(Rapinchuk talk) Over the last few years, the analysis of algebraic groups with good reduction has come to the forefront in the emerging arithmetic theory of linear algebraic groups over higher-dimensional fields. Current efforts are focused on finiteness conjectures for forms of reductive algebraic groups with good reduction that share some similarities with the famous Shafarevich Conjecture in the study of abelian varieties. Most results on these conjectures obtained so far have ultimately relied on finiteness properties of appropriate unramified cohomology groups. However, quite recently, methods based on building-theoretic techniques have emerged as a promising alternative approach. I will showcase some of these developments by sketching a new proof of a theorem of Raghunathan-Ramanathan concerning torsors over the affine line. Time-permitting, I will also discuss some connections to the genus problem, which deals with simple algebraic groups having the same isomorphism classes of maximal tori.

(Joshua talk) This concerns the following conjecture of Fabien Morel. Given a linear algebraic group G defined over a field k, with a maximal torus T, the conjecture was that a suitable form of the Euler Characteristic of G/N(T), where N(T) is the normalizer of T in G, is 1, when viewed as a class in the Grothendieck-Witt group of the field k. In a paper in 2023, together with P. Pelaez, we settled this conjecture in the affirmative for split groups G, provided the field k has a square root of -1. We also showed that this implies the same Euler characteristic is a unit in general. This has numerous applications to splitting in various forms of Borel-style equivariant cohomology theories, for example, equivariant K-theory, equivariant (higher) Chow groups, equivariant Brauer groups etc., and greatly facilitates the computation of all such equivariant cohomology theories. The talk will sketch a proof of the conjecture and outline some of the applications.

(Haburcak talk) The moduli space of algebraic curves is a central object of study, and its cohomology ring is deeply related to spaces of modular forms. The tautological ring is a subring of the cohomology of the moduli space of stable curves which contains most cohomology classes of geometric interest and is simpler to study. A question of interest is therefore how different the tautological ring is from the full cohomology. The first known example of a non-tautological algebraic class was discovered by Graber and Pandharipande, which was significantly generalized by van Zelm to find an infinite family of non-tautological classes. On the other hand, very little is known about non-tautological classes on the moduli space of smooth curves. In joint work with Arena, Canning, Clader, Li, Mok, and Tamborini, we show that there are non-tautological classes on the moduli space of smooth marked curves for an infinite family of genera.

(Scully talk) The Witt ring of a field F captures most of the essential information about the totality of finite-dimensional symmetric bilinear forms over F. In the 1960s, it was observed by A. Pfister and J. Milnor that many central questions in the study of symmetric bilinear forms over general fields depend on understanding a certain multiplicative filtration of the Witt ring known as the I-adic filtration. This led to a famous conjecture of Milnor predicting an identification of the graded ring associated to this filtration with what we now call mod-2 Milnor K-theory. Following the successful resolution of this conjecture by K.Kato (characteristic-2 case) and V. Voevodsky (characteristic-not-2 case), an important outstanding problem in the area is to understand the "low-dimensional" part of each piece of the I-adic filtration. In this talk, I will outline some aspects of this problem and discuss some of the known results. Towards the end, I will then discuss a recent proof of a conjectural classification of the elements of dimension 2ⁿ +2^n-1 in the nth piece of the filtration over fields of characteristic 2. Over fields of characteristic not 2, this conjecture remains wide open for all n ≥ 4.

(Berg talk) If a variety X over the rationals has p-adic (local) points for each p, then one might ask whether X has any (global) rational points. To start, we can impose conditions on the collection of all local points on X to narrow down the possible subset of global points, should any exist. One fruitful approach uses an algebro-geometric object called the Brauer group of X which defines an obstruction set; if this set is empty, then it guarantees the set of rational points is empty, too. For some nice classes of surfaces, if X is locally soluble for all p but does not have a rational point, then the Brauer group of X is conjectured to be the cause. In general, when such an obstruction occurs, it arises from a finite number of classes in the Brauer group. One might wonder whether properties of this finite subset can be determined in advance, i.e., without computing the obstruction set. In the case of cubic surfaces, for example, it is known that just one Brauer class is needed to detect an obstruction. In this talk, we'll discuss work that shows we cannot always hope to give such quantitative bounds; for any integer N > 0, we construct conic bundles over the projective line for which the Brauer group modulo constants is generated by N classes, all of which are required to witness an obstruction. (This is joint work with Pagano, Poonen, Stoll, Triantafillou, Viray, Vogt.)

(Balaji talk) Algebraic groups over local fields $K$ have been extensively studied in the past 60 years. Bruhat-Tits theory develops the study of bounded subgroups of the group $G(K)$ where $G$ is a connected reductive algebraic group under very general assumptions of being quasi-split over $K$. After the first big step in the paper by N. Iwahori and H. Matsumoto, this study was taken to Grothendieckian heights by F. Bruhat and J. Tits in their now well-known body of work, called Bruhat-Tits theory; their work extended over 20 years till the mid eighties. A main theme is to schematize these bounded groups and this is carried out in their papers in 1971 and 1984. In my paper with C.S. Seshadri (2015), a somewhat different geometrical approach was observed in the setting when $K=k((t))$. A novel yet foundational feature in my work with Seshadri is the theorem that any parahoric Bruhat-Tits group scheme over a complete discrete valuation ring $O$ can be realized as an "invariant direct image", i.e., obtained by taking Galois invariants of Weil restriction of scalars of a reductive group scheme, on a ramified cover of $O$. In my recent paper with Y. Pandey, we study the question of extending the schematic aspects of Bruhat-Tits theory to regular local rings. Such a study is already envisaged in the work of Bruhat and Tits. This paper provides a fairly complete higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. Application 1: I will briefly outline the solution to the problem of degeneration of moduli stacks of $G$-torsors on smooth projective curves when the curve is allowed to degenerate to a nodal curve. In fact, the higher dimensional theory arose out of this work. Application 2: I will briefly indicate constructions of new modular desingularizations of moduli spaces of semistable principal bundles over curves. This comprehensively generalizes work of Seshadri, Narasimhan-Ramanan.