Matt Larson

#### Stanford

### “Combinatorial algebraic geometry”

My work involves interactions between combinatorics and algebraic geometry in both directions. I’ve used combinatorics to study problems in singularity theory. The monodromy conjecture is a conjecture in singularity theory which dates back to the 90s and is due to Denef, Igusa, and Loeser. It predicts a relationship between two different-looking invariants of a hypersurface singularity, the eigenvalues of the monodromy action on the cohomology of the Milnor fiber and the motivic zeta function. With Payne and Stapledon, we verified the monodromy conjecture for a large class of singularities.

I’ve also used algebraic geometry to study combinatorial problems. Delta-matroids are certain polytopes which arise naturally in a variety of contexts. With Eur, Fink, and Spink, we used algebro-geometric methods to prove inequalities for a large class of delta-matroids. This involved constructing spaces associated to delta-matroids.

### ABSTRACT

Combinatorial algebraic geometry is an interdisciplinary field of math that sits between the fields of combinatorics, the study of finite objects, and algebraic geometry, the study of solutions to polynomial equations. My work reduces problems in algebraic geometry to more concrete combinatorial problems, and it uses deep tools from algebraic geometry to study combinatorics.

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