A finite set of elements in a field extension L/K defines a matroid via algebraic independence. From the perspective of algebraic geometry, every irreducible variety X in C^N therefore determines an algebraic matroid of rank dim(X) through the field extension C(X)/C. The bases of this matroid correspond precisely to coordinate projections that are branched covers.

This geometric perspective enriches the structure of algebraic matroids: bases and circuits carry natural degrees, and additionally, bases come equipped with a Galois (or monodromy) group. We describe these connections between algebraic matroids and algebraic geometry and make the case that numerical algebraic geometry is a powerful computational framework for handling such objects.

We end with two (open) examples of families of algebraic varieties whose matroids remain unknown.

A 3-dimensional subspace f of real polynomials defines a map f : P^1 -> P^2 whose image is a rational plane curve. It is maximally inflected when all of its flexes are real, equivalently, when its Wronski determinant has only real roots. We associate two a priori distinct signs (\pm 1) to f: the Welschinger invariant of the rational curve and the degree of the Wronski map at f. Extensive computation suggests that these signs coincide. While studying this conjecture we were led to a deeper conjecture: From f, we define a a function W : CP^1 -> CP^1 which encodes some real geometry of f and conjecturally gives an object called a web. We conjecture that known bijections between webs and standard Young tableaux and between tableaux and maximally inflected curves recovers the curve.

This talk will explain this picture with compelling evidence and beautiful pictures. It is joint work with Brazelton, Karp, Le, Levinson, McKean, Peltola, and Speyer.

Given two Chow motives M and N satisfying Rost nilpotence, a natural question is whether their direct sum has the same property. Rather obviously this question is closely related to the famous and old Kothe conjecture. If this conjecture is true the answer to above question is yes. However it seems that most ring theorists believe Kothe's conjecture does not hold. This leads to studying stronger versions of Rost nilpotence, which (surprisingly?) hold in (almost?) all cases where usual Rost nilpotence is known, as I will explain in my talk.

There are many results in topology showing that certain continuous maps from a space to itself have fixed points (most famously, Brouwer's fixed-point theorem). These results are often not accessible from the homotopy-theoretic point of view, since they usually depend on more than just the homotopy type of a given space, and many are also not constructive, making use of the law of excluded middle. After introducing the language of synthetic homotopy theory, we will see that studying fixed point properties in that setting provides a different, homotopy-invariant notion, and we will present (counter-)examples that can be obtained using constructive methods. For this talk, we also aim to avoid type-theoretic terminology, instead working from the topologist's perspective.