Algebraic and Geometric Aspects of Algebraic Groups and Homogeneous Varieties

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My research applies techniques from representation theory, algebraic groups, Galois cohomology and algebraic geometry to a variety of problems about torsors for algebraic groups over fields and curves. 1. Birational Invariant theory for tori: We study (stable) rationality problems and more generally (stable) birational classification problems for algebraic k-tori (torsors for split tori over a field) and quotients of split tori by finite group actions. The question of whether all stably rational algebraic k tori are rational is an interesting question, open since Voskresenskii's seminal work in the 70s. The rationality problem for quotients of split tori over a finite group was first studied by Emmy Noether [N] in her work on the inverse Galois problem. We also study the (stable) birational equivariant linearisation problem for split tori equipped with the action of a finite group. This is equivalent to asking whether the finite group is conjugate in the nth Cremona group, the group of birational automorphisms of projective n space to a subgroup of linear automorphisms. The Cremona group is a very mysterious group: the Cremona group of the plane is an important object of study; not much is known for n bigger than 3. 2. Essential Dimension of Moduli stacks of G bundles Essential dimension is an important numerical invariant of G torsors, as introduced by Reichstein (2010 ICM), Buhler and Merkurjev [BR,Re,Me]. With Dhillon, we will be interested in studying the essential dimension of the moduli stack of G-bundles over a curve, an important object in mathematical physics, generalising the moduli stack of vector bundles over a curve. 3. Hall Algebras and Hall modules: A Hall algebra of a small finitary abelian category encodes the structure of extensions between isomorphism classes of its objects.Important examples of Hall algebras are those of quivers (Ringel, Green) [Ri,Gr] and for coherent sheaves over a curve (Kapranov) [Kap2] (all considered over a finite field). With Dhillon and Sala [PDF1], we are interested in determining the Hall module (introduced by Young) of symplectic/orthogonal coherent sheaves over the projective line over Kapranov's Hall algebra. 4. Twisted projective homogeneous varieties: Twisted projective homogeneous varieties are forms of projective homogeneous varieties. Examples include Severi Brauer varieties (twisted forms of projective space) and Generalised Severi Brauer varieties, twisted forms of Grassmannians. The question of torsion in Chow groups of twisted projective homogeneous varieties is an important one and not well understood. With Junkins [PDF2] and Krashen, we wish to study this question for generalised Severi Brauer varieties. Important earlier work was done by Karpenko, Merkurjev [KM] on quadrics, and Karpenko [Kar]; Baek [Ba] on Severi Brauer varieties.