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03/06/2004, 11:00 — 12:00 — Room P4.35, Mathematics Building

Carlos Florentino, *Instituto Superior Técnico*

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Geometry and spectral theory of modular curves.

Modular curves and modular forms are defined, and used to describe
some important problems in geometry and number theory. We also
present some results about the spectrum of the Laplacian on these
algebraic curves, viewed as open Riemann surfaces.

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27/05/2004, 10:30 — 11:30 — Room P4.35, Mathematics Building

Ana Bela Cruzeiro, *Instituto Superior Técnico*

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Derivation formulae for heat kernels associated with some elliptic
operators.

Derivation (Bismut type) formulae for the heat kernel of some
elliptic operators on Riemannian manifolds are obtained using the
stochastic calculus of variations on the path space of these
manifolds.

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20/05/2004, 10:30 — 11:30 — Room P4.35, Mathematics Building

Ana Bela Cruzeiro, *Instituto Superior Técnico*

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Probabilistic approach to the existence of fundamental solutions
associated to hypoelliptic operators.

We describe the probabilistic interpretation for the fundamental
solutions of heat equations through their corresponding diffusions.
A probabilistic approach to their study can be done through
Malliavin Calculus: an idea of the methods is given.

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13/05/2004, 10:30 — 11:30 — Room P4.35, Mathematics Building

Jorge Silva, *Instituto Superior Técnico*

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A theorem on the spectral asymptotics of
${P}^{1/m}$, on a compact manifold, where
$P$ is an elliptic, self-adjoint linear partial differential operator of order
$m$

We will use the methods reviewed in the previous lectures, in order
to study the asymptotical behavior of the eigenvalues of a
particular elliptic, self-adjoint operator.

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29/04/2004, 10:30 — 11:30 — Room P4.35, Mathematics Building

Jorge Silva, *Instituto Superior Técnico*

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Fourier Integral Operators: the construction of parametrices for
strictly hyperbolic Cauchy problems

After having seen how to construct approximate fundamental
solutions for elliptic linear PDEs, in the first lecture of this
series, we will now review the construction of similar approximate
solutions, this time for the Cauchy problem in hyperbolic problems,
using Fourier Integral Operators. These operators can be seen as a
generalization of pseudo-differential operators, and bring into the
picture a strong symplectic geometry component, related to the
hamiltonian evolution in phase-space: the cotangent bundle of space
points and Fourier frequencies.

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22/04/2004, 10:30 — 11:30 — Room P4.35, Mathematics Building

Jorge Silva, *Instituto Superior Técnico*

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A review of methods for obtaining approximate fundamental solutions
to linear PDEs: Pseudodifferential Operators and Fourier Integral
Operators

We will give a fast overview of the Fourier Analysis methods for
obtaining parametrices for linear partial differential operators
with non-constant coefficients. These are approximate fundamental
solutions in the sense that, modulo smooth functions, they produce
the Dirac measure when acted by the operator. These methods have
become fundamental tools for the microlocal analysis of
distributions, accurate control of singularities of solutions for
linear PDEs with variable coefficients and spectral analysis of the
same operators.

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30/05/2003, 14:30 — 15:30 — Room P4.35, Mathematics Building

Diogo Gomes, *Instituto Superior Técnico*

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Semiclassical approximations, Wigner measures and quantum
Aubry-Mather theory

We present some tools to study semiclassical limits of
Schrodinger-type equations and try to relate them with classical
mechanics and Mather measures.

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23/05/2003, 14:30 — 15:30 — Room P4.35, Mathematics Building

Pedro Freitas, *Instituto Superior Técnico*

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Open problems in spectral geometry III

In the last of this series of seminars, we will consider the study
of properties of eigenfunctions. Some of the (very few) known
results will be presented, together with several open problems,
namely, the nodal line and hot spots conjectures.

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16/05/2003, 14:30 — 15:30 — Room P4.35, Mathematics Building

Pedro Freitas, *Instituto Superior Técnico*

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Open problems in spectral geometry II

In this second seminar we will begin by reviewing isoperimetric inequalities for the case of surfaces, starting with Herschs classical result for the sphere and moving on to problems on surfaces of higher genus. Examples of particular interest are compact surfaces with constant curvature. We will then move on to the other end of the spectrum, and consider some problems related to the asymptotic behaviour of eigenvalues, focusing mainly on several aspects related to Weyl asymptotics. As an example of a different type of problems, we will also mention the asymptotic behaviour of the spectrum for the damped wave equation.

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09/05/2003, 14:30 — 15:30 — Room P4.35, Mathematics Building

Pedro Freitas, *Instituto Superior Técnico*

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Open problems in Spectral Geometry I

This is the first in a series of seminars in Spectral Theory, whose
purpose is to present a basic introduction to the subject, with
emphasis on current research topics and their relation to other
areas of Mathematics, namely, Partial Differential Equations,
Geometry, Probability Theory, Number Theory, etc. The presentation
will be based upon a collection of open problems, and these first
three lectures will cover the following topics: I. Isoperimetric
inequalities II. Asymptotic behaviour of eigenvalues III.
Properties of eigenfunctions