We will first present the mathematical theory of Diffusion and Heat Propagation as seen from the classical point of view of Partial Differential Equations. This will allow us to mention concepts of great beauty and relevance in what follows. We will then introduce our favorite topic of Nonlinear Diffusion, embodied in equations like the Porous Medium Equation, and focused on the existence and properties of the very interesting geometrical objects called Free Boundaries. We will complete the presentation with a review of personal work on diffusion equations involving long distance interactions in the form of Fractional Laplacian Operators. This topic has been intensely developed in the last decade and is still in full bloom.

May 17, 2018, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

Cardiovascular diseases unfortunately represent one of the leading causes of death in Western countries. Mathematical models allow the description of the blood motion in the human circulatory system, as well as the interplay between electrical, mechanical and fluid-dynamical processes occurring in the heart. This is a classical environment where multiphysics processes have to be addressed. Appropriate numerical strategies can be devised to allow for an effective description of the fluid in large and medium size arteries, the analysis of physiological and pathological conditions, and the simulation, control and shape optimization of assisted devices or surgical prostheses. This presentation will address some of these issues and a few representative applications of clinical interest.

June 06, 2017, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

We present several stability results concerning smooth solutions of the compressible Euler and Navier-Stokes system in the class of measure-valued solutions. We show that strong and measure-valued solutions of these systems coincide as soon as the strong solution exists. The question if any measure-valued solution can be generated by a sequence of weak solutions will be also addressed.

May 31, 2016, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

A rubber band constrained to remain on a manifold evolves by trying to shorten its length, eventually settling on some minimal closed geodesic, or collapsing entirely. It is natural to try to consider a noisy version of such a model where each segment of the band gets pulled in random directions. Trying to build such a model turns out to be surprisingly difficult and generates a number of nice geometric insights, as well as some beautiful algebraic and analytical objects.

June 17, 2015, 16:00h. Getreidemarkt 9, 11th floor, TUtheSky

I discuss some recent progress in the study of stable self- similar blowup in energy-supercritical wave equations. In particular, I present a proof for the stability of the so- called ODE blowup for the energy-supercritical focusing wave equation without symmetry assumptions. This talk is based on joint work with Birgit Schörkhuber.

December 16, 2014, 14:00h. Wiedner Hauptstr. 8-10, green area, SEM 101C