Abstract. The cone of effective divisors in a general algebraic variety is usually not polyhedral. It may have infinitely many extremal rays, and it may even be round. For toric varieties, however, the effective cone is always polyhedral. In this talk I will discuss the effective cone of a toric variety blown up at a point. More specifically, I consider blowups of weighted projective planes P(a,b,c) and more general toric surfaces of Picard number 1. The problem here is to determine if the 2-dimensional effective cone of the blowup is closed or not. This problem relates to several classical problems about curves on algebraic surfaces. This is a joint work with Jose Gonzalez and Javier Gonzalez-Anaya.

Abstract. The algebraic form of Hilbert's 13th Problem asks for the resolvent degree rd(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + ... + a_n of degree n, where a_1, ..., a_n are independent variables. Here rd(n) is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with C(a_1, ..., a_n) and adjoining an algebraic function in <= d variables at each step. It is known that rd(n) = 1 for every n <= 5. It is not known whether or not rd(n) is bounded as n tends to infinity; it is not even known whether or not rd(n) > 1 for any n. Recently Farb and Wolfson defined the resolvent degree rd_k(G), where G is a finite group and k is a field of characteristic 0. In this setting rd(n) = rd_C(S_n), where S_n is the symmetric group on n letters and C is the field of complex numbers. In this talk I will define rd_k(G) for any field k and any algebraic group G over k. Surprisingly, Hilbert's 13th Problem simplifies when G is connected. My main result is that rd_k(G) <= 5 for an arbitrary connected algebraic group G defined over an arbitrary field k.

Abstract. Cluster Algebras, introduced in 2001 by Fomin and Zelevinsky, are a kind of commutative ring equipped with special combinatorial structure. They appear in a range of contexts, from representation theory to mirror symmetry. I will report on one aspect of recent joint work with Alfredo Nájera Chávez and Hipolito Treffinger. We show that for cluster algebras of finite cluster type, the cluster algebra with universal coefficients is equal to a canonically identified subfamily of the semiuniversal family for the Stanley-Reisner ring of the cluster complex.

Abstract. The construction of Kummer K3 surfaces from abelian surfaces can be generalized to yield higher dimensional varieties known as hyperkähler varieties of Kummer type. Hassett and Tschinkel showed that a portion of the middle cohomology of generalized Kummer 4-folds may be understood as fixed loci of symplectic involutions. In recent work with Sarah Frei, we have extended this result, allowing us to characterize the Galois action on the cohomology when working over non-closed fields, which has consequences for derived equivalences. These fixed loci also have an interesting geometry and I will discuss a particular example where the fixed locus contains an elliptically fibered K3 surface.

Abstract. The moduli space M_{0,n}-bar has a closed embedding into a product of projective spaces P^1 x ... x P^{n-3}, due to Keel–Tevelev and Kapranov. The composition of this map with the Segre map gives the log canonical embedding of M_{0,n}-bar, i.e. the embedding corresponding to the canonical bundle twisted by the class of the boundary of the moduli space. These maps are the most natural ways to realize M_{0,n}-bar as a projective variety.

Monin and Rana conjectured a set of equations defining the embedded image of M_{0,n}-bar in P^1 x ... x P^{n-3}, given by 2-by-2 minors of certain 2-by-k matrices, and verified their conjecture for n≤8 using Macaulay2. We prove this conjecture for all n. This is joint work with Maria Gillespie and Sean T. Griffin.

Abstract. Coulomb branches have recently been given a rigorous mathematical definition by work of Braverman–Finkelberg–Nakajima. We will discuss their geometric and categorical structure based on recent work with Harold Williams.

The Grothendieck groups of these categories recover previously studied algebras such as double affine Hecke algebras (DAHAs), certain open Richardson varieties in affine flag manifolds, multiplicative Nakajima quiver varieties, etc. One of our main results is that these categories carry a natural t-structure consisting of what we call Koszul-perverse coherent sheaves. The classes of such simples sheaves give a canonical basis of these algebras in a uniform way.

These Coulomb categories also carry (conjecturally) a cluster structure. We will survey some of these results as time (and interest) permits.

Abstract. Let E be an elliptic curve over a number field F. The group of F-rational is a finitely generated abelian group and the study of this "Mordell–Weil" group is among the centrepieces in the study of the arithmetic of elliptic curves. In this talk we will explain the role played by Iwasawa theory in studying the Mordell–Weil groups along certain infinite Galois extensions of the number field. We will also explain some of the recent results on the variation of certain invariants along families of Z_p-extensions of the number field.

Abstract. Buch, Mihalcea, Chaput, and Perrin proved that for cominuscule flag varieties, (T-equivariant) K-theoretic (3-pointed, genus 0) Gromov–Witten invariants can be computed in the (equivariant) ordinary K-theory ring. Buch and Mihalcea have a related conjecture for all type A flag varieties. In this talk, I will discuss work that proves this conjecture in the first non-cominuscule case—the incidence variety X=Fl(1,n-1;n). The proof is based on showing that Gromov–Witten varieties of stable maps to X with markings sent to a Schubert variety, a Schubert divisor, and a point are rationally connected. As applications, I will discuss positive formulas (an equivariant Chevalley formula and a non-equivariant Littlewood–Richardson rule) which determine the multiplicative structure of the (equivariant) quantum K-theory ring of X. If time permits, I will also discuss current joint work with Gu, Mihalcea, Sharpe, and Zhang on giving a presentation for this ring with inspiration from supersymmetric gauge theory. The talk is based on the arxiv preprint at https://arxiv.org/abs/2112.13036.

Abstract. Given a perfect field k with algebraic closure k' and a variety X over k', the field of moduli of X is the subfield of k' of elements fixed by elements s of the Galois group of k' over k such that the twist X_s is isomorphic to X. Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures.

In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety X with a smooth marked point p that ensure that the pair (X,p) is defined over its field of moduli.

Abstract. Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. Originating from game theory, a subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Algebraic curves, especially those defined by Redei-type polynomials, are powerful in studying blocking sets. One can reverse the engine and ask the following question: given an irreducible (or smooth) plane curve C in P^2, when does C(F_q) form a nontrivial blocking set? Alternatively, given d and q, does there exist an irreducible (or smooth) curve C with degree d defined over F_q that give rise to a nontrivial blocking set?

In this talk, I will give a partial answer to these questions using a mixture of tools from arithmetic geometry, arithmetic statistics, incidence geometry, and number theory. This is a joint work with Shamil Asgarli and Dragos Ghioca.