# SCIENTIFIC PROGRAMS AND ACTIVITIES

October 22, 2018

THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES

20th ANNIVERSARY YEAR

January–June 2013
Thematic Program on Torsors, Nonassociative Algebras and Cohomological Invariants

April-June, 2013 Torsors, motives and cohomological invariants
(organized by V. Chernousov, A. Merkurjev and K. Zainoulline)

Conference on Torsors, Nonassociative Algebras
and
Cohomological Invariants

June 10-14, 2013

 Registration on site Reimbursement information for funded participants Accommodation in Toronto Back to Main Index

Preliminary Schedule:

 Monday, June 10 Room 230, Fields Institute 8:45 - 9:15 On-site Registration 9:15 - 9:30 Welcome and Introduction 9:30 - 10:30 Roman Fedorov, Kansas State University A conjecture of Grothendieck and Serre on principal bundles 10:30-11:00 Coffee Break 11:00-12:00 Patrick Brosnan, University of Maryland Algebraic groups related to Hodge theory 12:00-14:00 Lunch Break 14:00-15:00 Igor Rapinchuk, Yale University On the conjecture of Borel and Tits for abstract homomorphisms of algebraic groups 15:00-15:30 Coffee Break 15:30-16:30 Ting-Yu Lee, Fields Institute The local-global principle for embeddings of maximal tori into reductive groups 16:45-17:15 Hernando Bermudez, Fields Institute/Emory University Degree 3 Cohomological Invariants of Split Quasi-Simple Groups 17:15 Reception Tuesday, June 11 Room 230, Fields Institute 9:30 - 10:30 Roman Fedorov, Kansas State University A conjecture of Grothendieck and Serre and affine Grassmannians 10:30-11:00 Coffee Break 11:00-12:00 Andrei Rapinchuk, University of Virginia The genus of a division algebra and ramification 12:00-14:00 Lunch Break 14:00-15:00 Stefan Gille, University of Alberta Permutation modules and motives 15:00-15:30 Coffee Break 15:30-16:30 Anastasia Stavrova, Fields Institute On the congruence kernel of isotropic groups over rings 16:45-17:15 Timothy Pollio, University of Virginia The Multinorm Principle Wednesday, June 12 Room 230, Fields Institute 9:30 - 10:30 David Saltman, CCR-Princeton Finite u Invariants and Bounds on Cohomology Symbol Lengths 10:30-11:00 Coffee Break 11:00-12:00 Alexander Merkurjev, University of California, Los Angeles On cohomological invariants of semisimple groups 12:00-14:00 Lunch Break 14:00-15:00 Eli Matzri, University of Virginia Symbol length over C_r fields 15:00-15:30 Coffee Break 15:30-16:30 Benjamin Antieau, UCLA Topological Azumaya algebras 16:45-17:15 Daniele Rosso, University of Chicago Mirabolic Convolution Algebras Thursday, June 13 Room 230, Fields Institute 9:30 - 10:30 Nicole Lemire, University of Western Ontario Equivariant Birational Aspects of Algebraic Tori 10:30-11:00 Coffee Break 11:00-12:00 Eric Brussel, California State Polytechnic University Arithmetic in the Brauer group of the function field of a p-adic curve 12:00-14:00 Lunch Break 14:00-15:00 Danny Neftin, University of Michigan, Ann Arbor Noncrossed products over Henselian fields and a Grunwald-Wang problem 15:00-15:30 Coffee Break 15:30-16:00 Mark MacDonald, Lancaster University (lecture notes) Reducing the structural group by using stabilizers in general position 16:10-16:40 Dinakar Muthiah, Brown University Some results on affine Mirkovic-Vilonen theory Friday, June 14 Room 230, Fields Institute 9:30-10:30 Jochen Kuttler, University of Alberta Tensors of bounded ranks are defined in bounded degree 10:30-11:00 Coffee Break 11:00-12:00 Nikolai Vavilov, St. Petersburg State University Commutators in Algebraic Groups

 Speaker & Affiliation Title and Abstract Antieau, Benjamin UCLA Topological Azumaya algebras I will describe how to use topological Azumaya algebras, or, equivalently, principal ${PU_n}$-bundles, to think about two problems in algebraic geometry: the period-index problem about the degrees of division algebras over function fields, and the problem of the existence of projective maximal orders in unramified division algebras. In particular, using topological methods, I will show that projective maximal orders do not necessarily exist, which solves an old problem of Auslander and Goldman. Brosnan, Patrick University of Maryland Algebraic groups related to Hodge theory In Hodge theory, there are several categories of objects that turn out to be (neutral) Tanakian, for example, split Hodge structures, mixed Hodge structures, variations of Hodge structure, etc. As such these categories are equivalent to the category of representations of their Tanakian galois groups. Unfortunately, most of these groups seem difficult to describe explicitly. However, there is an easy description of the category of split real Hodge structures. It is the category of representations of group Deligne called S: the Weil restriction of scalars from C to R of the multiplicative group. Deligne also described real mixed Hodge structures. But here the group involved is more complicated: it is the semi-direct product of S with a pro-unipotent group scheme U. Nilpotent orbits are certain variations of Hodge structure, which can be defined in terms of linear algerbaic data. The simplest of these are the SL2 orbits introduced by Schmid. It turns out that the category of SL2 orbits is equivalent to the category of representations of a certain real reductive algebraic group over R which is a semi-direct product of SL2 and Deligne's group S. I will describe this and a related group which controls certain nilpotent orbits. This gives a group-theoretic understanding of certain operations on variations of mixed Hodge structure, such as, taking the limit mixed Hodge structure. The content of this talk is joint work with Gregory Pearlstein. Brussel, Eric California State Polytechnic University Arithmetic in the Brauer group of the function field of a p-adic curve Joint work with: Kelly McKinnie and Eduardo Tengan. We present machinery that allows us to prove several results concerning the n-torsion subgroup of the Brauer group of the function field F of a p-adic curve, when n is prime to p. We prove that every class of period n is expressible as a sum of two Z/n-cyclic classes, and a more general statement relating symbol lengths of function fields of curves over a complete discretely valued field K and function fields of curves over the residue field of K. We also reprove Saltman's theorem that every division algebra of degree n (not p) over the function field of a p-adic curve is cyclic. Fedorov, Roman Kansas State University 1. A conjecture of Grothendieck and Serre on principal bundles Let R be a regular local ring, G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle on spec(R) is trivial if it has a rational section.This has been proved in many particular cases. Recently Fedorov and Panin, using previous results of Panin, Stavrova and Vavilov, gave a proof in the case, when R contains an infinite field. I will discuss the statement of the conjecture, some corollaries, and the strategy of the proof. 2. A conjecture of Grothendieck and Serre and affine Grassmannians This is a continuation of my previous talk. I will introduce affine Grassmannians parameterizing modifications of principal G-bundles on the projective line over a scheme. While the proof discussed in my first talk does not formally rely on affine Grassmannians, they were crucial in creating this proof. Then I will explain how one can use affine Grassmannians to construct certain "exotic" principal bundles. This will explain why certain "naive" attempts at a proof of Grothendieck-Serre conjecture failed. Gille, Stefan University of Alberta Permutation modules and motives We discuss how permutation resolutions of Chow groupscan be used to compute geometrically split motives (in some cases ). Kuttler, Jochen University of Alberta Tensors of bounded ranks are defined in bounded degree Tensor rank is a very classical notion, naturally arising in algebraic geometry, algebraic statistics, and complexity theory. In this context an old problem is to determine the rank of a given tensor, that is, to find defining equations for the variety of tensors of a given (border) rank k. In this talk I will report on joint work with Jan Draisma, where we prove qualitative results on the variety of p-tensors of border rank at most k. For example we show that this variety is defined by equations of degree at most d = d(k), independent of the number of tensor factors (or the dimension of each factor). Lee, Ting-Yu Fields Institute The local-global principle for embeddings of maximal tori into reductive groups Let G be a reductive group, T be a torus and ${\Psi}$ be a root datum associated with T. In this talk, I will discuss when we can embed T to G as a maximal torus with respect to the root datum ${\Psi}$. Over local fields, the existence of such embedding is determined by the Tits indices of G and ${\Psi}$. Then I will use this to construct an example where the local-global principle for the embedding fails. I will also explain the relation between the embeddings of root data into reductive groups and embeddings of étale algebras with involution into central simple algebras with involution. The latter was discussed in G. Prasad and A. Rapinchuk's paper. Lemire, Nicole University of Western Ontario Equivariant Birational Aspects of Algebraic Tori We examine the equivariant birational linearisation problem for algebraic tori equipped with a finite group action. We also study bounds on the degree of linearisability, a measure of the obstruction for such an algebraic torus to be linearisable. We connect these problems to the question of determining when an algebraic group is (stably) Cayley - that is (stably) equivariantly birationally isomorphic to its Lie algebra. We discuss joint work with Popov and Reichstein on the classification of the simple Algebraic groups which are Cayley and on determining bounds on the Cayley degree of an algebraic group, a measure of the obstruction for an algebraic group to be Cayley. We also relate this to recent work with Borovoi, Kunyavskii and Reichstein extending the classification of stably Cayley simple groups from the algebraically closed characteristic zero case to arbitrary fields of characteristic zero. Lastly, we investigate the stable rationality of four-dimensional algebraic tori and the associated equivariant birational linearisation problem. Matzri, Eli University of Virginia Symbol length over C_r fields A field, F, is called C_r if every homogenous form of degree n in more then n^r variables has a non-trivial solution. Consider a central simple algebra, A, of exponent n over a field F. By the Merkurjev-Suslin theorem assuming F contains a primitive n-th root of one, A is similar to the product of symbol algebras, the smallest number of symbols required is called the length of A denoted l(A). If F is C_r we prove l(A) \leq n^{r-}-1. In particular the length is independent of the index of A. Merkurjev, Alexander University of California, Los Angeles On cohomological invariants of semisimple groups Neftin, Danny University of Michigan, Ann Arbor Noncrossed products over Henselian fields and a Grunwald-Wang problem A finite dimensional division algebra is called a crossed product if it contains a maximal subfield which is Galois over its center, otherwise a noncrossed product. Since Amitsur settled the long standing open problem of existence of noncrossed products, their existence over familiar fields was an object of investigation. The simplest fields over which they occur are Henselian fields with global residue field (such as Q((x)), where Q is the field of rational numbers). We shall describe the "location" of noncrossed products over such fields by proving the existence of bounds that, roughly speaking, separate crossed and noncrossed products. Furthermore, we describe those bounds in terms of Grunwald-Wang type of problems and address their solvability in various cases. (joint work with Timo Hanke and Jack Sonn) Rapinchuk, Igor Yale University On the conjecture of Borel and Tits for abstract homomorphisms of algebraic groups The conjecture of Borel-Tits (1973) states that if $G$ and $G'$ are algebraic groups defined over infinite fields $k$ and $k'$, respectively, with $G$ semisimple and simply connected, then given any abstract representation $\rho \colon G(k) \to G' (k')$ with Zariski-dense image, there exists a commutative finite-dimensional $k'$-algebra $B$ and a ring homomorphism $f \colon k \to B$ such that $\rho$ can essentially be written as a composition $\sigma \circ F$, where $F \colon G(k) \to G(B)$ is the homomorphism induced by $f$ and $\sigma \colon G(B) \to G'(k')$ is a morphism of algebraic groups. We prove this conjecture in the case that $G$ is either a universal Chevalley group of rank $\geq 2$ or the group $\mathbf{SL}_{n, D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic 0 and $n \geq 3$, and $k'$ is an algebraically closed field of characteristic 0. In fact, we show, more generally, that if $R$ is a commutative ring and $G$ is a universal Chevalley-Demazure group scheme of rank $\geq 2$, then abstract representations over algebraically closed fields of characteristic 0 of the elementary subgroup $E(R) \subset G(R)$ have the expected description. We also describe some applications of these results to character varieties of finitely generated groups. Rapinchuk, Andrei University of Virginia The genus of a division algebra and ramification Let $D$ be a finite-dimensional central division algebra over a field $K$. The genus $\mathbf{gen}(D)$ is defined to be the set of the Brauer classes $[D'] \in \mathrm{Br}(K)$ where $D'$ is a central division $K$-algebra having the same maximal subfields as $D$. I will discuss the ideas involved in the proof of the following finiteness result: {\it Let $K$ be a finitely generated field, $n \geqslant 1$ be an integer prime to $\mathrm{char} \: K$. Then for any central division $K$-algebra $D$ of degree $n$, the genus $\mathbf{gen}(D)$ is finite.} One of the main ingredients is the analysis of ramification at a suitable chosen set of discrete valuations of $K$. Time permitting, I will discuss generalizations of these methods to absolutely almost simple algebraic groups. This is a joint work with V.~Chernousov and I.~Rapinchuk. Saltman, David Princeton University Finite u Invariants and Bounds on Cohomology Symbol Lengths We answer a question of Parimala's showing that fields with finite u invariant have bounds on the symbol lengths in their $\mu_2$ cohomology in all degrees. Stavrova, Anastasia Fields Institute On the congruence kernel of isotropic groups over rings We discuss an extension of a recent result of A. Rapinchuk and I. Rapinchuk on the centrality of the congruence kernel of the elementary subgroup of a Chevalley (i.e. split) simple algebraic group to the case of isotropic groups. Namely, we prove that for any simply connected simple group scheme G of isotropic rank at least 2 over a Noetherian commutative ring R, the congruence kernel of its elementary subgroup E(R) is central in E(R). Along the way, we define the Steinberg group functor St(-) associated to an isotropic group G as above, and show that for a local ring R, St(R) is a central covering of E(R). Vavilov, Nikolai St. Petersburg State University Commutators in Algebraic Groups (based on joint work with Roozbeh Hazrat, Alexei Stepanov and Zuhong Zhang) In an abstract group, an element of the commutator subgroup is not necessarily a commutator. However, the famous Ore conjecture, recently completely settled by Ellers-Gordeev and by Liebeck-O'Brien-Shalev-Tiep, asserts that any element of a finite simple group is a single commutator. On the other hand, from the work of van der Kallen, Dennis and Vaserstein it was known that nothing like that can possibly hold in general, for commutators in classical groups over rings. Actually, these groups do not even have bounded width with respect to commutators. In the present talk, we report the amazing recent results which assert that exactly the opposite holds: over any commutative ring commutators have bounded width with respect to elementary generators, which in the case of SL_n are the usual elementary transformations of the undergraduate linear algebra course. Technically, these results are based on a further development of localisation methods proposed in the groundbreaking work by Quillen and Suslin to solve Serre's conjecture, their expantion and refinement proposed by Bak, localisation-completion, further enhancements implemented by the authors (R.H., N.V, and Z.Zh.), and the terrific recent method of universal localisation, devised by one of us (A.S.) Apart from the above results on bounded width of commutators, and their relative versions, these new methods have a whole range of further applications, nilpotency of K_1, multiple commutator formulae and the like, which enhance and generalise many important results of classical algebraic K-theory. In fact, our results are already new for the group SL_n, and time permitting I would like to mention further related width problems (unipotent factorisations, powers, etc.) and connections with geometry, arithmetics, asymptotic group theory, etc. 30-minute Talks Bermudez, Hernando Fields Institute, Emory University Degree 3 Cohomological Invariants of Split Quasi-Simple Groups In this talk I will discuss the results of a recent joint work with A. Ruozzi on degree 3 cohomological invariants of groups which are neither simply connected nor adjoint. Using recent results of A. Merkurjev we obtain a description of these invariants and we show how our results relate to previous constructions. We also obtain further applications to algebras with orthogonal involution. MacDonald, Mark Lancaster University Reducing the structural group by using stabilizers in general position For a reductive linear algebraic group G (over the complex numbers), all linear representations have the property that on an open dense subset of V, the stabilizers are all conjugate to each other. This is a result of Richardson and Luna. If H is an element of that conjugacy class, then any G-torsor (over a field extension of the complex numbers) is induced from an N_G(H)-torsor; in other words, we can reduce the structural group from G to the normalizer of H. This implies that the essential dimension of G is bounded above by that of N_G(H). I will discuss how this extends to more general base fields, in particular those of prime characteristic. The examples of G=F_4 and G=E_7 will be considered. Muthiah, Dinakar Brown University Some results on affine Mirkovic-Vilonen theory MV (Mirkovic-Vilonen) polytopes control the combinatorics of a diverse array of constructions related to the representation theory of semi-simple Lie algebras. They arise as the moment map images of MV cycles in the affine Grassmannian. They describe the combinatorics of the PBW construction of the canonical basis. And they control the submodule behavior of modules for preprojective algebras and KLR algebras. Recently, there has been much work toward extending this picture to the case of affine Lie algebras. I will give a brief overview of the current state of affairs, focusing on some rank-2 results (joint with P. Tingley) and some type A results on MV cycles. Pollio, Timothy University of Virginia The Multinorm Principle The multinorm principle is a local-global principle for products of norm maps which generalizes the Hasse norm principle. Let L_1 and L_2 be finite separable extensions of a global field K. We say that an element of the multiplicative group of K is a local multinorm if it can be written as a product of norms of ideles from L_1 and L_2 and we say that such an element is a global multinorm if it can be written as a product of norms of field elements from L_1 and L_2. Then the pair of extensions L_1, L_2 satisfies the multinorm principle if every local multinorm is a global multinorm. Two basic problems are to determine which pairs of extensions satisfy the multinorm principle and to describe the obstruction to the multinorm principle which is defined as the group of local multinorms modulo the group of global multinorms. I will discuss what is known about each problem. In particular, I will sketch the computation of the obstruction for pairs of abelian extensions using class field theory, group cohomology, and the theory of Schur multipliers. I will also outline a purely cohomological approach to the multinorm problem which is based on the identification of the obstruction with the Tate-Shafarevich group of the associated multinorm torus. Rosso, Daniele University of Chicago Mirabolic Convolution Algebras Several important algebras in representation theory, like Iwahori-Hecke algebras of Weyl groups and Affine Hecke Algebras, can be realized as convolution algebras on flag varieties. Some of these constructions can be carried over to the 'mirabolic' setting to obtain other interesting algebras. We will discuss the convolution algebra of GL(V)-invariant functions on triples of two flags and a vector, which was first described by Solomon, and its connections to the cyclotomic Hecke algebras of Ariki and Koike.

Participants as of June 6, 2013
* to be confirmed

 Full Name University/Affiliation Antieau, Benjamin University of California, Los Angeles Bermudez, Hernando Emory University Brosnan, Patrick University of Maryland Brussel, Eric California State Polytechnic University Burda, Yuri University of British Columbia Chang, Zhihua University of Alberta Chernousov, Vladimir University of Alberta Chintala, Vineeth Tata Institute of Fundamental Research Duncan, Alexander University of Michigan Fedorov, Roman Kansas State University Ferguson, Tom Southwestern Assemblies of God University Gille, Stefan University of Alberta Jacobson, Jeremy The Fields Institute Junkins, Caroline University of Ottawa Kuttler, Jochen University of Alberta Lee, Ting-Yu The Fields Institute Lemire, Nicole Western University Lian, Annie York University Lieblich, Max University of Washington Liu, Dongwen University of Connecticut MacDonald, Mark Lancaster University Mathews, Bryant Azusa Pacific University Matzri, Eliyahu Virginia University McFaddin, Patrick University of Georgia McKinnie, Kelly University of Montana Merkurjev, Alexander University of California, Los Angeles Minác, Ján Western University Muthiah, Dinakar Brown University Neftin, Danny University of Michigan, Ann Arbor Neher, Erhard University of Ottawa Neshitov, Alexander University of Ottawa Pianzola, Arturo University of Alberta Plaumann, Peter Universität Erlangen-Nürnberg Pollio, Timothy University of Virginia Rapinchuk, Andrei University of Virginia Rapinchuk, Igor Yale University Rosso, Daniele University of Chicago Roth, Michael Queen's University Ruozzi, Anthony Emory University Saltman, David J CCR-Princeton Schwartz, Joshua University of Virginia Stavrova, Anastasia The Fields Institute Sun, Jie University of California, Berkeley Turbow, Maren University of Georgia Vavilov, Nikolai St. Petersburg State University Weekes, Alex University of Toronto Wong, Wanshun The Fields Institute Yahorau, Uladzimir University of Alberta Zainoulline, Kirill University of Ottawa Zhang, Yichao University of Connecticut Zhong, Changlong The Fields Institute