The talk will be devoted to D-affinity of smooth projective varieties over fields of positive characteristic. First, I will recall the notion of a D-affine variety and justify its importance. Then, I will explain how in positive characteristic this notion relates to other properties defined in terms of the Frobenius morphism: the tilting property of Frobenius pushforwards of the structure sheaf, GFFRT, etc. In the last part of the talk I will present the results from my recent preprint https://arxiv.org/abs/2601.13340.

Let T be a recursively axiomatizable first-order theory. We say that T is relatively decidable if, for any model of T, the atomic diagram of that model can compute the full elementary diagram. For example, if T is model complete, then there is a uniform decision procedure which works for any model of T. We characterize the complete relatively decidable theories by showing that they have a sort of conservative extension which is model complete. The proof combines a standard theorem from computable structure theory with an intricate but elementary model-theoretic argument.

A semi-classical question of Avramov asks whether embedded deformations of a local ring correspond exactly to central elements in the homotopy Lie algebra of the ring. In this talk, I will explain the question and some recent insights. The latter is based on joint work with Briggs, Grifo, and Walker.

The forced 2D Euler equations exhibit non-unique solutions with vorticity in L^p, p > 1, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of "resolving" the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity and consider epsilon-size perturbations of his initial datum. We discover a uniqueness threshold below which the vanishing viscosity solution is unique and radial, and at which certain vanishing viscosity solutions converge to non-unique, non-radial solutions. Joint work with Maria Colombo and Giulia Mescolini (EPFL).

TBA

It is well known that asset returns usually do not follow a normal distribution, rather, they have long and fat tails. This paper focuses on the quantile portfolio methodology, which considers the whole distribution of asset returns and employs expected loss as a risk measurement. In particular, we explore statistical properties of tau risk and propose related theories of quantile portfolio optimization. We also introduce portfolio performance terms for the quantile portfolio framework.