To expose students to an modern and exciting research topics in optimization.
In this course will survey recent topics relating to semidefinite programming, with a particular focus on the Parillo-Lasserre framework for sums of squares relaxations. We will explore applications in combinatorial optimization, computational complexity, and machine learning. This course is inspired by the course "Sum-of-Squares: proofs, beliefs, and algorithms" taught by David Steurer & Boaz Barak at Harvard.
- Basic knowledge of semidefinite and linear programming & probability theory (Chernoff bounds, etc.).
- Mathematical maturity.
Introduction to the theory and practice of Bayesian statistics.
Bayesian statistics is routinely used in various fields of applied sciences. In the Bayesian approach one first specifies a so called prior distribution on the parameter space representing the initial belief or expert knowledge about the problem. Then using the obtained data updates this preliminary belief by computing the conditional distribution of the parameter given the data, the so called posterior distribution.
The aim of this course is to give a rigorous, measure theoretic introduction to Bayesian statistical procedure and investigate its performance in view of the classical, frequentist framework, in which it is assumed that the data are generated according to a given parameter. We are usually concerned with the question whether the posterior is able to reconstruct this parameter, for instance if the amount of data would increase indefinitely. We shall study specific examples of prior distributions (both parametric and nonparametric ones, including the Dirichlet process and Gaussian processes) and the corresponding posteriors. We will investigate the behaviour of Bayesian point estimators, Bayesian confidence sets and tests. Standard sampling algorithms (MCMC method) will be also considered and implemented.
Probability theory, some knowledge of statistics.
Measure-theoretic probability and asymptotic statistics are recommended.
Aim: Basic topological tools for studying and understanding nonlinear partial differential equations and dynamical systems
Content: In this course a variety of topological techniques are discussed that are important in the modern treatment of partial differential equations and dynamical systems. Among these are degree theory (finite and infinite dimensional), nonlinear Fredholm maps, variational techniques, Morse theory and Conley Index theory. In this course the techniques are explained and motivated via applications to numerous examples in nonlinear differential equations.
Prerequisites: Basic knowledge of ordinary and partial differential equations, basic topology and functional analysis.
- Docent: Robert van der Vorst