On Markov's conjecture

Markov numbers are triples of positive integers (x,y,z) satisfying the Diophantine equation x²+y²+z²=3xyz. They appear in many fields of mathematics. Conjecturally, each triple is uniquely determined by max(x,y,z). I will talk about an algebro-geometric approach to this conjecture.

Moduli of Wildly Ramified Covers of Curves

One of many hazards in the jungle of characteristic p algebraic geometry is the presence of wild ramification, which is in a loose sense what happens when objects possess automorphisms of order divisible by p. In this talk, I will tell an incomplete story of wild ramification for algebraic curves in characteristic p, starting with the fundamental example of Artin-Schreier curves. I will also describe some work in progress towards a description of the moduli stack of Artin-Schreier(-Witt) covers of curves, part of which relies on results in a recent preprint (arXiv:1910.03146).

The Geometry of Hilbert’s 13^th Problem

The goal of this talk is to explain how enumerative geometry can be used to simplify the solution of polynomials in one variable. Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert's geometric reformulation of solving polynomials, explain the gaps in Hilbert's sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.

Severi dimensions for unicuspidal curves

We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. cusps; and to do so, we stratify cusps according to value semigroup. We show that generalized Severi varieties of maps P¹ → Pⁿ with images of fixed degree and arithmetic genus are often reducible whenever n is at least 3. We also prove that the Severi variety of degree-d maps with a hyperelliptic cusp of δ-invariant g << d is of codimension at least (n-1)g inside the space of degree-d holomorphic maps P¹ → Pⁿ; and that for small g, the bound is exact, and the corresponding space of maps is the disjoint union of unirational strata. Finally, we conjecture a generalization of this picture for unicuspidal rational curves associated to an arbitrary value semigroup. This work is joint with Vinícius Lara Lima and Renato Vidal Martins (Federal University of Minas Gerais).

Higher Prym representations and canonical curves

I'll explain an approach to some conjectures of Putman-Wieland and Ivanov on representations of mapping class groups of surfaces, via Hodge theory and an analysis of low rank quadrics containing a canonical curve of genus g. This is a report on (very early) joint work in progress with Aaron Landesman.