Related papers: An introduction to the analysis of gradients syste…
For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such…
These lecture notes concern information-theoretic notions of entropy. They are intended for, and have been successfully taught to, undergraduate students interested inresearch careers. Besides basic notions of analysis related to…
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
We have studied the gradient-flow equations in information geometry from a point-particle perspective. Based on the motion of a null (or light-like) particle in a curved space, we have rederived the Hamiltonians which describe the…
The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not…
These informal notes briefly discuss some basic topics involving Lipschitz functions, connectedness, and Hausdorff content in particular.
The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and…
The classical tools which ensure the completeness of vector fields and second order differential equations for mechanical systems are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach and Hilbert…
These lecture notes are based on the second course in a series of lectures at the Spring school "Non-archimedean geometry and Eigenvarieties" in March 2023 in Heidelberg. The objective of the first three courses was to give an introduction…
These notes are the written version of my lectures at the Banach Center mini-school "Schubert Varieties" in Warsaw, May 18-22, 2003. Their aim is to give a self-contained exposition of some geometric aspects of Schubert calculus.
We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach…
The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish…
We are interested in existence of gradient flows for shape functionals especially for first Laplacian eigenvalues. We introduce different techniques to prove existence and use different formulations for gradient flows. We apply a…
In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as…
In this work, we investigate links between the formulation of the flow of marginals of reversible diffusion processes as gradient flows in the space of probability measures and path wise large deviation principles for sequences of such…
These lecture notes contain an introduction to some of the fundamental ideas and results in analysis and probability on infinite-dimensional spaces, mainly Gaussian measures on Banach spaces. They originated as the notes for a topics course…
To improve models of the structure and evolution of stellar and planetary interiors, it is important to quantify transport by strongly stratified turbulence in low Prandtl number fluids. Recent numerical studies have shown evidence for…
Gradient descent-ascent (GDA) flows play a central role in finding saddle points of bivariate functionals, with applications in optimization, game theory, and robust control. While they are well-understood in Hilbert and Banach spaces via…
We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c.~energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals…
This is a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019. It is supposed to be a concise (and therefore not comprehensive) and pragmatic course on applied holography and…