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).

I will start to prove Saracino’s theorem, that every countably categorical theory has a model companion.

The Beck-Fiala Conjecture asserts that any set system of n elements with degree k has combinatorial discrepancy $O(\sqrt{k})$. A substantial generalization is the Komlós Conjecture, which states that any m by n matrix with columns of unit Euclidean length has discrepancy O(1). In this talk, we describe an $\tilde{O}(\log^{1/4} n)$ bound for the Komlós problem, improving upon the $O(\log^{1/2} n)$ bound due to Banaszczyk from 1998. We will also see how these ideas can be used to resolve the Beck-Fiala Conjecture for $k \geq \log^2 n$, and give a $\tilde{O}(k^{1/2} + \log^{1/2} n)$ bound for smaller k, which improves upon Banaszczyk's $O(k^{1/2} \log^{1/2} n)$ bound. These results are based on a new technique of "Decoupling via Affine Spectral Independence" in designing rounding algorithms, which might also be useful in other contexts. This talk is based on joint work with Nikhil Bansal (University of Michigan).

The space of rational curves on a Fano variety X serves as a powerful tool for probing the geometry of X. Even for hypersurfaces, characterizing these spaces is difficult; however, work by Riedl–Yang established they are irreducible and have the expected dimension. In this talk, I will discuss another aspect, namely the singular locus. Specifically, I will show the singular locus of the moduli space of smooth degree e curves on a general low-degree hypersurface is small, i.e. has codimension growing linearly with e. This turns out to use a weird combination of 1. Lehmann–Riedl–Tanimoto's recent work on geometric Manin’s conjecture and 2. Sawin's work on Waring's problem from analytic number theory.

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.

TBA

tba