Algebraic Geometry Seminar Spring 2017

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)	Equations of Kalman varieties	local (Steven)
March 31	Jie Zhou (Perimeter Institute)	Gromov-Witten invariants of elliptic curves and moments of Weierstrass P-function	Andrei
April 7	Vladimir Dokchitser (Warwick)	Arithmetic of hyperelliptic curves over local fields	Jordan
April 14	Laurentiu Maxim (UW-Madison)	TBA	local	April 21	Vladimir Sotirov	TBA	local

Abstracts

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

We work with some moduli spaces of (parabolic) Higgs bundles which come in infinite families indexed by rank. I'll give some motivation for the study of parabolic Higgs bundles, but the main problem will be to describe the moduli spaces. By applying some integral transforms, most importantly the Fourier-Mukai transform associated to the Poincare line bundle, we are able to reduce the rank of the problem and eventually get a good presentation of the moduli spaces. One fun technique involved in the argument deals with the spectrum of a one-parameter family of linear operators. When such an operator degenerates to one that is diagonalizable with repeated eigenvalues, the spectrum of the operator admits a scheme-theoretic refinement in a certain blowup which carries more information than simply the eigenvalues with multiplicity.

Amy Huang

Equations of Kalman Varieties

Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. We will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. This long exact sequence is first conjectured by Sam in 2011. Time permitting we will also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Jie Zhou

Gromov-Witten invariants of elliptic curves and moments of Weierstrass P-function

I will talk about a joint work with Si Li on the computation of higher genus Gromov-Witten invariants of elliptic curves using mirror symmetry.

The Gromov-Witten theory for elliptic curves is proved by Si Li, basing on the works of Bershadsky-Cecotti-Ooguri-Vafa and Costello-Li, to be equivalent to a quantum field theory on the mirror elliptic curve. Taking the Feynman graph integrals as the definition of the quantum field theory, I will explain the computations on the integrals (which are closely related to moments of the Weierstrass P-function). I will also discuss the quasi-modularity and the modular completion of the integrals. The Hodge-theoretic interpretations of all of these will also be explained.

Vladimir Dokchitser

Arithmetic of hyperelliptic curves over local fields

Let C:y^2 = f(x) be a hyperelliptic curve over a local field K of odd residue characteristic. We show how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of f(x).