University of Utah


Click on the title of a talk for the abstract (if available).
Date  Speaker  Title  
January 12 

Chiral Hecke algebra and representations of affine KacMoody Lie algebras
I will discuss the conjectural role that the Chiral Hecke algebra plays in the local geometric Langlands correspondence.
If time permits, I will sketch a possible approach to the unramified case using BRST reduction.


January 19 

The Universal Formulas for Counting Nodal Curves on Surfaces
The problem of counting nodal curves on algebraic surfaces
has been studied since the nineteenth century. On the projective
surface, it asks how many curves defined by homogeneous degree d
polynomials have only nodes as singularities and pass through points
in general position. On K3 surfaces, the number of rational nodal
curves was predicted by the famous YauZaslow formula. Goettsche
conjectured that for sufficiently ample line bundles L on algebraic
surfaces, the numbers of nodal curves in L are given by universal
polynomials in four topological numbers. Furthermore, based on the
YauZaslow formula he gave a conjectural generating function in terms
of quasimodular forms. The formula is consistent with many existing
results on projective surface, K3, and curves with at most 8 nodes on
general surfaces. In this talk, I will discuss how degeneration
methods can be applied to count nodal curves and sketch my proof of
Goettsche's conjecture.




The automorphic groups of general type varieties
(Joint with C. Hacon and J. McKernan)
In this talk, I will sketch a workinprogress, which is aimed to show that the
order of the automorphic group of a general type varietie, which is
wellknown to be finite, is linearly bounded by the volume of K_X.
This is a generalization of the classical Hurwitz's theorem in the 1 dimensional case,
and G. Xiao's theorem in the 2 dimensional case.


January 26  No seminar today 


February 2  Y.P. Lee 
Algebraic Cobordism of Bundles on Varieties


February 9  No seminar today 




Noncommutative desingularizations and clustertilting objects over CohenMacaulay rings
Let M be a reflexive module over a CohenMacaulay local ring R and A = Hom(M,M). If A has finite global dimension, then it has been proposed to serve as a noncommutative analogue of desingularizations of Spec(R). In this talk, we will survey what is known about when such module M exists, and discuss the connections with birational geometry and representation theory of commutative rings. Part of the new results are joint work with Craig Huneke.


February 16  No seminar today 




Filtered linear series and Okounkov bodies
I will present joint work with Huayi Chen where we associate to
certain filtrations on the ring of sections of a big line bundle on a
projective variety a concave function on the Okounkov body of the line
bundle, in such a way that the image of the Lebesgue measure under
this function describes the asymptotic distribution of the jumps of
the filtration. Combining this with results of GilletSoulé we get a
natural and simple proof of the existence of the arithmetic volume as
a limit as well as the Fujita approximation theorem in the context of
Arakelov geometry.


March 2  Aaron Bertram 
Quiver moduli are Bridgeland moduli on the projective plane
Le Potier constructed the moduli spaces of stable torsionfree
sheaves on P^2 by converting them (via the Beilinson construction) into
representations of a quiver. This conversion generalizes to allow geometric
invariant theory constructions of all moduli spaces of Bridgelandstable
objects on P^2. This talk is based on work of Bayer and Macri.


March 9 

Classifying maps of Lagrangian fibrations
Fibrations on holomorphic symplectic manifolds are extremely
restricted. Matsushita proved that the fibres must be complex tori which are
Lagrangian with respect to the holomorphic symplectic form. When the total
space X is projective, Hwang proved that the base of a fibration must be
projective space P^n. We therefore obtain a map from (an open subset of) P^n
into the moduli space of ndimensional abelian varieties. In this talk I
will describe how this "classifying map" can be used to obtain finiteness
results for the number of Lagrangian fibrations, up to deformation.




Stability conditions for the local projective plane
I will talk about the space of stability conditions for the total space of the cotangent bundle on P^2. Its geometry is related to classical questions on Chern classes of stable vector bundles P^2, to the modular group Gamma_1(3), and, via mirror symmetry, to the moduli space of elliptic curves with Gamma_1(3)level structure. This is based on joint work with Emanuele Macri.


March 16  ChingJui Lai 
Varieties fibered by good minimal models


March 30 

Curvature restricitions for (almost) complex manifolds
Consider the following open questions:
Is S^6 a complex manifold?
If a compact M admits a hyperbolic structure, can it be also a complex
manifold?
Perhaps the answer to both questions is no. And perhaps the reason it is so
is because these manifolds admit metrics of constant nonzero curvature.
In studying a (almost) complex manifold, there is no loss in assuming that
it also carries a compatible Riemannian metric (compatibility in the sense
that the almostcomplex structure is orthogonal with respect to the metric).
With it one can use tools of Riemannian geometry to study these objects.
In this talk I will give (very) partial answers to the above questions,
which ultimately do depend on the choice of particular metrics. Nevertheless, they
could provide some insight into the general questions.


April 6 

Mustafin Varieties
A Mustafin variety is a degeneration of projective space induced by a point
configuration in a BruhatTits building. The special fiber is reduced and
CohenMacaulay, and its irreducible components form interesting
combinatorial patterns. For configurations that lie in one apartment, these
patterns are regular mixed subdivisions of scaled simplices, and the
Mustafin variety is a twisted Veronese variety
built from such a subdivision. This connects our study to tropical and
toric geometry. For general configurations, the irreducible components of
the special fiber are rational varieties, and any blowup of projective
space along a linear subspace arrangement can arise. A detailed study of
Mustafin varieties is undertaken for configurations in the BruhatTits tree
of PGL(2) and in the twodimensional building of PGL(3). The latter yields
the classification of Mustafin triangles into 38 combinatorial types.


April 13  Rob Easton 
RiemannRoch Theorem for graphs and tropical curves


April 20  Stefano Urbinati 
Discrepancies on nonQGorenstein varieties
I am going to give an example of a nonQGorenstein variety
which is canonical but not klt, and whose canonical divisor has an
irrational valuation. I will also give an example of an irrational
jumping number and I will prove that there are no accumulation points
for the jumping numbers of normal nonQGorenstein varieties with
isolated singularities.

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