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.

The absolute Galois group $\operatorname{Gal}(F)$ of a field $F$ is the Galois group of its algebraic closure $\overline{F}$ relative to $F$, containing precisely those automorphisms of $\overline{F}$ that fix $F$ itself pointwise. Even for a field as simple as the rational numbers $\mathbb{Q}$, $\operatorname{Gal}(\mathbb Q)$ is a complicated object. Indeed (perhaps counterintuitively), $\operatorname{Gal}(\mathbb Q)$ is among the thorniest of all absolute Galois groups normally studied. When $F$ is countable, $\operatorname{Gal}(F)$ usually has the cardinality of the continuum. However, it can be nicely presented as the set of all paths through an $F$-computable finite-branching tree, built by a procedure uniform in $F$. We will first consider the basic properties of this tree, which depend in some part on $F$. Then we will address questions about the subgroup consisting of the computable paths through this tree, along with other subgroups similarly defined by Turing ideals. One naturally asks to what extent these are elementary subgroups of $\operatorname{Gal}(F)$ (or at least elementarily equivalent to $\operatorname{Gal}(F)$). This question is connected to the computability of Skolem functions for $\operatorname{Gal}(F)$, and also to the arithmetic complexity of definable subsets of $\operatorname{Gal}(F)$. When $F=\mathbb Q$, we have many questions and a few answers, partly due to joint work with Debanjana Kundu. In the simpler situations of the absolute Galois group of a finite field, and of the Galois group of the cyclotomic field over $\mathbb Q$, much more is known, thanks in part to joint work by Jason Block and the speaker.

Thurston revolutionized the study of diffeomorphisms of Riemann surfaces with a myriad of tools: hyperbolic geometry, geometric group theory, Teichmüller theory, and much more. One goal of this work is to see how much of his vision persists in higher dimensions. In this talk, I will tell a few stories about 4-manifolds and their mapping class groups, with an eye towards the Nielsen realization problem for certain algebraic surfaces. Some of the results I plan to share are joint work with Seraphina Lee and Tudur Lewis.

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