# The Use of Linear Algebraic Groups in Geometry and Number Theory (15w5016)

## Organizers

Skip Garibaldi (University of California at Los Angeles)

Nicole Lemire (University of Western Ontario)

Raman Parimala (Emory University)

Kirill Zainoulline (University of Ottawa)

## Objectives

The last years can be characterised as a boom of research activity in the area of linear algebraic groups and its applications. We should mention here recent results by

1. Fedorov and Panin on the proof of the geometric case of the Grothendieck-Serre conjecture using the theory of affine Grassmannians.

We recall that the geometric case of the Grothendieck-Serre conjecture stated in the mid-1960's says that if a G-torsor (defined over a smooth algebraic variety X) is rationally trivial, then it is locally trivial (in Zariski topology), where G is a smooth reductive group scheme over X.

This conjecture has a long history: It was proven for curves and surfaces for quasi-split groups by Nisnevich in the mid-1980's and for arbitrary tori by Colliot-Thelene and Sansuc in the late 1980’s. If G is defined over a field, then the conjecture is known as Serre's conjecture and was proven by Colliot-Thelene, Ojanguren and Raghunathan in the beginning of the 1990's. Serre’s conjecture was proven for most classical groups in the late 1990's by Ojanguren-Panin-Suslin-Zainoulline and for isotropic groups in 2009 by Panin-Stavrova-Vavilov. However, no general proof was known up to now. Very recently, Fedorov and Panin found an original approach that proves the conjecture using the theory of affine Grassmannians coming from Langlands' program.

2. Merkurjev on computations of the group of degree 3 cohomological invariants using new results concerning motivic cohomology.

According to J.-P. Serre, by a cohomological invariant one means a natural transformation from the first Galois cohomology with coefficients in an algebraic group G (the pointed set which describes all G-torsors) to a cohomology functor h(-), where h is a Galois cohomology with torsion coefficients, a Witt group, a Chow group with coefficients in a Rost cyclic module M, etc. The ideal result here would be to construct enough invariants to classify all G-torsors/linear algebraic groups. The question was put on a firm foundation by Serre and Rost in the 1990's, allowing the proof of statements like "the collection of cohomological invariants of G is a free module over the following cohomology ring" for certain groups G; this theory is expounded in the 2003 book by Garibaldi-Merkurjev-Serre.

Using this theory one obtains a complete description of all cohomological invariants landing in degree 1 Galois cohomology for all algebraic groups, in degree 2 for connected groups, and in degree 3 for simply connected semisimple groups (Rost). In a breakthrough recent development, Merkurjev provided a complete description of degree 3 invariants for semisimple groups, solving a long standing question. This has ignited new activity, with researchers trying to understand the full power of his new methods, as well as to understand the new invariants discovered as a corollary of his results. Merkurjev, in some cases, describes concretely the group of degree 3 invariants as a finite group, but in most cases, there is no such concrete description of the members of this group (which are morphisms of functors).

3. Prasad and Rapinchuk on length spectra of locally symmetric spaces.

The answer to the question "Can you hear the shape of a drum?" is famously no, but variations of the problem such as restricting the collection of spaces under consideration or strengthening the hypothesis has led to situations where the answer is yes. In a remarkable 2009 Pub. Math. IHES paper, Prasad and Rapinchuk introduced the notion of weak commensurability of semisimple elements of algebraic groups and of arithmetic groups and used this new concept to address the question of when arithmetically defined locally symmetric spaces have the same length spectrum. In this paper they also settled many cases of the long-standing question of when algebraic groups with the same maximal tori are necessarily isomorphic. This paper and the stream of research stemming from it connects algebraic groups and their Galois cohomology -- the central subject of this conference -- with arithmetic groups, with geometry, and even with transcendental number theory.

4. Baek-Garibaldi-Gille-Queguiner-Zainoulline on applications of the algebraic cycles and Grothendieck gamma filtration to the invariants of torsors and algebras with involutions.

Let X be the variety of Borel subgroups of a simple linear algebraic group G over a field k. In a series of papers it was proven that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is closely related to the torsion of the Chow group and hence to the group of cohomological invariants in degree 3 computed recently by Merkurjev. As a byproduct of this new striking connection, one obtains an explicit geometric interpretation/description of various cohomological invariants in degree 3 as well as new results concerning algebraic cycles and motives of projective homogeneous spaces.

5. Ananyevski-Auel-Garibaldi-Zainoulline on new links between the Steinberg basis for Grothendieck K_0 of twisted flag varieties and the celebrated question by Beilinson on exceptional collections of bundles in the derived categories of flag varieties.

The existence question for full exceptional collections in the bounded category of coherent sheaves Db(X) of a smooth projective variety X was initiated by foundational results of Beilinson and Bernstein–Gelfand–Gelfand. In a recent paper, Ananyevski-Auel-Garibaldi-Zainoulline address the following question:

Let X be the variety of Borel subgroups of a split semisimple linear algebraic group G and fix a partial order P on the Weyl group W of G. Does Db(X) have a P-exceptional collection of the expected length consisting of line bundles?

The methods they use come from the theory of torsors and results concerning Grothendieck K_0 of twisted flag varieties by Panin.

6. Brosnan-Reichstein-Vistoli on genericity theorems for the essential dimension of algebraic stacks and their applications.

Techniques from the theory of algebraic stacks are used to prove genericity theorems to bound their essential dimension which is then applied to finding new bounds for the more classical essential dimension problems for algebraic groups, forms and hypersurfaces. These genericity theorems have also been used in particular by Biswas, Dhillon and Lemire to find bounds on the essential dimension of stacks of (parabolic) vector bundles over curves.

7. Borovoi-Kunyavskii-Lemire-Reichstein on the classification of simple stably Cayley groups.

A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples of Cayley groups with his Cayley map back in 1846. Over an algebraically closed field of characteristic 0, Cayley and stably Cayley simple groups were classified by Lemire-Popov-Reichstein in 2006. In 2012, the classification of stably Cayley simple groups was extended to arbitrary fields of characteristic 0 by Borovoi-Kunyavskii-Lemire-Reichstein.

8. Beauville and Duncan on classifications of finite groups of low essential dimension.

Duncan used the classification of minimal models of rational G-surfaces in his classification of the finite groups of essential dimension 2 over an algebraically closed field of characteristic 0. Beauville more recently used Prokhorov's classification of rationally connected threefolds with an action of a simple group to classify the finite simple groups of essential dimension 3.

All these topics, especially their applications and links to different areas of mathematics, will form the subject of the workshop.

As most of these achievement are very recent (some of them exist at the level of prepublications only) and have not been intensively discussed yet, it is reasonable to expect that a large interdisciplinary meeting of 5-days workshop format can contribute a lot to the development of these areas; can bring together specialists and young researchers (PhDs and PDFs); can help to establish new links and projects.