## November 2017

Abstract: I will discuss the application of the method of compensated compactness to the compressible, isentropic Euler equations under certain geometric assumptions, e.g. the case of fluid flow in a nozzle of varying cross-sectional area or the assumption of planar symmetry under special relativity. Under these assumptions, the equations reduce to the classical (or relativistic) one-dimensional isentropic Euler equations with additional geometric source terms. In this talk, I will explain how the classical strategy of DiPerna, Chen et. al. can be adapted to handle these more complicated systems and will highlight some of the difficulties involved in extending the techniques to the relativistic setting.

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS

A T-design is a collection of subsets of a set X of size k, called “blocks”, such that every subset of X of size T is contained in lambda blocks. Examples of designs are magic and latin squares, Hadamard matrices, Steiner systems, finite projective planes, to name just a few.

From a T-design I will show how to construct a T-dimensional tropical variety of degree lambda. Then using this construction, I will interpret some operations on designs through an algebro-geometric lens. This work is very much in progress and comments/suggestions are welcome.

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2017/Berlin-LE-WS/index.html

Semi-algebraic sets of integer points.

We look at sets of integer points in the plane,

and discuss possible definitions of when such a set

is “complicated” — this might be the case if it

is not the set of integer solutions

to some system of polynomial equations and inequalities.

Let’s together work out lots of examples,

and on the way let’s try to develop criteria

and proof techniques …

The examples that motivated our study come

from polytope theory: Many question of the type

“What is the possible pairs of

(number of vertices, number of edges)

for 5-dimensional polytopes?”

have been asked, many of them with simple and complete

answers, but in other cases the answer looks complicated.

Our main result says: In some cases it IS complicated!

(Joint work with Hannah Sjöberg.)

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2017/Berlin-LE-WS/index.html

The topic of the talk is the Hardy space theory of compensated compactness quantities which originated in the seminal paper of Coifman, Lions, Meyer and Semmes from 1993. I concentrate on quantities relevant to fluid dynamics and discuss the relation between compensated compactness and Hardy space integrability. I also present an application to uniqueness of weak solutions with vanishing Cauchy data in 2D magnetohydrodynamics, which is joint work with Daniel Faraco.