Algebraic Geometry Seminar Spring 2017

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The seminar meets on Fridays at 2:25 pm in Van Vleck B113.

Here is the schedule for the previous semester.

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Spring 2017 Schedule

date speaker title host(s)
January 20 Sam Raskin (MIT) W-algebras and Whittaker categories Dima
January 27 Nick Salter (U Chicago) Mapping class groups and the monodromy of some families of algebraic curves Jordan
March 3 Robert Laudone (UW Madison) The Spin-Brauer diagram algebra local (Steven)
March 10 Nathan Clement (UW Madison) Parabolic Higgs bundles and the Poincare line bundle local
March 17 Amy Huang (UW Madison) TBA local (Steven)
March 31 Jie Zhou (Perimeter Institute) TBA Andrei
April 7 Vladimir Dokchitser (Warwick) TBA Jordan


Sam Raskin

W-algebras and Whittaker categories

Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention because of Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and for its Langlands dual through a subtle construction.

The purpose of this talk is threefold: 1) to introduce a ``stratification" of the category of modules for the affine W-algebra, 2) to prove an analogue of Skryabin's equivalence in this setting, realizing the categoryof (discrete) modules over the W-algebra in a more natural way, and 3) to explain how these constructions help understand Whittaker categories in the more general setting of local geometric Langlands. These three points all rest on the same geometric observation, which provides a family of affine analogues of Bezrukavnikov-Braverman-Mirkovic. These results lead to a new understanding of the exactness properties of the quantum Drinfeld-Sokolov functor.

Nick Salter

Mapping class groups and the monodromy of some families of algebraic curves

In this talk we will be concerned with some topological questions arising in the study of families of smooth complex algebraic curves. Associated to any such family is a monodromy representation valued in the mapping class group of the underlying topological surface. The induced action on the cohomology of the fiber has been studied for decades- the more refined topological monodromy is largely unexplored. In this talk, I will discuss some theorems concerning the topological monodromy groups of families of smooth plane curves, as well as families of curves in CP^1 x CP^1. This will involve a blend of algebraic geometry, singularity theory, and the mapping class group, particularly the Torelli subgroup.

Robert Laudone

The Spin-Brauer diagram algebra

Schur-Weyl duality is an important result in representation theory which states that the actions of [math]\displaystyle{ \mathfrak{S}_n }[/math] and [math]\displaystyle{ \mathbf{GL}(N) }[/math] on [math]\displaystyle{ \mathbf{V}^{\otimes n} }[/math] generate each others' commutants. Here [math]\displaystyle{ \mathfrak{S}_n }[/math] is the symmetric group and [math]\displaystyle{ \mathbf{V} }[/math] is the standard complex representation. In this talk, we investigate the Spin-Brauer diagram algebra, which arises from studying an analogous form of Schur-Weyl duality for the action of the spinor group on [math]\displaystyle{ \mathbf{V}^{\otimes n} \otimes \Delta }[/math]. Here [math]\displaystyle{ \mathbf{V} }[/math] is again the standard [math]\displaystyle{ N }[/math]-dimensional complex representation of [math]\displaystyle{ {\rm Pin}(N) }[/math] and [math]\displaystyle{ \Delta }[/math] is the spin representation. We will give a general construction of the Spin-Brauer diagram algebra, discuss its connection to [math]\displaystyle{ {\rm End}_{{\rm Pin}(N)}(V^{\otimes n} \otimes \Delta) }[/math] and time permitting we will mention some interesting properties of the algebra, in particular its cellularity.

Nathan Clement

Parabolic Higgs bundles and the Poincare line bundle