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Analysis, Geometry, and Dynamical Systems Seminar   RSS

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19/10/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Imperial College, London

The geometry of Penrose tilings: projection and substitution

In the early 1970's, R. Penrose constructed a set of two tiles that can tile the plane only nonperiodically. His proof uses a renormalization argument based on the existence of substitution rules. In 1981, N. de Bruijn showed that Penrose tiling also can be obtained by projection of a discrete plane in R 5 (with vertices in Z 5 ) to the nearest two-dimensional hyperplane. In this talk we show that projection tilings of this type admit (a countable infinity of different) substitution rules if and only if there exists a "quadratic" (partially) hyperbolic lattice automorphism that commutes with the projection. As the latter condition is very easy to verify, we obtain a simple characterization (and many new examples) of such renormalizable projection tilings. This is joint work with Edmund Harriss.

12/10/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Katrin Gelfert, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Multifractal analysis for Lyapunov exponents

28/09/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Attractors for Semilinear Parabolic Equations on the Circle

27/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Carnegie Mellon University, Pittsburgh

Second order variational problems

20/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Giovanni Leoni, Carnegie Mellon University, Pittsburgh

Optimal regularity for free boundary problems and a conjecture of De Giorgi

08/07/2004, 11:00 — 12:00 — Room P3.10, Mathematics Building
, Universidade de São Paulo

Discretization of Unidimensional Functional Partial Equations

06/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Columbia University

A classification of affine smooth spherical varieties

F. Knop has translated Delzant's conjecture (1990) on compact multiplicity free spaces (also known as noncommutative completely integrable systems) into a conjecture on affine smooth spherical varieties. If G is a reductive algebraic group (over C ) and X is an affine G -variety (over C ), then X is called spherical if its coordinate ring C [X] is multiplicity free as a G -module. Knop's conjecture states that if moreover X is smooth, the module structure of C [X] determines the G -variety X up to isomorphism. As a first step towards proving the conjecture, we have classified these varieties, "up to pesky tori and connected components".

29/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Pierre Martinetti, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Renormalization and Hopf algebras

22/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Philippe Monnier, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

A cohomology associated to a function

15/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Stability of nonautonomous differential equations

01/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, University of Bristol

Ergodic theory and the distribution of n 2 x modulo one

I will discuss some recent results on the distribution of certain unipotent orbits on hyperbolic surfaces and their implication on the randomness of the fractional parts of the sequence n 2 x .

18/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
João Lopes Dias, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Floquet solutions for linear ODEs with quasiperiodic coefficients

We investigate the local reducibility of linear analytic multifrequency cocycles on SL(2,R) using renormalisation. In this way we show that some linear ODEs with coefficients depending quasiperiodically on time can be conjugated to others with constant coefficients. Much in the same way as Floquet theory for periodic coefficients.

11/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

A lower bound to the spectral threshold in curved tubes

Motivated by the theory of quantum waveguides, we consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. This is a joint work with Pavel Exner and Pedro Freitas.

04/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Université Paris 13

Dispersion for Schrödinger equations with variable coefficients and applications

27/04/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Generalized theta functions

20/04/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Integral equations, positive matrices, and reproducing kernel inequalities

05/04/2004, 16:00 — 17:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Godeaux surfaces with an involution

30/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Instituto Superior Técnico

Ergodic theory and cohomology

23/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
José Francisco Rodrigues, Universidade de Lisboa

The N-Membranes Problem for Nonlinear Degenerate Systems

We consider the elliptic variational inequality associated with quasilinear nonlinear systems, including those of p-Laplacian type, and with the ordering constraint of the N-membrane problem. We extend the Lewy-Stampacchia inequalities for the solution, obtaining new regularity results for the derivatives of the solution, including in the case of linear operators ( p=2) integrability of second derivatives for all q>1. Considering the N-membrane problem as coupled (N-1)-obstacle problem we obtain also the corresponding conditions for the stability of the coincidence sets for the variation of external forces.

16/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
Miguel Ramos, Universidade de Lisboa

Asymptotic properties of the ground-state solutions of singularly perturbed elliptic Hamiltonian systems

We study the shape of the positive solutions of an elliptic system of the form
- ε2 Δu+u=g(v),- ε2 Δv+v=f(u)

as the parameter ε goes to zero, namely the appearance of "spike-layer patterns" for the solutions of the system having minimal energy. Here the nonlinear terms are assumed to have superlinear and subcritical growth at infinity and we consider both cases of Neumann and Dirichlet boundary conditions over a given bounded open set Ω Rn . The proofs combine standard estimates in elliptic equations with new ideas in the calculus of variations (variational methods and Morse theory). This is joint work with J. Yang (to appear in Trans. Amer. Math. Soc.) and with A. Pistoia (to appear in J. Differential Equations).

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