(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.

(Krishna talk) In this talk, I shall recall a well known relation between the Brauer group and the group 0-cycles on a smooth projective variety over a field. I shall then consider the special case of this relation over a local field and present some new results I recently obtained in a joint work with Jitendra Rathore and Samiron Sadhukhan.

(Cassady talk) Abstract: Given a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one could ask are: *Does q have a non-trivial zero (is q isotropic)? *Which non-zero elements of k are represented by q? *Does q represent all non-zero elements of k (is q universal)? Over global fields F, the Hasse-Minkowski theorem, which is one of the first examples of a local-global principle, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we'll explore when the local-global principle for isotropy holds over more general fields k, as well as connections between this local-global principle and universal quadratic forms over k.

(Hotchkiss talk) Abstract: The period-index problem for the Brauer group of a field asks, briefly, how the size of a division algebra is constrained by its order in the Brauer group. I will describe a Hodge-theoretic interpretation of the problem for complex function fields, and explain some consequences for the integral Hodge conjecture and for Brauer groups.

(Sankar talk) Abstract: The question of whether derived categories determine a variety has been studied widely, particularly in the case when the varieties have trivial canonical bundles. Antieau, Krashen and Ward (AKW) studied the question of when two genus 1 curves are derived equivalent. Their work showed, in particular, that this question is intimately related to the arithmetic of genus 1 curves. In joint work with Libby Taylor, we explore the question of when two G_m gerbes over a genus 1 curves are derived equivalent, and what that says about the arithmetic of these curves. In this talk, I will give some background on derived equivalences of varieties, how they relate to derived equivalences of stacks and then talk about some extensions of the results of AKW. The talk does not assume familiarity with derived categories or stacks.

(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.