Geometric Stories Seminar
Stewart Library 2-4pm on Thursdays
April 5, 2007
Jacob Mostovoj, UNAM, Cuernavaca
Generalized homology and the Dold-Thom theorem
I shall speak about the ways to define generalized homology
theories. We shall see how to interpret K-homology and stable
homotopy in the spirit of singular homology, and how configuration
spaces of labelled points give rise to homology theories.
March 19, 20, 2007
|Geometric Stories Mini Course
Andrei Losev, ITEP
(Topological) quantum mechanics and field theories and enumerative
Mar 15, 2007
Ludmil Katzarkov, Univ of Miami
Homological Mirror Symmetry for Manifolds of general type
We will discuss a prospective of HMS which has been less looked
at in Physics papers
Mar 15, 2007
| Alexander Karp, Columbia
University, Teachers College
"Euler on Squaring the Circle: in Life and Literature
How much do nonmathematicians know about what mathematicians
do, and how do they come to know it? This talk will address
this issue using Euler's biography as an example. The starting
point for the discussion will be a novel, written at the end
of the 1830s in Russia, in which
Euler appears as a character.
Mar. 1, 2007
|Mohammed Abouzaid, Institute
for Advanced Study
Tropical Geometry and Homological Mirror Symmetry for Toric
I will begin by explaining the statement of the Homological
Mirror Symmetry conjecture for Fano toric varieties and outline
how Lefschetz fibrations have been used to prove the conjecture
in some cases. I will then show how tropical geometry can be
used to prove half of the homological mirror conjecture for
all smooth projective toric varieties (dropping the Fano condition!).
Feb. 1, 2007
|Stephen Kudla, University
Ball quotients and their supersingular loci (after Vollard)
Quotients of the complex n-ball by certain arithmetic are moduli
spaces for abelian varieties with additional structure. This
interpretation allows one to extend these quotients to schemes
over the p-adic integers. The structure of the reduction of
these schemes modulo p, and in particular the supersingular
locus, has a beautiful combinatorial structure. I will attempt
to give a glimpse of this theory for non-specialists.
Jan. 18, 2007
|Selman Akbulut, East Lansing
Topology of Manifolds with Exceptional Holonomy
We will discuss G_2 and Spin(7) manifolds, and the deformations
of associative sub-manifolds in a G_2 manifold, and discuss
various dualities related to mirror symmetry.
Nov. 30, 2006
Nov. 23, 2006
|H. Markwig, IMA Minneapolis
Counting plane elliptic tropical curves with fixed j-invariant
In tropical geometry, usual algebraic varieties are replaced
by certain degenerations which are piece-wise linear. There
is hope that the study of algebraic geometry becomes easier
using the tropical degenerations, as they are piece-wise linear.
In this talk, I want to present an example for a result which
can easily be derived within tropical geometry, whereas the
proof within usual algebraic geometry is hard: the counting
of plane elliptic curves with fixed j-invariant.
Nov. 16, 2006
|S. Payne, Clay Mathematical Institute
Polyhedral complexes for tropical geometryI will discuss
some ideas for describing tropical varieities as thickenings
of polyhedral complexes with integral structure, building upon
earlier ideas of Kempf, Knudsen, Mumford, and Saint-Donat as
well as recent work of Mikhalkin, Gathmann, Markwig, Konstevich,
Soibelman, and many others.
Nov. 9, 2006
Nov. 2, 2006
|Oleg Viro, Uppsala University
Roads that Mathematics did not like to take.
The shapes that mathematical theories acquire while finding
their ways to mainstream mathematical curriculums depend on
accidental circumstances. This costs losses of many bright opportunities.
For example, speaking on differentiable manifolds, one usually
pretends that they have no legitimate singular siblings. This
causes lots of inconveniences. Another example: finite topological
spaces are not familiar to most of mathematicians. Topology
appears to feel ashamed of its finite objects, despite of their
beauty and usability. These and other examples will be considered.
Oct. 26, 2006
|Grigory Mikhalkin, University of Toronto
Amoebae, algae and log-fronts
This talk can be viewed as a continuation of the talk "Amoebaa,
algae, shifts and phases" at 12:10 at the Graduate Student
Seminar. In this second part we'll continue the study of amoebae
and algae. As an example of their applications we'll look at
the geometry of the so-called log-fronts (that appear as frozen
boundaries in statistical physics).
Oct. 12, 2006
Assymtotic cones of semi-simple groups and De Concini-Procesi
First we recall the two approaches to toric varieties. From
one point of view a toric variety is an equivariant compactification
of a complex torus (this is the point of view due to e.g. Khovansky).
From another one a toric variety is a geometric quotient for
an action of a torus on an affine space, thus it is a generalization
of the complex projective space (this point of view is due to
e.g. Cox). Next we describe the De Concini-Procesi compactification
of a semi-simple group G. There are two approaches to this as
well. From one point of view the compactification is just a
nice G x G - equivariant projective variety with open orbit
isomorphic to G. From the second point of view the De Concini-Procesi
compactification is the geometric quotient of a certain affine
cone called the assymptotic cone of a certain extension of G
by the action of a complex torus.
Oct. 28, 2006
|A. Braverman (Brown),
From the Hitchin fibration to the geometric Langlands correcposndence
The talk will consist of two parts. First, I will explain some
geometry of the Hiching integrable system (a.k.a. Hitchin fibration).
Next I will try to explain in what sense one would like to quantize
this integrable system; such a quantization is closely related
to the so called geometric Langlands conjecture.
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