# Spectral Theory Seminar

## Past sessions

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### Open problems in spectral geometry II

In this second seminar we will begin by reviewing isoperimetric inequalities for the case of surfaces, starting with Herschs 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.

### 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