### 02/12/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Ioannis Parissis, *KTH, Stockholm*

### Logarithmic dimension bounds for the maximal function along a
polynomial curve

Let $\mathcal{M}$ denote the maximal function along the polynomial
curve $({\gamma}_{1}t,\dots ,{\gamma}_{d}{t}^{d})$:
$$\mathcal{M}(f)(x)=\underset{r>0}{\mathrm{sup}}\frac{1}{2r}{\int}_{\mid t\mid \le r}\mid f({x}_{1}-{\gamma}_{1}t,\dots ,{x}_{d}-{\gamma}_{d}{t}^{d})\mid \mathrm{dt}.$$We show that the
${L}^{2}$ norm of this operator grows at most logarithmically with the
parameter $d$: $$\parallel \mathcal{M}f{\parallel}_{{L}^{2}({\mathbb{R}}^{d})}\le c\mathrm{log}d\parallel f{\parallel}_{{L}^{2}({\mathbb{R}}^{d})},$$where $c>0$ is an absolute
constant. The proof depends on the explicit construction of a
"parabolic" semigroup of operators which is a mixture of stable
semigroups.

### 23/09/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Henry van Roessel, *University of Alberta, Edmonton, Canada*

### Some exact solutions to the coagulation equation

### 17/07/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Roman Hric, *Matej Bel University and Slovak Academy of Sciences, Banska Bystrica, Slovakia*

### Dense orbits and some misunderstandings around them

### 24/06/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Lennard Bakker, *Brigham Young University, Salt Lake City*

### The Conjugacy Problem for Torus Automorphisms

### 05/06/2008, 11:00 — 12:00 — Room P3.10, Mathematics Building

Rachid El Harti, *University Hassan I, Morocco*

### On Pro-C*-algebras

### 22/04/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jonathan Wattis, *University of Nottingham*

### The effects of grinding chiral crystals and similarity solutions of
Becker-Doring equations

### 15/02/2008, 11:00 — 12:00 — Room P3.10, Mathematics Building

Orlando Lopes, *IMECC, Universidade de Campinas*

### Simetria radial de minimizadores de problemas variacionais com
termos não locais

### 12/02/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Carlos Tomei, *PUC, Rio de Janeiro*

### The subtle convergence of Wilkinson's method

Wilkinson's iteration is frequently used to compute eigenvalues of
symmetric matrices. Decades of experience led to believing that the
algorithm performed extremely fast, Indeed, recently Nicolau
Saldanha (PUC, Rio de Janeiro), Ricardo Leite (UFES) and I proved
that this is so, for matrices whose spectrum does not contain three
eigenvalues forming an arithmetic progression. Things may go
slightly slower otherwise. The argument uses techniques from the
theory of completely integrable systems, a new class of inverse
variables for tridiagonal matrices and the construction of some
Lyapunov functions. The counter-examples arise from an unexpected
property related to the iteration of a discontinuous function on
the plane.

### 29/01/2008, 15:00 — 16:00 — Room P3.10, Mathematics Building

Rafael Souza, *Universidade Federal do Rio Grande do Sul*

### Stationary Markov Chains on $[\mathrm{0,1}]$ that maximize a Potential:
Unicity and a L.D.P.

Given a continuous potential $A:[\mathrm{0,1}{]}^{\mathbb{N}}\to \mathbb{R}$,
defined in the Bernoulli space $\Omega =[\mathrm{0,1}{]}^{\mathbb{N}}$, where
$[\mathrm{0,1}]=\{x\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}0\le x\le 1\}\subset \mathbb{R}$, and supposing
that $A({x}_{1},{x}_{2},{x}_{3},...)=A({x}_{1},{x}_{2})$ depends only on the two first
coordinates, we are interested in finding stationary Markov
probabilities ${\mu}_{\mathrm{\infty}}$ on $[\mathrm{0,1}{]}^{\mathbb{N}}$ that maximize the
value $\int A(x,y)d\mu $, among all stationary Markov
probabilities $\mu $ on $\Omega =[\mathrm{0,1}{]}^{\mathbb{N}}$. This problem
corresponds in Statistical Mechanics to the zero temperature case
for the interaction described by the potential $A$. The main
purpose is to show, under the hypothesis of uniqueness of the
maximizing probability, a Large Deviation Principle for a family of
absolutely continuous Markov probabilities ${\mu}_{\beta}$ which weakly
converges to ${\mu}_{\mathrm{\infty}}$. The probabilities ${\mu}_{\beta}$ are
obtained via an information we get from a Perron operator and they
satisfy a variational principle similar to the pressure in
Thermodynamic Formalism. Considering that $A$ depends only on the
first two coordinates, much of the work can be done using measures
defined in the space $[\mathrm{0,1}{]}^{2}$. We also show that, in the sense of
Mañé, the maximizing probability is unique. It means that, if we
perturb the observable $A(x,y)$ by adding a second term that
depends only on the variable $x$, and could be seen as a magnetic
term, then generically we have the unicity of the maximizing
probability. If we consider $\sigma $-invariant measures in
$[\mathrm{0,1}{]}^{\mathbb{N}}$, where $\sigma $ is the shift map
$\sigma (({x}_{1},{x}_{2},{x}_{3},...))=({x}_{2},{x}_{3},...)$, then stationary Markov
probabilities are a special case of $\sigma $-invariant measures. We
show that the maximizing problem, now performed in a much larger
class, still attains its maximum in a stationary Markov
probability. The unicity of maximizing measures still remains, but
only after projection on $[\mathrm{0,1}{]}^{2}$.

### 24/01/2008, 11:00 — 12:00 — Room P3.10, Mathematics Building

Joana Mohr, *Universidade Federal do Rio Grande do Sul*

### Large deviation principle for the Mather measure

We present the rate function and a Large Deviation Principle for
the Entropy Penalized Method when the Mather measure is unique.
More explicitly, under some natural assumptions about the
Lagrangian $L(x,v)$, $x$ in the torus ${\mathbb{T}}^{N}$, there
exists a sequence of measures ${\mu}_{\epsilon ,h}$ converging to the
Mather measure $\mu $, when $\epsilon ,h\to 0$. We show a LDP of the
kind ${\mathrm{lim}}_{\epsilon ,h\to 0}\epsilon \mathrm{ln}{\mu}_{\epsilon ,h}(A),$ where
$A\subset {\mathbb{T}}^{N}\times {\mathbb{R}}^{N}$. The measures
${\mu}_{\epsilon ,h}$ minimizes the entropy penalized problem:
$$\mathrm{min}\{{\int}_{{\mathbb{T}}^{N}\times {\mathbb{R}}^{N}}L(x,v)d\mu (x,v)+\epsilon S[\mu ]\},$$where the entropy $S$ is
given by $$S[\mu ]={\int}_{{\mathbb{T}}^{N}\times {\mathbb{R}}^{N}}\mu (x,v)\mathrm{ln}\frac{\mu (x,v)}{{\int}_{{\mathbb{R}}^{N}}\mu (x,w)\mathrm{dw}}\mathrm{dxdv}.$$and the minimization is performed over the space of probability
densities on ${\mathbb{T}}^{N}\times {\mathbb{R}}^{N}$ that satisfy the
holonomic constrain. We also show some dynamical properties of the
discrete time Aubry-Mather problem.

### 20/11/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building

Hossein Tehrani, *University of Nevada, Las Vegas*

### New and old results on semilinear elliptic equations with logistic
type nonlinearities and harvesting

### 13/11/2007, 15:00 — 16:00 — Room P3.10, Mathematics Building

Julien Keller, *Institute for Mathematical Sciences, Imperial College*

### Kähler-Ricci flow: infinite and finite dimensional approach

In Kähler geometry, the Kähler-Ricci flow is a powerful tool to
study the existence of Kähler-Einstein metrics. I will discuss how
one can think of the Kähler-Ricci flow in terms of so-called
balanced metrics, introduced by S. K. Donaldson. In the
case of toric Fano-Einstein manifolds, we obtain a simple algorithm
to compute an approximation of the Kähler-Einstein metric. For
manifolds with negative first Chern class, the algorithm is related
to the work of H. Tsuji. Finally, our work is related to the
problem of G. I. T stability for a smooth projective
manifold.

### 09/10/2007, 14:30 — 15:30 — Room P3.10, Mathematics Building

Bernold Fiedler, *Freie Universität Berlin*

### Planar attractors of Sturm type: dynamics and graph theory

### 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
${H}^{s}$-scale

The Cauchy problem for the cubic nonlinear Schrödinger equation $${\mathrm{iu}}_{t}+{u}_{\mathrm{xx}}=\mid u{\mid}^{2}u,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u(x\mathrm{,0})={u}_{0}(x)$$ is considered, with
emphasis on the case of one space dimension, i.e., $x\in R$. 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

Benjamin Steinberg, *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

Radoslaw Czaja, *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.