###
09/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques, Paris*

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Arithmetics: motives, periods and motivic Galois group

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08/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques, Paris*

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Geometry: sheaves, topos and fields

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05/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques, Paris*

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Geometry: the notions of spectrum and scheme

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08/05/2009, 14:00 — 15:00 — Room P4.35, Mathematics Building

Emma D'Aniello, *Seconda Universitá degli Studi di Napoli*

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Topological dynamical systems and odometers

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10/03/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Alex Vladimirsky, *Cornell University*

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Homogenization and multiobjective optimization - computational
challenges

I will present two recent projects related to front propagation and
optimal control.
The first of these (joint with A. Oberman and R. Takei) deals
with 2-scale and 3-scale computations in geometric optics. We
propose a new and efficient method to homogenize first-order
Hamilton-Jacobi PDEs. Unlike the prior cell-problem methods, our
algorithm is based on homogenizing the related Finsler metric. We
illustrate by computing the effective velocity profiles for a
number of periodic and "random" composite materials.

The second project (joint with A. Kumar) deals with multiple
criteria for optimality (e.g., fastest versus shortest
trajectories) and optimality under integral constraints. We show
that an augmented PDE on a higher-dimensional domain describes all
Pareto-optimal trajectories. Our numerical method uses the
causality of this PDE to approximate its discontinuous viscosity
solution efficiently. The method is illustrated by problems in
robotic navigation (e.g., minimizing the path length and exposure
to an enemy observer simultaneously).

###
17/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Andreas Döring, *Imperial College London*

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A topos approach to quantum theory

I will report on work with Chris Isham on the application of topos
theory to physics. The Kochen-Specker theorem shows that a naive
realist description of quantum systems is impossible. This can be
understood as the inapplicability of Boolean logic to quantum
systems. In order to arrive at a more realist description, one can
use the internal logic of a certain topos of presheaves. The choice
of this topos is directly motivated from the Kochen-Specker
theorem. I will show which structures within this topos are of
physical significance and how propositions about physical
quantities are assigned truth-values. The relation with
constructive Gel'fand duality is sketched.

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10/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Alexandre Baraviera, *Universidade Federal do Rio Grande do Sul*

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Iterated function systems, classical and quantum

An iterated function system (IFS) is a collection of maps that are
chosen and iterated according to some probability. The geometric
and measurable structure of the limit set is usually very
interesting and studied with the use of the well known technique of
transfer operator. In this talk we will review some results about
this problem and its quantum version, where the maps are unitary
operators acting on some suitable Hilbert space.

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03/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Florin Radulescu, *Università di Roma-Tor Vergata*

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Type
${\mathrm{II}}_{1}$ Von Neumann Representations for Hecke Operators on Maass Forms and Inequalities for their Eigenvalues

We prove that classical Hecke operators on Maass forms are a
special case of completely positive maps on II${}_{1}$ factors,
associated to a pair of isomorphic subfactors. This representation
induces several matrix inequalities on the eigenvalues of the Hecke
operators on Maass forms. These inequalities are the consequence of
a "double" action of the Hecke algebra which can be seen only in
the type II${}_{1}$ representation. The classical Hecke operators are
then a "diagonal" of this double action.

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27/01/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Josep Sardanyés, *Complex Systems Lab, Universitat Pompeu Fabra, Barcelona*

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Theoretical models of cooperation with ODEs: from prebiotic
evolution to complex ecosystems

The dynamics of cooperation can be studied using mean field models
given by nonlinear ordinary differential equations (ODEs). Such a
dynamics is modeled considering heterocatalytic feedback loops,
with density-dependent enhancement of reproduction rates between
cooperating replicators. These models are used to study the
dynamics of catalytic networks named hypercycles which have a
particular graph architecture and which has been suggested to be of
importance in earlier stages of prebiotic evolution. In this talk
we will introduce the mathematical formalism used for the dynamical
study of such networks, focusing on the origin of life problem and
on the dynamics of ecological systems. We will analyze several
low-dimensional catalytic networks providing their stability and
bifurcation scenarios. We will also discuss some interesting
dynamical phenomenon associated to the bifurcations implying the
transition from survival to extinction phases. We will then also
compare the results obtained with the ODEs for these kind of
networks with other computational tools typical of complex systems
analysis.

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13/01/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building

Artur Lopes, *Universidade Federal do Rio Grande do Sul*

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Transport and the involution kernel in Ergodic Theory

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

Ioannis Parissis, *KTH, Stockholm*

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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*

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Some exact solutions to the coagulation equation

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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*

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Dense orbits and some misunderstandings around them

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

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

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The Conjugacy Problem for Torus Automorphisms

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

Rachid El Harti, *University Hassan I, Morocco*

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On Pro-C*-algebras

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

Jonathan Wattis, *University of Nottingham*

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

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

Orlando Lopes, *IMECC, Universidade de Campinas*

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Simetria radial de minimizadores de problemas variacionais com
termos não locais

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

Carlos Tomei, *PUC, Rio de Janeiro*

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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.

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

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

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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 $[\mathrm{0,1}{]}^{2}$.

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

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

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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.