99-00/99.11.10.AGS.10.30.am
Algebra/Geometry Seminar
5-127
Mark de Cataldo
SUNY at Stony Brook
Hilbert schemes and Heisenberg algebras
Abstract:  I will discuss recently uncovered connections between
geometry and represenation theory.
In particular the one between the so-called Hilbert schemes
(do not worry: they are nice smooth configuration spaces) 
and Heisenberg algebras.
The plan is to explain results and how they fit together to the
non-expert.
 


99-00/99.11.17.AGS.10.30.am
Algebra/Geometry Seminar
5-127
Mark de Cataldo
SUNY at Stony Brook
Hilbert Schemes and Heisenberg algebras, II
Abstract:  I will discuss the geometry of the configuration spaces
introduced last time. In particular, how intersection cohomology
plays a role in this picture. I will also discuss how
the Heisenberg algebra acts via this geometric picture.
 


99-00/99.11.24.AGS.10.30.am
Algebra/Geometry Seminar

No Meeting This Week Due To Holiday


Abstract:   


99-00/99.12.1.AGS.10.30.am
Algebra/Geometry Seminar

Cancelled


Abstract:   


99-00/99.12.8.AGS.10.30.am
Algebra/Geometry Seminar
5-127
Meeyoung Kim
SUNY at Stony Brook
On low degree coverings of homogeneous spaces
Abstract:  The Classical Barth-Lefschetz theorem asserts that SMALL codimensional
subvarieties of projective space ``resemble'' the ambient projective space.
Similar phenomena occur for small codimensional subvarieties of more
general homogeneous spaces.
Results of Lazarsfeld assert that a similar situation occurs for
LOW degree branched coverings of projective space.
We obtain Barth-Lefschetz type theorems for low degree branched coverings
of homogeneous spaces.

This is joint work with L. Manivel.
 


00.2.2.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
Amy Ksir
SUNY at Stony Brook
Algebraic Integrable Systems
Abstract:  I will discuss two important examples of algebraically completely
integrable systems.  The first will be Hitchin's system on the
moduli space of principal bundles on a compact Riemann surface.  The
second will be the moduli space of principal bundles on an elliptic
K3 surface.  Both of these moduli spaces are relevant to dualities in
string theory.





00.2.9.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
A. Ksir
SUNY at Stony Brook
Algebraic Integrable Systems (continued)
Abstract:  
I will discuss two important examples of algebraically completely
integrable systems.  The first will be Hitchin's system on the
moduli space of principal bundles on a compact Riemann surface.  The
second will be the moduli space of principal bundles on an elliptic
K3 surface.  Both of these moduli spaces are relevant to dualities in
string theory.
 






00.2.16.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
A. Kirillov, Jr.
SUNY at Stony Brook
Category 0 and geometry of flag variety I
Abstract:   








 


00.2.23.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
A. Kirillov, Jr.
SUNY at Stony Brook
Category 0 and geometry of flag varieties, II
Abstract:   


00.2.23.AGS.4.30.pm
Algebra/Geometry Seminar
Common
George Tomanov
Univ. of Lyon
On the Auslander Conjecture
Abstract:   



00.3.1.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
E. Markman
U. Mass.-Amherst
Hyperkahler Varieties: Their Reflections and Monodromy
Abstract: Reflections of holomorphic symplectic varieties
are certain symplectic birational (surgery) operations.
They play a central role in the study of K3 surfaces and higher dimensional
holomorphic symplectic varieties.



1) We construct holomorphic-symplectic reflections modeled after
dual pairs of Springer resolutions of nilpotent coadjoint orbits.

2) We derive a Picard-Lefschetz formula, describing the reflection
on the level of cohomology rings.

3) We apply these results to study the monodromy representation on the
cohomology ring of Hilbert scheme of points on a K3 surface (and other moduli
spaces of vector bundles on a K3).




00.3.8.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
A. Kirillov
SUNY at Stony Brook
Category 0 and the geometry of flag varieties, III
Abstract:   




00.3.9.AGS.11.00.am
Algebra/Geometry Seminar
S-240
M. Thaddeus
Columbia Univ.
On The Cohomology Ring of The Moduli Space of Higgs Bundles
Abstract:  I will describe joint work with Tamas Hausel.  We show that the
moduli 
space of Higgs bundles on a curve -- or equivalently, representations of
the fundamental group into GL(n,C) -- has cohomology ring generated by
universal classes in the style of Newstead, Atiyah and Bott.  Moreover, we
can give a complete set of explicit relations between these classes when
the rank is 2.





00.3.13.AGS.4.00.pm
Algebra/Geometry Seminar
5-127
Weiqiang Wang
NCSU
McKay correspondence
Abstract:  We will discuss different but
closely related (geometric and finite group theoretic) 
approaches of the McKay correspondence which relates 
the finite subgroups of $SL_2(C)$ to representations 
of affine Lie algebras of ADE types. The main geometric objects
involved are holomorphic symplectic surfaces which are the
resolutions of simple singularities and Hilbert schemes
of points on a surface which is a resolution of orbifold singularities.
The finite groups involved are certain generalized symmetric groups.


00.3.22.AGS.2.30.pm
Algebra/Geometry Seminar

No Meeting This Week


Abstract:   










00.3.31.AGS.11.00.am
Algebra/Geometry Seminar
5-127
Ian Morrison
Fordham Univ.
Mori cones of moduli spaces of pointed stable curves
Abstract:  The talk will be aimed at a fairly broad audience --- a modest
familiarity 
with moduli spaces of curves will be helpful but not essential. I will
first
review basic facts about divisors and other classes on moduli spaces of
pointed stable curves.  I will then give a conjectural description of the
Mori cones (cones of effective curves) of these spaces, discuss a theorem
which describes the cones of numerically effective (nef) divisors assuming
the conjecture, and summarize the evidence for the conjecture including a
few partial results.




00.4.5.AGS.3.00.pm
Algebra/Geometry Seminar
P-131
Barak Weiss
SUNY at Stony Brook
Dynamics of subgroup actions on homogeneous spaces--algebraic applications
Abstract:   
Let G be a Lie group and H and J closed subgroups. There is a natural
"subgroup action" of H on the homogeneous space G/J, which may be studied
from the point of view of dynamics of group actions. Over the past decade,
application of the dynamical point of view has yielded spectacular
solutions to some classical questions in algebra and number theory.

In this survey talk I will illustrate these developments by describing
how Eskin, Mozes and Shah used dynamical ideas to solve the following
elementary question:

Let p(x) be a polynomial of degree n with integer coefficients. What are
the asymptotics (in T) of the number of integer matrices with norm at most
T and characteristic polynomial p(x)?

The talk is aimed at a wide audience and no previous acquaintance with
this field is assumed.






00.4.12.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
Ionut Chiose
SUNY at Stony Brook
Holomorphic Morse Inequalities on Covering Manifolds
Abstract:  The existence of $L^2$ holomorphic sections of invariant line
bundles over 
coverings of (not necessarily compact) complex manifolds depends upon the
positivity of the curvature of the line bundle. The von Neumann dimension
of the space of $L^2$ holomorphic sections is bounded below under
reasonable curvature conditions. Applications of this result include a
weak Lefshetz theorem and a criterion for a strongly pseudoconcave
manifold to be Moishezon.
 


00.4.19.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
Indranil Biswas
TATA and U. Chicago
Projective structures on a Riemann surface
Abstract:  A projective structure on a Riemann surface
is defined by giving a holomorphic coordinate
atlas such that the transition functions are
all Mobius transformations. There are many
equivalent ways of defining projective
structures. For a  Riemann surface $X$ equipped with
a projective structure, the space of differential
operators on $X$ admits a simple description. The
space of equivariant immersions of the universal
cover of $X$ into ${\bbb CP}^n$ satisfying certain
nondegeneracy condition can be identified with
the space of projective structures on $X$ and
higher order differentials (of order from
3 to $n+1$).
 


00.4.26.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
Shaun Martin
SUNY at Stony Brook
An overview of Equivariant Cohomology I and II
Abstract: Abstracts: A 2-lecture introduction to equivariant cohomology.
I.   Equivariant Cohomology
 -   functorial properties
 -   topological definition
 -   equivariant differential forms
 -   references to relevant literature
II.  Fixed point localization formulas
 -   approach via equivariant differential forms
 -   approach via pushforward homomorphisms
III. Equivariant K-theory
 -   definition
 -   relation to index theory and representation theory
 -   localization in equivariant K-theory
IV.  Generalized Equivariant cohomology theories
 -   properties
 -   localization and pushforwards
  


00.5.10.AGS.2.30.pm
Algebra/Geometry Seminar
P-131
Shaun Martin
SUNY at Stony Brook
An overview of Equivariant Cohomology I and II
Abstract:  Abstracts: A 2-lecture introduction to equivariant cohomology.

I.   Equivariant Cohomology

 -   functorial properties

 -   topological definition

 -   equivariant differential forms

 -   references to relevant literature

II.  Fixed point localization formulas

 -   approach via equivariant differential forms

 -   approach via pushforward homomorphisms

III. Equivariant K-theory

 -   definition

 -   relation to index theory and representation theory

 -   localization in equivariant K-theory

IV.  Generalized Equivariant cohomology theories

 -   properties

 -   localization and pushforwards