| DATE |
SPEAKER |
TOPIC |
| Sep 17 |
Benedict Gross
Harvard University |
Integral zeta values and the number of automorphic representations
Abstract
|
| Sep 24 |
David Donoho
Stanford University |
Counting Faces of
Randomly-Projected Polytopes,
with applications to Compressed Sensing, Error-Correcting
Codes, and Statistical Data Mining.
Abstract
|
| Oct 1 |
Alan Reid
University of Texas |
The geometry and topology of
arithmetic hyperbolic 3-manifolds
This talk will discuss recent
advances in regards to some
of the main open problems about hyperbolic 3-manifolds in the context
of arithmetic hyperbolic 3-manifolds.
|
| Oct 8 |
No colloquium
|
Yom Kippur
|
| Oct 15 |
David Vogan
MIT |
Unitary representations of simple Lie groups
By 1950, work of Gelfand and others had led to a general program for
"non-commutative harmonic analysis": understanding very general
mathematical problems (particularly of geometry or analysis) in the
presence of a (non-commutative) symmetry group G. A first step in
that program is classification of unitary representations - that is,
the realizations of G as automorphisms of a Hilbert space. Despite
tremendous advances from the work of Harish-Chandra, Langlands, and
others, completing this first step is still some distance away.
Since functional analysis is not as fashionable now as it was in 1950,
I'll explain some of the ways that Gelfand's problem can be related to
algebraic geometry (particularly to equivariant K-theory). I'll also
discuss the (closely related) question of whether computers may be
able to help solve these problems.
|
| Oct 22 |
Frans Oort
Utrecht and Columbia |
Three conjectures in arithmetic
geometry
We discuss the Manin-Mumford conjecture (about the closure of any set of
torsion points in an abelian variety),
the Andr\'e-Oort conjecure (about the closure of any set of CM-points in a
moduli space)
and the Hecke Orbit Conjecture (about the closure of the Hecke orbit of a
moduli point). These conjectures,
on the borderline of geometry and arithmetic, seem to be (have been)
solved. We explain the similarities.
We will discuss the motivation for these conjectures, and in some cases we
will say something about methods of proofs.
|
| Oct 29 |
No colloquium
|
Fall break
|
| Nov 5 |
János Kollár
Princeton University |
Cremona transformations and
homeomorphisms of topological surfaces
Abstract
|
| Nov 12 |
Bao Châu Ngô
IAS |
Fundamental lemma and Hitchin fibration
The fundamental lemma is an identity of orbital integrals on p-adic reductive groups which
was stated precisely by Langlands and Shelstads as a conjecture in the 80's. We now have
a proof due to the efforts of many peoples with many ingredients. I will only explain how
a certain particular type of geometry like affine Springer fibers and Hitchin were helpful in this
proof.
|
| Nov 19 |
Valery Alexeev
University of Georgia |
Higher-dimensional generalizations of stable curves
I will speak about recent progress in extending the notion of stable
curves, stable maps,
and Gromov-Witten invariants to higher dimensions, in which the
curve is replaced by a surface,
a 3-fold, etc.
The particular case I will describe concerns projective
spaces with n
hyperplanes in it, which would be an analogue of a curve with n
points. Depending on the choice of additional parameters
("weights"), the moduli space of
such "stable pairs" could be either a very simple space, such as a
particular toric variety corresponding
to a fiber polytope, or even a product of projective spaces, or it
could be arbitrarily singular and complicated.
I will explain the connections between this case and tropical
geometry, as well as applications to
compactifications of moduli spaces of certain surfaces of general type.
|
| Nov 26 |
No colloquium
|
Thanksgiving
|
| Dec 3 |
Bruce Kleiner
Yale University |
A new proof of Gromov's theorem on groups of polynomial growth
|
| Dec 10 |
Kai Behrend
University of British Columbia |
The geometry underlying Donaldson-Thomas theory
Donaldson-Thomas invariants are virtual counts of certain kinds of sheaves
on three-dimensional complex oriented manifolds. Ideally, the
moduli spaces giving rise to these invariants should be critical sets of
"holomorphic Chern-Simons functions" and thus the Donaldson-Thomas
invariants should be the number of critical points of this function,
counted correctly. Currently, such holomorphic Chern-Simons functions
exist at
best locally (see my seminar talk on Monday), and it is unlikely that they
exist globally. I will describe geometric structures on the moduli
spaces (some conjectural) that exist globally and reflect the fact that
the moduli spaces look as if they were the zero loci of holomorphic maps.
|