January 20
Paul Bressler (Arizona)
Deformation theory I
I will give an introductory lecture on basic notions in formal
deformation theory with a view towards deformations of complex
manifolds, homological mirror symmetry and present numerous examples.
January 27
Paul Bressler (Arizona)
Deformation theory II
February 3
David Glickenstein (Arizona)
An introduction to Perelman's work on
the Poincare conjecture
This talk is intended as an introduction to the field of Ricci flow and
Perelman's first paper, whose aim is to solve the Poincare conjecture.
We plan to slowly go through the details of Perelman's first paper and
much of the background related to it (comparison geometry, log-Sobolev
inequalities, etc.) No background other than a rudimentary understanding
of Riemannian curvature will be assumed. In this talk, I plan to give a
(very) brief overview of Hamilton's Ricci flow, some of Perelman's aims,
and a vague outline of Perelman's first paper.
February 10
Haru Pinson (Arizona)
Section 1 of Perelman's paper
We will go through section 1 of Grisha Perelman's paper titled "The
entropy formula for the Ricci flow and its geometric applications". In
particular, we will go through the derivation of some variational
formulas and an upper bound for a certain functional.
February 17
Doug Pickrell (Arizona)
Perelman's entropy paper, III
In the first part of this talk we will discuss Perelman's basic
observation that the Ricci flow is equivalent to a gradient flow (if
there is interest, we may briefly discuss the apparent motivation for
this from physics). In the second part we will show that ``breathers are
solitons'': if one thinks of the Ricci flow as a dynamical system on the
space of metrics modulo automorphisms and rescaling, then all periodic
points are actually fixed points (sections 2 and 3 of Perelman's paper).
February 24
Doug Pickrell (Arizona)
Perelman's entropy paper IV:
breathers are solitons
March 2
Paul Bressler (Arizona)
Non-commutative Fourier-Mukai
transform
March 9
Xiaowei Wang (UCLA)
Moment map, Futaki invariant and
Stability of projective manifolds
In this talk, we will see how various of notions arised in study the
stability of projective manifolds (e.g. Futaki invariants, balanced
condition) can be described in a rather simple way from the moment map
point view.
March 16
Spring break
March 23
Tie Luo (University of
Texas at Arlington)
Holomorphic forms on varieties of
general type
As topology having influence on the zeros of a vector field over a
compact differentiable manifolds, the Kodaira dimension of a complex
projective manifold should affect the zeros of holomorphic forms. The
relation between the Kodaira dimension and number of nowhere vanishing
1-forms is proposed and verified in dimension 3 (using a little bit of
Minimal Model Theory). Partial results in higher dimensions are also
discussed.
March 25, 2pm
John Millson (University of
Maryland)
NOTE SPECIAL DAY AND TIME
The generalized ideal triangle
inequalities in symmetric spaces and buildings with applications to
Hecke algebras and representation rings.
In last year's colloquium I presented joint work with Misha Kapovich
and Bernhard Leeb on the Generalized Triangle Inequalities.
A geodesic segment in a symmetric space of noncompact type and rank m
is determined up to isometry by m real numbers (not just the usual
length). The generalized triangle inequalities are a system of
homogeneous linear inequalities that give conditions on 3 m-tuples of
real numbers that are necessary and sufficient in order that one can
assemble three geodesic segments with these parameters into a triangle.
It is a remarkable fact that the triangle inequalities play a
fundamental role in some important problems from algebra.
In the colloquium I presented four equivalent algebra problems for
GL(n)
1. Eigenvectors of a sum.
2. Singular values of a product.
3. Invariant factors of a product (Hecke algebra structure constants).
4. Irreducible constituents of a tensor product.
and related their solutions to the triangle inequalities. I then
explained how these problems generalize to an arbitrary Chevalley group.
This year I will present a geometry problem (constructing ideal
triangles in a symmetric space) and relate it to four new algebra
problems which can be roughly stated as
1. Kostant/Heckman linear convexity.
2. Kostant nonlinear convexity.
3. The constant term map (associated to a parabolic subgroup) defined
on the spherical Hecke algebra.
4. Branching from Chevalley group G to a Levi subgroup M.
This will be an informal (to some degree speculative) talk about joint
work with Tom Haines, Misha Kapovich and Bernhard Leeb.
March 30
Jaroslaw Wlodarczyk (Purdue University)
Simple Hironaka resolution
Building upon work of Villamayor and Bierstone-Milman we give a proof
of the canonical Hironaka principalization and desingularization. We
introduce here the idea of the homogenized ideals which gives a priori
the canonicity of the algorithm and radically simplifies the proof.
April 6
Yi Hu (Arizona)
Stable
Configurations of Linear Subspaces and Quotient Coherent Sheave
This talk will be very elementary in nature.
Graduate students, including the first year students, are very welcome
to the talk. In specific and elementary (linear algebra) terms, we will
provide some "stability" criteria for systems of linear
subspaces, and for systems of quotient coherent sheaves. Among
several other things, we will explain that in some sense, a system of
linear subspaces (hence also coherent sheaves) is stable if it can be
moved by a linear transformation to a 'balanced' position: the balanced
position is defined by the balance equation which comes from the
vanishing of symplectic moment map. Connection with related fields such
as generalized Gelfand-MacPherson correspondence and generalized Gale
transforms will be discussed. Along the way, we will also mention
some further problems if time permits.
April 13
Philip Foth (Arizona)
Toric degenerations of flag manifolds
Toric degeneration is a procedure of making out of a given variety a
toric variety, which can be readily understood using polytopes. The
good thing is that a flat degeneration has the same Hilbert polynomial
as the original variety and carries other important information about
it. In recent years, this topic gained a significant momentum due to
applications to integrable systems, mirror symmetry, and representation
theory. The whole talk will be rather introductory in nature. I will
define the main objects that we will be dealing with, like toric
varieties, flag manifolds, canonical bases, flat families, sagbi
bases, etc. and explain a general method due to Caldero and
Alexeev-Brion of degenerating a flag manifold to a toric variety. An
explicit 3-dimensional example will be discussed.
April 20
April 27
May 4