Graduate Student Math Conference


Brown University

Tuesday, April 26th 2016

Titles & Abstracts


10:30 - 11:20

David Stapleton

Stony Brook

Hilbert schemes of points and their tautological bundles

The Hilbert scheme of n points on a smooth curve or surface parametrizes scheme-theoretic limits of embedded n-tuples of points. For curves the Hilbert scheme is the nth symmetric power, but in the case of surfaces Fogarty showed the Hilbert scheme is a smooth irreducible crepant resolution of the nth symmetric product of the surface. It is an extremely concrete and bountiful source of examples. For example Hilbert schemes of points on surfaces encompass nearly all known higher dimensional examples of compact hyperkahler manifolds. We will introduce the Hilbert scheme of points along with certain tautological bundles on them. We will pose some open problems in the area and give a new construction using the spectral curves of Beauville-Narasimhan-Ramanan showing that every vector bundle on a curve C is the pullback of a tautological bundle along an embedding of the curve into the Hilbert scheme of points on ℙ2​.

11:30 - 12:20

Representations of p-adic groups via geometric invariant theory

The structure of a reductive p-adic group arises from the interaction of Euclidean geometry and the arithmetic of p-adic fields. Reeder and Yu have drawn upon this interaction to construct certain “epipelagic” representations using geometric invariant theory. In recent work, Jessica Fintzen and I have built on their methods to find new supercuspidal representations. For each of these representa- tions, the Local Langlands Correspondence predicts the existence of a corresponding field extension of Qp, whose Galois theory reflects the structure of the representation. In my talk, I will give explicit examples of representations and corresponding field extensions for the group G2.

12:30 - 1:30

Lunch

Lunch will be provided for those who attend.

1:30- 2:20

Alex Perry

Harvard

Derived Categories

I will discuss the derived categories of Fano varieties of Picard number 1, degree 10, and coindex 3. In particular, I will describe an interesting semiorthogonal component of the derived category of such a variety, and discuss its behavior for some birationally special families of fourfolds. This is joint work with Alexander Kuznetsov.

2:30 - 3:20

Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.

3:30 - 4:00

Tea


4:00 - 4:50

Interpolation for normal bundles of general curves.

This talk will address the following question: When does there exist a curve of given degree d and genus g, passing through n general points p_1, p_2,..., p_n in P^r?

5:00 - 5:50

Aditya Karnataki

Boston University

Canonical Models for Symmetric Spaces

We show that canonical models can be defined over number fields for certain locally symmetric spaces arising from some unitary groups.



Organized by Ken Ascher, Dori Bejleri, Mamikon Gulian, and Laura Walton at the Brown Math Department

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