L4

 In the first part of the talk, after revisiting some classical models for dilute polymeric fluids, we show that thermodynamically 
consistent models for non-isothermal flows of such fluids can be derived in a very elementary manner. Our approach is based on identifying the 
energy storage mechanisms and entropy production mechanisms in the fluid of interest, which in turn leads to explicit formulae for the Cauchy 
stress tensor and for all the fluxes involved. Having identified these mechanisms, we first derive the governing system of nonlinear partial 
differential equations coupling the unsteady incompressible temperature-dependent Navier–Stokes equations with a 
temperature-dependent generalization of the classical Fokker–Planck equation and an evolution equation for the internal energy. We then 
illustrate the potential use of the thermodynamic basis on a rudimentary stability analysis—specifically, the finite-amplitude (nonlinear) 
stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

In the second part of the talk, we show that sequences of smooth solutions to the initial–boundary-value problem, which satisfy the 
underlying energy/entropy estimates (and their consequences in connection with the governing system of PDEs), converge to weak 
solutions that satisfy a renormalized entropy inequality. The talk is based on joint results with Miroslav Bulíček, Mark Dostalík, Vít Průša 
and Endré Süli.

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.

When doing arithmetic geometry, it is helpful to have invariants of the objects which we are studying that see both the arithmetic and the geometry. Motivic homotopy theory allows us to produce new invariants which generalise classical topological invariants, such as the Euler characteristic of a variety. These motivic invariants not only recover the classical topological ones, but also provide arithmetic information. In this talk, I'll review the construction of a motivic Euler characteristic, then study its arithmetic properties, and mention some applications. I'll then talk about work in progress with Ran Azouri, Stephen McKean and Anubhav Nanavaty which studies a "higher Euler characteristic", allowing us to produce an invariant of automorphisms valued in an arithmetically interesting group. I'll then talk about how to relate part of this invariant to a more classical invariant of quadratic forms.

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.