University of Utah


Click on the title of a talk for the abstract (if available).
Date  Speaker  Title  
September 8  Roi Docampo Álvarez 
Arcs on determinantal varieties


September 15 

Holomorphic maps between ball quotients
A recent theorem of Koziarz and Mok shows that there can be no
holomorphic submersions between compact ball quotients, apart from totally
geodesic coverings.
After sketching the proof of their result, I will describe the known
examples of holomorphic surjections between ball quotients, namely the Livne
fibrations and the MostowToledo maps, that illustrate the failure of
superrigidity of the automorphism group of complex hyperbolic space. These
two classes of examples can be understood in terms of "forgetful maps"
between DeligneMostow ball quotients, which allows to extend the list of
examples a little.


September 22 

BPS invariants for resolutions of polyhedral singularities
Let G be a finite subgroup of SO(3). Y=GHilb(C^3) gives the preferred
CalabiYau resolution of singularities of C^3/G. We use two different
approaches to determine the BPS invariants of Y: GromovWitten theory, and
a specific moduli space of torsion sheaves on Y. The result is expressed
in terms of an ADE root system canonically associated to G.


September 29 

Speculations on cubic surfaces: Hodge and Galois theory
I will discuss work in progress with Domingo Toledo. (a) What
can one say about the locus of cubic surfaces with periods
certain number fields? (Periods to defined: not what you might think they
are). (b) What can one say about their Galois representations?
The main object of study is a certain natural ladic
representation whose image we hope to prove is large.


October 6  Yunfeng Jiang 
Counting invariants for O+O(2)quiver representations


October 20  Davide Fusi 
Rationality and related problems


October 2425  WAGS / Fall 2009  
October 27 

Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles
We will motivate the work by an important example of the Strange Duality conjecture for moduli spaces of sheaves on the projective plane, in close relation with Barth morphism. The technique used to attack the conjecture in this case, due to Le Potier, He and Danila, uses moduli spaces of coherent systems in order to interpolate the moduli space M_n (of semistable sheaves of rank 2, degree 0 and second Chern class n) with a Hilbert scheme of points in P_2; this procedure allows to reduce the computation of the space of global sections of the determinant line bundle on M_n to the understanding of the cohomology H^*(P_2^[m] , S^q L^[m] ) of the Hilbert scheme of points on P_2 with values in symmetric powers of a tautological bundle L^[m], associated to a certain line bundle L on P_2 . Danila's results on the cohomology of S^q L_[m] yield the strange duality for 0 < n < 19.
We will show how to improve and generalize Danila's results on the cohomology of the Hilbert scheme of points on a smooth surface X with values in representations of tautological bundles. By making use of the derived McKay correspondence of BridgelandKingReid, adapted by Haiman in the case of the Hilbert scheme X^[m], we prove general formulas for the cohomology of X^[m] with values in the double tensor power and general exterior powers of tautological bundles. Finally, we will sketch the work in progress on higher symmetric powers of tautological bundles, directly useful for the strange duality conjecture on the projective plane. (arxiv: 0710.3072)


November 3 

Topology of compactified tropicalizations
I will introduce compact tropicalizations of subvarieties of toric
varieties over nonarchimedean fields, and discuss how the topology of
these compactified tropicalizations relate to analytifications and
weight filtrations.


November 10 

Elliptic K3 surfaces with p^ntorsion sections
We classify K3 surfaces in positive characteristic p, which come with an elliptic fibration and a section of order p^n. Since such elliptic fibrations pull back from the universal elliptic curve over the
Igusa curve, which is the characteristic panalog of the modular curves, we get explicit equations. In particular, we can check arithmetic conjectures and find beautiful
connections between arithmetic and geometry.


November 17 

The coherentconstructible correspondence for toric varieties
This is a talk on joint work with Bohan Fang, ChiuChu Melissa Liu,
and Eric Zaslow. An equivariant ample line bundle on a toric variety
determines a polytope in a real vector space. I will discuss work we
have done to extend this familiar correspondence to an equivalence of
categories, between the derived category of coherent sheaves on the
toric variety and a derived category of polyhedrallyconstructible
sheaves on the real vector space. Many famous aspects of the
dictionary between toric geometry and polyhedral combinatorics are
visible in this equivalence. Work of Nadler and Zaslow relates the
second category to a Fukaya category of Lagrangian submanifolds of a
symplectic vector space. Thus our correspondence can be viewed as a
verificationfor toric varietiesof a version of Kontsevich's
homological mirror symmetry conjecture.


November 24  No seminar today  
December 1  Jimmy Dillies 
Invariants of Dessins


December 8 

Categorical Representations and Mirror Symmetry
Following ideas of KontsevichSoibelman, Tamarkin and Kajiura we study certain categorical representations of the (affine) symplectic group. Our main application is to a conjecture of BenBassat regarding homological mirror symmetry for torus bundles over tori.

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