Complex ball quotients and cubic hypersurfaces.

It is known since Gauss that the projective equivalence classes of plane
cubic curves are naturally parametrized by the orbits of the modular group
acting in the unit disk. I will discuss some recent work on the  generalization
of this construction to the case of cubic hypersurfaces of higher dimension.
 

Susan Tolman (Urbana-Champaign)

Hamiltonian loop group actions and symplectic quotients.

Based on joint  work with R. Bott and J. Weitsman. If a Lie group G acts in a
Hamiltonian fashion on a symplectic manifold M, there are many tools which
allow one to use information about the topology of M to compute information
about the topology of the symplectic quotient. I will discuss new results which
extend some of these techniques to the case when the loop group LG acts in
a Hamiltonian fashion on a symplectic Banach manifold X.
One reason that these spaces are important is that moduli spaces
of stable vector bundles arise as symplectic quotients in this way.
 

Jun Li (Stanford)

Vanishing of Chern classes of the moduli of vector bundles over a curve.

Let C be a genus g smooth Riemann surface and M(r,d) be the moduli space
of stable vector bundles on C of rank r and degree d. When r=1, the space M(1,d)
is a component of the Picard group of C, and thus is diffeomorphic to a
g-dimensional complex torus. Hence   c_g-i( M(1,d) )=0      for    i < g.
When r=2 and d odd, M(2,d) is a smooth 4g-3 dimensional variety. Newstead
conjectured and Gieseker proved the following vanishing result
c_4g-3-i( M(2,d) )=0   for   i < 2g-1.
In this talk, we will discuss how to generalize this to higher rank cases.
 

Tara Holm (Berkeley)

Symplectic quotients and real loci

Let M be a compact, connected symplectic manifold with a Hamiltonian
action of a compact n-dimensional torus T. Suppose that M is equipped
with an anti-symplectic involution Sigma compatible with the T-action.
The real locus of M is the fixed point set M^Sigma of Sigma.  Duistermaat
introduced real loci, and extended several theorems of symplectic geometry
to real loci. We extend another classical result to real loci: the Kirwan
surjectivity theorem.  In addition, we compute the kernel of the real Kirwan
map.  We will mention several salient examples.  This is joint work with
Rebecca Goldin (George Mason University).
 

Brent Doran (Princeton)

Moduli Spaces as Ball Quotients via Hypergeometric Functions of Deligne-Mostow Type.

In the past few years a number of classical GIT moduli spaces have been shown
to admit complex hyperbolic structures (ball quotients).  Most notable are the
moduli spaces of Del Pezzo surfaces, in particular those of cubic surfaces (Allcock,
Carlson, and Toledo) and of rational elliptic surfaces (Heckman and Looijenga).
The techniques invariably involve carefully chosen auxillary period maps tailored
to each case. A large class of moduli space/ball quotient examples is due to Deligne
and Mostow, in their exploration of moduli spaces of points on P¹and hypergeometric
functions.  At first glance, the new examples are not directly of Deligne-Mostow type,
since the discrete groups do not appear on the various lists of Deligne-Mostow and
Thurston.  However, we show, by taking a view of hypergeometric functions based on
intersection cohomology valued in local systems, that there are a host of such examples
"functorially" obtained as locally symmetric subvarieties of the Deligne-Mostow
examples.  Furthermore, we show there are two "ancestral" Deligne-Mostow examples
with associated "ancestral" hyperball quotients admitting a natural GIT description.
 

Zoran Skoda (Indiana)

Noncommutative quotients via Hopf algebras and localization.

Study of a number of problems in geometry and physics, e.g. quantization, deformation
problems and moduli space compactifications suggests viewing these problems in
larger categories of noncommutative spaces. We focus on comodule algebras over
Hopf algebras as noncommutative generalizations of affine G-varieties. Algebraic
literature in the past has focused on algebras which correspond to their affine quotients.
We pursue a geometric picture instead, using noncommutative localizations to simulate
topology and sheaf theory, what allows the introduction of noncommutative quotient
spaces which are not affine. Locally trivial principal bundles, associated bundles,
coherent states, realization of induction functor via global section functor, and some
other basic constructions generalize to this noncommutative setting. Flat resolutions of
noncommutative quotient spaces which are not necessarily coming from localizations
are also sketched.
 

Megumi Harada (Berkeley)

The symplectic geometry of the Gel'fand-Cetlin basis for representations of the symplectic group.

The Gel'fand-Cetlin canonical basis for a finite-dimensional representation V_lambda of U(n,C)
can be constructed by successive decompositions of the representation by a chain of subgroups
U(1,C) < U(2,C) <...< U(n-1,C) < U(n, C). A key point in the analysis is that the decomposition
of an irreducible representation of U(k,C) under the subgroup U(k-1,C) is multiplicity-free.
Guillemin and Sternberg constructed in the 1980s the Gel'fand-Cetlin integrable system on the
coadjoint orbits O_lambda of U(n,C), which is the symplectic geometric version, via geometric
quantization, of the Gel'fand-Cetlin canonical basis for V_lambda.
For G = U(n,H) the compact symplectic group (also described as the quaternionic unitary group),
however, the decompositions are not multiplicity-free. However, in recent years, Molev et al. have
found a Gel'fand-Cetlin type basis for representations of the symplectic group, using essentially
new ideas, including the Yangian Y(2), an infinite-dimensional quantum group, and a subalgebra
called the twisted Yangian Y^-(2). In this talk I will explain the symplectic and Poisson geometry
underlying the canonical basis for finite-dimensional irreducible representations of U(n,H). In
particular, I will construct an integrable system on the symplectic reductions of the coadjoint orbits
of U(n,H) by U(n-1,H) and explain its correspondence with Molev et al.'s work.
 

Reyer Sjamaar (Cornell)

Singularities of symplectic quotients.

Symplectic reduction frequently gives rise to singular spaces.  This talk will be
a survey of some recent work on the topological and analytical invariants
(Betti numbers, arithmetic genera, etc.) of such singular quotients.
 

Sean Keel (Texas)

Oorts conjecture for A_g.

Sadun and I recently proved that A_g (or M_g^c) does not contain a compact analytic
subvariety of codimension g for g>2, over the complex numbers. I'll explain the
motivation for considering the question (which comes from Fabers conjectures
on the cohomology of M_g), and the ideas behind the proof.
 

Milen Yakimov (Cornell)

Complex simple Poisson Lie groups

For arbitrary triangular simple Poisson Lie group we describe a procedure for
stratification of it into Poisson submanifolds obtained by symplectic reduction.
This leads to an explicit parameterization of the leaves in a Poisson Lie group in
this class. Quasitriangular Poisson strictures on simple Lie groups were classified
by Belavin and Drinfeld. We relate the problem of studying their leaves to works
of Dulfo and Moeglin-Rentschler on primitive spectrum of universal enveloping
algebras of general Lie algebras that are neither solvable, nor semisimple. This
link gives a parameterization of the leaves of an arbitrary Poisson Lie group in
Belavin-Drinfeld's class, fundamentally different from special case of the
standard Poisson structure which was previously studied.
 

Andre Henriques (Berkeley)

The presentability of non-effective orbifolds and gerbes as global quotients.

It is a known fact that every compact effective orbifold is a global quotient of a
manifold by a Lie group. We present another class of orbifolds for which the same
result holds: A-gerbes. An A-gerbe is an orbifold that locally looks like the
quotient of a manifold by a trivial action of a finite abelian group A. We will also
discuss the case of S¹-gerbes (which are not orbifolds), for which the result is false.
 

Markus Pflaum (Germany)

The deformation quantization of symplectic orbispaces.

In the first part of this talk we will provide a geometrically oriented approach to the
theory of orbispaces which originally had been introduced by Chen. We explain the
notion of a vector orbibundle and characterize the good sections of a reduced vector
orbibundle as the smooth stratified sections. In the second part of the talk we  will
elaborate on the quantizability of a symplectic orbispace. By adapting Fedosov's method
to the orbispace setting we show that every symplectic orbispace has a deformation
quantization. As a byproduct we obtain that every symplectic orbifold possesses a star product.
 
 

Mark Roberts (U.K.)

A fibration theorem for momentum maps.

We prove that under reasonable assumptions, the image of the orbit momentum map
of a Hamiltonian action of a compact Lie group can be stratified in such a way that the
map defines a fibre bundle over each stratum, with the reduced phase spaces as fibres.
Some properties of the stratification will also be described. (Joint work with James Montaldi.)
 
 

Dmitri Millionschikov (Moscow State)

Symplectic nilmanifolds and graded Lie algebras.

We study symplectic structures on nilmanifolds (compact quotient spaces of nilpotent Lie groups
over lattices) that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal
length of the descending central sequence. We give a complete classification of filiform Lie algebras
that possess the basis e₁, ..., e_n,   [e_i,e_j]=c_ije_i+j   (they are examples of  N-graded Lie  algebras). In
particular we describe the space of symplectic cohomology classes for each algebra  of the list. It is
proved that a symplectic filiform Lie algebra g is a deformation of some  N-graded symplectic
filiform Lie algebra g₀. But this condition is not sufficient. A spectral sequence is constructed in
order to answer the question whether a given deformation of an N-graded symplectic filiform Lie
algebra g₀ admit a symplectic structure or not. Other applications and examples are discussed.
 
 

Brendan Hassett (Rice)

Flipping the moduli space of curves.

Using methods of Geometric Invariant Theory, we constuct the first flip of
the moduli space of stable curves and discuss how it may be interpretted
as a moduli space in its own right. (Joint work with David Hyeon).
 

Jan Segert (Missouri)

A GIT approach to isomonodromic deformations.

Isomondromic deformations of a meromorphic connection with simple poles
are usually described by differential equations  that do  not respect the
symmetries of the system.  Painleve VI, the Schlesinger equations, and
the semisimple Frobenius manifold equations are all examples of such
differential equations.  We reformulate the isomonodromic deformation
problem using ideas from GIT.  This leads to a simple unified framework for
understanding the known results about Painlve VI using elementary algebraic
geometry, and suggests methods for studying more difficult isomonodromic
deformation problems such as semisimple quantum cohomology.
 
 

Ivan Arzhantsev (Moscow)

On stability of diagonal actions.

We prove that for any action of a semisimple group G on an affine variety M
there exists an integer n such that the diagonal action of G
on the m-fold product M x  M x ... x M is stable for all m> n.
 
 

Anvar Mavlyutov (Indiana)

Embedding of Calabi-Yau deformations  into toric varieties.

Calabi-Yau hypersurfaces in toric varieties have two types of deformations:
"polynomial" - peformed inside the toric variety and "non-polynomial" -
leading outside the ambient space. Recently, we constructed the non-polynomial
deformations of Calabi-Yau hypersurfaces by regluing certain Zariski open subsets.
This generalized examples of non-polynomial deformations of Calabi-Yau
hypersurfaces in some weighted projective spaces due to S. Katz and D. Morrison.
While the latter construction worked only in certain cases, it gave an explicit
embedding of the deformations into products of weighted projective spaces.
We found that our construction also has such an embedding into a toric variety.
More precisely, we show that all deformations of semiample Calabi-Yau
hypersurfaces in complete simplicial toric varieties can be realized as complete
intersections in higher dimensional toric varieties.
 
 

Nicholas Proudfoot (Berkeley)

Hypertoric varieties.

Hypertoric (or toric hyperkahler) varieties are quaternionic analogues of toric varieties,
introduced a few years ago by Bielawski and Dancer.  Like toric varieties, hypertoric
varieties are combinatorially defined, and many of their elegant geometric
properties can be read off from simple combinatorial diagrams.  In this
talk we will discuss some of these properties, emphasizing the differences
between the combinatorics in the toric and hypertoric settings.