# Studying subschemes of affine/projective space via matroids

### Rob Silversmith (Northeastern)

15-May-2020, 19:00-20:00 (19 months ago)

Abstract: Given a homogeneous ideal $I$ in a polynomial ring, one may apply the following combinatorial operation: for each degree $d$, make a list of all subsets $S$ of the set of degree-$d$ monomials such that $S$ is the set of nonzero coefficients of an element of $I$. For each $d$, this set of subsets is a combinatorial object called a matroid. As $d$ varies, the resulting sequence of matroids is called the tropicalization of $I$.

I will discuss some of the many questions one can ask about tropicalizations of ideals, and how they are related to some classical questions in combinatorial algebraic geometry, such as the classification of torus orbits on Hilbert schemes of points in $\mathbb{C}^2$. Some unexpected combinatorial objects appear: e.g. when studying tropicalizations of subschemes of $\mathbb{P}^1$, one is led to Schur polynomials and binary necklaces.

algebraic geometrycombinatorics

Audience: researchers in the topic

Series comments: This seminar requires both advance registration, and a password. Register at stanford.zoom.us/meeting/register/tJEvcOuprz8vHtbL2_TTgZzr-_UhGvnr1EGv Password: 362880

If you have registered once, you are always registered for the seminar, and can join any future talk using the link you receive by email. If you lose the link, feel free to reregister. This might work too: stanford.zoom.us/j/95272114542

More seminar information (including slides and videos, when available): agstanford.com

 Organizer: Ravi Vakil* *contact for this listing

Export talk to