UC San Diego

Abstract: Equivariant compactifications of reductive groups, such as toric varieties, can often be described using combinatorial data. In contrast, equivariant compactifications of unipotent groups generally lack such a combinatorial description. I will introduce equivariant compactifications of vector groups, known as additive varieties, and I will discuss some aspects of their birational geometry.

Abstract: Given nef divisors D_1, ..., D_n on a smooth projective variety Y of dimension d, their volume polynomial is the homogeneous polynomial

f_Y(x_1, ..., x_n) = (1/d!) (x_1 D_1 + ... + x_n D_n)^d

whose coefficients are intersection numbers. A natural question: which polynomials arise this way? Brändén and Huh singled out a purely combinatorial class — Lorentzian polynomials — that contains every volume polynomial and also captures log-concavity phenomena for matroids. I will survey recent work of June Huh and collaborators clarifying the gap between these two classes: the supports of volume polynomials are exactly algebraic polymatroids, and the containment is strict in dimension at least 3. I will illustrate with projection areas of convex bodies in R^4 and the Fano matroid over characteristic 2, and, time permitting, mention the triangular hyperfield and open problems.

Abstract: In this talk, we will discuss Cayley-Bacharach conditions, correspondences with null traces and their connections to some birational complexities. Using techniques developed by Bastianelli and Hodge theoretic inputs by Lazarsfeld and Martin, we will compute joint correspondence degree of a pair of symmetric squares of curves that are Hodge independent. Also, we will determine exactly degree of irrationality of symmetric squares of some special curves. Lastly, time permitting, we will discuss some questions towards computing auto correspondence degree of symmetric squares of curves and some open questions in this field. This is joint work with my advisor Prof. Elham Izadi.

Abstract: Algebraic de Rham cohomology is a cohomology theory for smooth proper morphisms of schemes. It is built from differential forms and, over the complex numbers, agrees with relative singular cohomology. This comparison suggests that it should satisfy relative Poincaré duality over an arbitrary base. This is rather tricky to prove, owing to the non-linearity of the de Rham differential.

We will describe a simple new proof of Poincaré duality in this setting, combining ideas from motivic homotopy theory and differential operators, due to Clausen and de Jong respectively. The method is robust enough to lift to Poincaré duality for a notion of differential complexes, and is closely related to a result of Saito in characteristic 0. The latter result also partially answers a suggestion of Grothendieck.

This is joint work with Caleb Ji, generalizing the joint work of both authors with Casimir Kothari, Oliver Li, Sveta Makarova, and Sridhar Venkatesh.

Abstract:

Abstract:

Abstract: