathematical basis of this selection principle and describe certain applications to the origin of metabolic diseases.

See also

04/06/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building
Christian Le Merdy, Université de Franche-Comté

Dilation of operators on $L^p$-spaces

Let $1\lt p \lt\infty$, let $(\Omega,\mu)$ be a measure space and let $T : L^p(\Omega)\to L^p(\Omega)$ be a bounded operator. We say that it admits a dilation (in a loose sense) when there exist another measure space $(\Omega',\mu')$, an invertible operator $U$ on ${L^p}(\Omega')$ such that $\{U^n:n\in\mathbb{Z} \}$ is bounded and two bounded operators $J : L^p(\Omega) \to L^p(\Omega')$ and $Q : L^p(\Omega')\to L^p(\Omega)$ such that $T^n=Q {U^n} J$ for any integer $n\ge 0$. When $p=2$, this property is equivalent to $T$ being similar to a contraction. The main question considered in this talk is to characterize operators with this property when $p\ne 2$. Our results give partial answers and strong connections with functional calculus properties. The talk will include motivation for this dilation question. (Joint work with C. Arhancet.)

14/05/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building
Filippo Cagnetti, University of Sussex

A new method for large time behavior of convex Hamilton-Jacobi equations

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

16/04/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building
Florin Radulescu, Università di Roma - Tor Vergata

Ramanujan-Petersson conjectures and Operator Algebras

Apoio do Instituto Cultural Romeno

02/04/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jorge Ferreira, Universidade Federal Rural de Pernambuco

On the asymptotic behaviour of nonlocal nonlinear problems

This lecture deals with nonlocal nonlinear problems. Our main results concern existence, uniqueness and asymptotic behavior of the weak solutions of a nonlinear parabolic equation of reaction-diffusion nonlocal type by an application of the Faedo-Galerkin approximation and Aubin-Lions compactness result. Moreover, we prove continuity with respect to the initial values, the joint continuity of the solution and a result on the existence of the global attractor for the problem \[ \begin{cases} u_{t}-a(l(u))\Delta u+| u|^{\rho }u=f(u) \phantom{.} \text{ in} \phantom{.} \Omega \times (0,T), \\ u(x,t)=0 \phantom{.} \text{ on }\phantom{.} \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) \phantom{.}\text{ in }\phantom{.} \Omega , \end{cases} \] when $0\lt \rho \le 2/(n-2)$ if $n\ge 3$ and $0\lt \rho \lt \infty $ if $n=1,2$, where $u=u(x,t)$ is a real valued function, $\Omega \subset \mathbb{R}^{n}$ is a bounded smooth domain, $n\ge 1$ with regular boundary $\Gamma =\partial \Omega $, $p\geq 2$. Moreover, $a$ and $f$ are continuous functions satisfying some appropriate conditions and $l: L^{2}(\Omega )\to\mathbb{R}$ is a continuous linear form.

14/12/2012, 11:00 — 12:00 — Room P3.10, Mathematics Building
Rafael de la Llave, Georgia Tech.

Manifolds on the verge of a regularity breakdown

There are two main stabibility arguments for solutions in dynamical systems: the theory of normal hyperbolicity and the Kolmogorov Arnold Moser theory of perturbations. In recent times, there has been progress in developing versions of the theory that are well suited for computations. The theory does not require that the system is close to integrable, but rather uses geometric identities. The theorems prove that approximate solutions satisfying some non-degeneracy assumptions correspond to a true solution. Furthermore, the proofs lead at the same time to very efficient algorithms. Implementing these algorithms, leads to some conjectural insights on the phenomena that happen at breakdown. They turn out to be remarkably similar to phenomena that were observed in phase transition and "renormalization group" provides a unifying point of view. Nevertheless, many questions remain open.

27/11/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Jinjun Li, Instituto Superior Técnico

Topological entropy of refined irregular sets

16/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
David Owen, Carnegie Mellon University

Multiscale Geometrical Changes in Continua: Structured Deformations and Some Questions in Analysis

Classes of injective deformations of a continuous body that are stable under composition and under the taking inverses are central to the description of geometrical changes of a continuous body at both the macroscopic and microscopic levels. Classical, simple, and structured deformations are described and assessed from the point of view of continuum mechanics and of variational problems. The natural way of assigning an energy to a structured deformation, through relaxation of an energy assigned to a smaller class of deformations, leads to various alternative, "variationally friendly" notions of structured deformations. Several of these alternative notions are examined from the point of view of approximation by deformations in a smaller class and of relaxation of an initial energy defined on the smaller class of deformations. Questions are raised as to whether or not different notions of structured deformations lead to different relaxed energies for a structured deformation that satisfies the defining requirements of two or more notions. A recent example of an explicit formula for a relaxed energy is used to illustrate these questions.

10/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Roman Hric, Matej Bel University, Banska Bystrica, Slovakia

Dense orbits of homeomorphisms, flows and their time maps

24/04/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Henrique Oliveira, Instituto Superior Técnico

New Results on the Collatz Problem

03/04/2012, 14:00 — 15:00 — Room P3.10, Mathematics Building
Saber Elaydi, Trinity University, San Antonio, USA

Application of singularity theory in planar discrete dynamical systems and applications to competition models

13/03/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
George Sell, University of Minnesota

How to use numerical techniques to study the dynamics inside the attractor

16/02/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
José Ferreira Alves, Universidade do Porto

Gibbs-Markov structures vs statistical properties in dynamical systems

In a classical approach to dynamical systems one frequently uses certain geometric structures of the system to deduce statistical properties, such as invariant measures with stochastic-like behaviour, large deviations or decay of correlations. Such geometric structures are generally highly non-trivial and thus a natural question is the extent to which this approach can be applied. We show that in many cases stochastic-like behaviour itself implies that the system has certain geometric properties, which are therefore necessary as well as sufficient conditions for the occurrence of the statistical properties under consideration.

18/10/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Bernold Fiedler, Freie Universität Berlin

Schoenflies spheres in Sturm attractors

22/06/2011, 16:00 — 17:00 — Room P3.10, Mathematics Building
Diogo Oliveira e Silva, University of California, Berkeley

On trilinear oscillatory integrals

We examine a certain class of trilinear integral operators which incorporate oscillatory factors e iP, where P is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations. Tools include sublevel set estimates, higher dimensional versions of van der Corput's lemma and corresponding multilinear analogues. This is joint work with Michael Christ.

24/05/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Felipe Rivero, Universidad de Sevilla

Structure stability of pullback attractors

Attractors help us to understand the dynamics of a system thanks to the asymptotic information given about the solutions. But any mathematical object needs a robust definition, that is, needs consistence under perturbations, because any physical model might experiment any kind of natural or artificial changes. Our aim in this talk is to show how a gradient-like pullback attractor has a stable structure under small perturbations and show some nontrivial examples where the perturbation can be found in the time dependent operator or in the nonlinearity.

15/03/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building
Bernold Fiedler, Institute of Mathematics, Free University of Berlin

Negative Bendixson-Dulac criteria in several dimensions? A chemostat example

The negative criterium of Bendixson and Dulac states that area contracting planar vector fields cannot possess periodic orbits. A physics student asked in class why this would not generalize to three dimensions, obviously, using the curl.

We give a higher-dimensional example, arising from chemostats, where a suitable differential form excludes periodicity. This is joint work with Sze-Bi Hsu.

07/12/2010, 15:00 — 16:00 — Room P3.10, Mathematics Building
Manuel de León, ICMAT-Consejo Superior de Investigaciones Cientificas

Generalized Hamiltonian systems and applications to nonholonomic mechanics

Over twenty years ago Alan Weinstein proposed to formulate mechanics in the framework of Lie algebroids. The use of Lie algebroids in mechanics has recently permitted two unexpected applications: the development of a Hamilton--Jacobi theory and a simple reduced scheme for nonholonomic mechanical systems.

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