Courses

Lie Symmetries of Differential and Difference Equations

1. Lie symmetries of differential equatios and their extensions and generalizations 2. Lie point symmetries of Difference Equations; their derivation and their applications.
3. From Point Symmetries to Generalized Symmetries for Difference Equations 4. Generalized Symmetries from the Integrability of Difference Equations 5. Formal Symmetries and Integrable Lattice Equations
 

TIME SERIES ANALYSIS

1. We will recall the basic principles of applied and numerical Fourier analysis: Fourier series and transform, energy and power spectrum, mutual and autocorrelation and their numerical computation.
2. Impulse and harmonic response of a system.
3. Filtering of a time series.
4. Time series as sampling of a continuous signal.

Out of equilibrium quantum statistical mechanics

Quantum statistical mechanics describes maximum entropy states in equilibrium ensembles, but the question how quantum systems evolving under unitary dynamics relax to equilibrium remains to a large extent open. We will review recent progress in understanding the emergence of statistical mechanics from microscopic quantum dynamics. After a general introduction to different approaches to out of equilibrium quantum physics, we will focus on two types of problems: quantum quenches and quantum transport in one dimensional lattice systems. Quantum quenches are abrupt changes of dynamical parameters in a quantum system. We will show that a quantum quench in the transverse field Ising model leads to local relaxation to a non-thermal statistical ensemble. Next we will study the problem of quantum transport when a system that is initially split in two halves at different temperature, is joined and evolves under a common Hamiltonian. We will show that in quantum spin chains the system approaches a non-equilibrium steady state.
 

Mixing of Markov Chains on Manifolds

Lecture 1 
Mixing Properties of Hamiltonian Monte Carlo
 
Abstract: Hamiltonian Monte Carlo (HMC) and its variants are extremely popular Markov chain Monte Carlo algorithms in statistical computing. In practice, they seem to outperform classical algorithms such as the Gibbs sampler and the Metropolis-Hastings algorithm. Unfortunately, the characteristic long moves made by HMC also make it difficult to analyze. I will survey some existing results on HMC and competitor algorithms, including some recent results of myself and Mangoubi, and present open problems.
 
Lecture 2
Convex Sets, Conductance and Curvature.
 
Abstract: Geometric random walks are popular tools for sampling from the interior or boundary of convex sets. I will introduce some popular geometric random walks and describe the mathematical techniques developed by Dieker, Kannan, Lovasz, Vempala and others to analyze them. I will then describe how some recent work takes advantage of ideas from geometry and convex optimization to propose and analyze algorithms that can have improved performance.  
  
 Lecture 3
Kac's Walks and Couplings.
 
Abstract: Kac's walks on the sphere and on the special orthogonal group. introduced in 1953 and 1970, have long histories in the statistical physics and computational statistics literatures. I will describe the history of these walks and review some of the many results on the mixing properties of these processes. I then present some recent work of myself and Pillai, which uses connections to random matrix theory to obtain substantially improved bounds on the mixing times of these two walks. 



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Introduction to the Finite Elements Method

The purpose of this course is to give a brief introduction to the Finite ELements Method (FEM),
a gold standard for the numerical solution of PDEs systems. 
Oddly enough, the widespread use of the FEM is not accompained by an adeguate knowledge of the mathematical framework underlying the method.
 
This course, starting from the weak formulation of balance equations, will give an overview of 
the techniques used to reduce a differential problem into an algebraic one.
During the course, some selected problems in mechanics and physics will be solved, 
covering the three main types of equations: elliptic, parabolic and hyperbolic.  
 
The course will cover the following topics:
- Applied Linear Algebra.
- Boundary Value Problems.
- Initial Value Problems
 
Moreover, the students will be introduced to the use COMSOL Multiphysics, 
a scientific software for numerical simulations based on the Finite Element Method.
 
Suggested Reading
Mark S. Gockenbach,
Understanding and Implementing the Finite Elements Method,
SIAM 2006
 
Gilbert Strang
Computational Sciences and Engineering
Wellesley-Cambridge Press, 2008 
 

 

Liouville equations (minicourse)

We consider a class of Liouville equations and systems which arise in differential geometry when prescribing the Gaussian curvature of a surface, in models of mathematical physics describing stationary Euler flows and self-dual Chern-Simons equations or in spectral and string theory.
We discuss methods, variational in nature, to derive general existence results from suitable improvements of the Moser-Trudinger inequality combined with topological methods.
 
Tentative program
1. Introduction and variational theory
2. Min-max construction
3. Singular Liouville equations
4. Toda systems
5. Liouville equations in spectral theory
 

Geometric aspects of Newton-Okounkov bodies (Minicourse)

Abstract:

Recent years have witnessed a new way to introduce convex geometric methods to areas of mathematics around algebraic geometry: based on earlier works of Newton and Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata defined convex bodies (so-called Newton-Okounkov bodies), which capture  the  vanishing behaviour of sections of line bundles.
 
As a first approximation, the theory of Newton-Okounkov bodies is an attempt to create a correspondence between line bundles and convex 
bodies    known from toric geometry, except that in the absence of a 
large torus action, one has to make do with an infinite collection of bodies for every line bundle.
 
This point of view has been fairly succesful in that Newton-Okounkov bodies has been shown to encode positivity of line bundles, and they also serve as targets for completely integrable systems analogous to moment maps.
 
The theory has exciting connections with symplectic geometry, representation theory, and combinatorics for instance, nevertheless, in these lectures we will focus on its applications to projective geometry.
 
After introducing Newton-Okounkov bodies and their basic theory, we will discuss the case of surfaces, where there is a particularly satisfying theory, and the connection to (local) positivity of line bundles.  With this done, we will proceed to applications to higher syzygies of line bundles on abelian surfaces, and an exciting connection with Diophantine approximation.
 

TROPICAL LINEAR SERIES (Minicourse)

In this series of talks, we will discuss the basic combinatorial theory of divisors on graphs and its relationship to the theory of divisors on algebraic curves. We will cover several concrete examples and applications to problems in algebraic geometry.