Lectures at the 45. Heidelberg Graduate Days

October 05-09, 2020

Field theory of dissipative quantum systems

Jamir Marino, Johannes Gutenberg University, Mainz

Abstract: The lectures aim at providing a crash course knowledge of Keldysh field theory for dissipative quantum many body systems. We will start with an overview of the dynamics of open quantum models in Lindblad form, to construct systematically a field theory suited to describe non-equilibrium phenomena whose evolution is non unitary. The core of the lectures is represented by a broad number of applications ranging from quantum many body optics to solid state physics, including dynamics of condensed matter systems. In particular, we will analyse dissipative phase transitions in light-matter coupled models, dissipative quantum spin chains, and the dynamics of one-dimensional strongly correlated systems subject to local or global dissipative channels. The lectures will be in the format of a blackboard talk; focus will be on the mathematical constructions and on the physical effects, while some elementary technical aspects will be left to the audience as exercises.

The homology of data

Nina Otter, UCLA - University of California, Los Angeles

Abstract: Techniques and ideas from topology - the mathematical area that studies shapes - are being applied to the study of data with increasing frequency and success.
In this lecture series we will explore how we can use homology, a technique in topology that gives a measure of the number of holes of a space, to study data. The most well-known method of this type is persistent homology, in which one associates a one-parameter family of spaces to a data set and studies how the holes evolve across the parameter space. A more recent and less well-known technique is magnitude homology, which one can think of as giving a measure of the "effective number of points" of a metric space. In this course we will introduce the theoretical background for persistent and magnitude homology, and then dive into applications using software implementations and statistical analysis tools on real-world data sets.