Algebraic and Geometric Aspects of Algebraic Groups and Homogeneous Varieties

本文应用表示论、代数群、Galois上同调和代数几何的方法研究了域和曲线上代数群的扭量问题。*1.环的旋转不变理论：*我们研究了代数k-环面(域上分裂环面的扭量)和有限群作用下分裂环面的商的(稳定)合理性问题和更一般的(稳定)二元分类问题。是否所有稳定的有理代数k环面都是有理的，这是一个有趣的问题，自从Voskresenskii在70年代的开创性工作以来一直是一个有趣的问题。有限群上分裂环面的商的合理性问题最早是由Emmy Noether[N]在她的Galois逆问题上研究的，我们还研究了有限群作用下分裂环面的(稳定)二元等变线性化问题。这等价于询问有限群在第n个Cremona群中是否共轭，第n个Cremona群是射影n空间到线性自同构子群的双态自同构群。Cremona群是一个非常神秘的群：平面的Cremona群是一个重要的研究对象；对于n大于3的情况，我们所知的并不多。2.G丛的模叠叠的本质维度*本质维度是G的一个重要的数值不变量，如Reichstein(2010 ICM)，Buhler和Merkurjev[BR，Re，Me]所介绍的。对于Dhillon，我们将有兴趣研究曲线上G丛的模堆的本质维度，这是数学物理中的一个重要对象，将曲线上的向量丛的模堆推广。*3.霍尔代数和霍尔模：一个小的有限阿贝尔范畴的霍尔代数编码了它的对象的同构类之间扩张的结构。重要的霍尔代数的例子是箭图(Ringel，Green)[Ri，Gr]和曲线上的凝聚层(Kapranov)[Kap2](都考虑在有限域上)。利用Dhillon和Sala[PDF1]，我们有兴趣确定了Kapranov Hall代数上投影线上的辛/正交凝聚层的霍尔模(由Young引入)。例子包括Severi Brauer变种(射影空间的扭曲形式)和广义Severi Brauer变种(Grassmannians的扭曲形式)。扭曲射影齐次簇的Chow群中的挠率问题是一个重要的问题，目前还没有得到很好的理解。有了Junkins[PDF2]和Krashen，我们希望研究推广的Severi Brauer变种的这个问题。重要的早期工作是由Karpenko，Merkurjev[KM]在二次曲面上，以及Karpenko[Kar]；Baek[BA]在Severi Brauer品种上。*

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.*** **