(Lee Talk) In 1993, Demeyer and Ford computed the Brauer group of a smooth toric variety over an algebraically closed field of characteristic zero. One may pose the same question to the toric varieties over any field of positive characteristic. Another interesting question is what will happen if we replace the base field by a discrete valuation ring, thereby replacing smooth toric varieties by smooth toric schemes over a discrete valuation ring in the sense of Kempf-Knudsen-Mumford-Saint-Donat. In this talk, I am going to discuss the answers to these questions. This is a joint work with Roy Joshua.

(Kobin talk) Solutions to many problems in number theory can be described using the theory of algebraic stacks. In this talk, I will describe a few Diophantine equations, such as the ``generalized Fermat equation'' $Ax^{p} + Bx^{q} = Cz^{r}$, whose integer solutions can be found using an appropriate stacky curve: a curve with extra automorphisms at prescribed points. I will also describe how stacky curves can be used to study rings of modular forms both classically and in characteristic $p$. Parts of the talk are joint work in progress with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang, and separately with David Zureick-Brown.

(Gazaki talk) The Chow group of zero-cycles is a generalization to higher dimensions of the Picard group of a smooth projective curve. When $X$ is a curve over an algebraically closed field $k$ its Picard group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the $k$-points of the Jacobian variety of $X$. In higher dimensions however the situation is much more chaotic, as the Abel-Jacobi map in general has a kernel, which over large fields like $\mathbb{C}$ can be enormous. On the other extreme, a famous conjecture of Beilinson predicts that if $X$ is a smooth projective variety over $\overline{\mathbb{Q}}$, then this kernel is zero. For a variety $X$ with positive geometric genus this conjecture is very hard to establish. In fact, there are hardly any examples in the literature. In this talk I will discuss joint work with Jonathan Love where we make substantial progress on this conjecture for an abelian surface $A$. First, we will describe a very large collection of relations in the kernel arising from hyperelliptic curves mapping to $A$. Second, we will show that at least in the special case when $A$ is isogenous to a product of two elliptic curves, such hyperelliptic curves are plentiful. Namely, we will describe a construction that produces for infinitely many values of $g\geq 2$ countably many hyperelliptic curves of genus $g$ mapping birationally into A.

(Klagsbrun talk) We show that the $\phi$-Selmer ranks of twists of an elliptic curve $E$ with a point of order two are distributed like the ranks of random groups in a manner consistent with the philosophy underlying the Cohen-Lenstra heuristics. If $E$ has a point of order two, then the distribution of $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ) - dim_{\mathbb{F}_2} Sel_{\hat\phi}(E^{\prime d}/\QQ)$ tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2} \log \log X)$ as $X \rightarrow \infty$. We consider the distribution of $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ) - dim_{\mathbb{F}_2} Sel_{\hat\phi}(E^{\prime d}/\QQ)$ has a fixed value $u$. We show that for every $r$, the limiting probability that $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ)= r$ is given by an explicit constant $\alpha_{r,u}$ introduced in Cohen and Lenstra's original work on the distribution of class groups. If time permits, we will also speculate on the distribution of $dim_{\mathbb{F}_2} \Sel_2(E^d/\QQ)$ in the case where $dim_{\mathbb{F}_2} \Sel_\phi(E^d/\QQ) - dim_{\mathbb{F}_2} Sel_{\hat\phi}(E^{\prime d}/\QQ)$ has a fixed value $u$ as well. This is (mostly) joint work with Daniel Kane.

(Williams talk) Abstract: This is joint work with Uriya First. If R is a ring and n is a natural number, then an Azumaya algebra of degree n over R is an R-algebra A that becomes isomorphic to the nxn matrix algebra Mat_n(R) after some faithfully flat extension of scalars. An involution of an Azumaya algebra is an additive self map of order 2 that reverses the multiplication. We will say that an Azumaya algebra with involution (A,s) is semiordinary if there it becomes isomorphic to some algebra-with-involution (End(P), t) after a faithfully flat extension of scalars. Although this is an extremely broad class of Azumaya algebras with involution, I will show that it is not all of them: there exist Azumaya algebras with truly extraordinary involutions. The method is to find obstructions to being semiordinary in equivariant algebraic topology.

(Leep talk) Abstract: Let K be a p-adic field and let Q1, Q2 ∈ K[x1, . . . , xn] be quadratic forms in n variables with coefficients in K. Heath-Brown proved that if n = 8, Q1, Q2 have a nontrivial common zero defined over K, and the variety defined by {Q1, Q2} is nonsingular, then there exist a, b ∈ K, not both zero, such that aQ1 + bQ2 splits off at least 3 hyperbolic planes. This rather technical theorem, whose proof is very long, was an im- portant ingredient to Heath-Brown’s proof that nonsingular pairs of quadratic forms in 8 variables defined over a number field satisfy the Hasse Principle. More concretely: Suppose that {Q1, Q2} is a pair of quadratic forms in 8 variables defined over a number field F and as- sume that the variety defined by {Q1, Q2} is nonsingular. If Q1, Q2 have a nontrivial zero defined over each nonarchimedean completion of F and also over each real completion of F (if any), then Q1, Q2 have a nontrivial zero defined over F. The above theorem of Heath-Brown on pairs of quadratic forms over p-adic fields raised many questions in my mind, including why his proof was so difficult. Also, it seemed that this result for n = 8 should be part of a bigger theorem for general values of n. This talk will explore the situation for general values of n. This talk is based on joint work with my Ph.D. student John Hall.

(Singh talk) A Higgs bundle over an algebraic curve is a vector bundle with a twisted endomorphism, which can be thought of as a matrix of $1$-forms. An important question is to calculate the volume of the groupoid of Higgs bundles over finite fields. In 2014, Olivier Schiffmann succeeded in finding the corresponding generating function and together with Mozvogoy reduced the problem to counting pairs of a vector bundle and a nilpotent endomorphism. It was generalized recently by Anton Mellit to the case of Higgs bundles with regular singularities. An important step in Mellit's calculations is the case of $\mathbb{P}^{1}$ and two marked points, which allows him to relate the corresponding generating function with the Macdonald polynomials. It is a natural question to generalize Mellit's calculations to arbitrary reductive groups. In the talk we will consider the case of $\mathbb{P}^{1}$ with two points for an arbitrary split connected reductive group $G$ over $\mathbb{F}_{q}$. Firstly, we give an explicit formula for the number of $\mathbb{F}_{q}$-rational points of generalized Steinberg varieties of $G$. Secondly, for each principal $G$-bundle over $\mathbb{P}^1$, we give an explicit formula counting the number of triples consisting of parabolic structures at $0$ and $\infty$ and compatible nilpotent sections of the associated adjoint bundl

(Ghorpade talk) Abstract: Consider the Grassmann variety with its canonical Plucker embedding, or more generally a Schubert variety in a Grassmannian with its nondegenerate embedding in a subspace of the Plucker projective space. We can cut it by linear subspaces of a fixed dimension of the ambient projective space, and ask which of the linear sections are ”maximal”. The term ”maximal” can be interpreted in several ways and we will be particularly interested in maximality with respect to the number of rational points over a given finite field. In general, this is an open problem. This problem is also closely related to questions in the study of linear error correcting codes. We will quickly outline the relevant background, explain the connection with coding theory, and then describe some of the known results and problems. If time permits, we will also discuss the case of flag varieties over finite fields and related questions concerning them.