We discuss the concept of dissipative weak solution to the Euler and Navier Stokes systems describing the motion of a general compressible and heat conducting fluid. We present examples of ill-posedness in the context of the Euler system and develop the theory of weak and measure-valued solutions that comply with a certain form of the Second law of thermodynamics. We show applications of the theory to various problems including singular limits, convergence of numerical schemes, and problem driven by stochastic forcing.

Let S be a compact surface of negative Euler characteristic and G a semisimple group. Higher Teichmueller theory aims at singling out connected components of the variety of G-representations of the fundamental group of S which are formed of representations with geometric significance, echoing the case of PSL(2,R) where the relevant component is the classical Teichmueller space classifying hyperbolic structures on S. One of the outstanding problems is to construct compactifications of such components on which the mapping class group acts and having good topological properties, like the Thurston compactification does for classical Teichmueller space. In this talk we discuss such compactifications in the case when G=Sp(2n, R),constructed using representations over non-archimedean ordered fields, and relate these compactifications to classical objects like the space of geodesic currents on the surface S. This is joint work with A. Iozzi, A. Parreau, B.Pozzetti.

The rigor of mathematics lies in its systematic organization that makes for conclusive proofs of assertions on the basis of assumed principles. Proofs are constructed through thinking, but can also be taken as objects of mathematical investigation; that was the key insight underlying Hilbert’s call for a theory of the “specifically mathematical proof” in 1917. This pivotal thought was rooted in revolutionary developments of mathematics and logic, but it also shaped the new field of mathematical logic. It grounded, in particular, Hilbert’s proof theory. The emerging investigations led over time to computability theory, artificial intelligence and cognitive science. Within this broad framework, I will describe first the pursuit of Hilbert’s proof theory with “reductive” foundational goals and then some recent, tentative steps towards a theory of the “specifically mathematical proof”. Those steps have been made possible by a confluence of proof theoretic investigations in the tradition of Gentzen’s work on natural deduction and computer implementations of mechanisms in which proofs can be (automatically) constructed. Here, at the intersection of automated proof search and interactive verification, is a promising avenue for exploring the structure of mathematical thought.