A general real rational plane curve C of degree d has 3(d-2) flexes and (d-1)(d-2)/2 complex double points. Those double points lying in RP^2 are either nodes or solitary points. The Welschinger sign of C is (-1)^s, where s is the number of solitary points. When all flexes of C are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to C we may associate the local degree of the Wronskii map, which is also 1 or -1. My talk will discuss work with Brazelton and McKean towards a possible conjecture that that these two signs associated to C agree, and the challenges to gathering evidence for this.

The variety of sums of r powers (VSP(F,r)) of a homogeneous degree d polynomial F is the closure in the Hilbert scheme of the set of all those r-tuples of points of the d-th Veronese variety that contain F in their linear span. As a main ingredient in their border apolarity theory, Buczyńska and Buczyński introduced the notion of a border variety of sums of powers. During the talk I will compare VSP(F,r) (and a similarly defined subset of the Hilbert scheme) with the border variety of sums of powers. The talk is based on a joint work with Emanuele Ventura.

see https://franksottile.github.io/conferences/TAGS24/index.html

Classically, a partial differential equation is said to be Darboux integrable if its general solution can be found by only solving a system of ordinary differential equations. In this talk, we describe a transformation group-theoretic approach to the study of Darboux integrable equations and highlight how this approach can be used to solve the equivalence problem for so-called $f$-Gordon equations of the form $u_{xy} = f(x,y,u,u_x,u_y)$.

A polynomial f(x_1, … x_n) over a finite field has a large bias if its output distribution is far from uniform. It has rank `r' if we can write `f' as a function of polynomials g_1, …, g_r that each have smaller degree. Bias measures the amount of randomness, and rank measures the amount of structure. It is known that if `f' has small rank, it must have large bias. Green and Tao proved an inverse theorem stating that if `f' is significantly biased, its rank is bounded. Their bound was qualitative, however, and several authors gave quantitative improvements. We prove an optimal inverse theorem: the rank and the log of the bias are equivalent up to linear factors (over large enough fields). Our techniques are very different from the usual methods in this area, we rely on algebraic geometry rather than additive combinatorics. This is joint work with Guy Moshkovitz.