Difference between revisions of "Geometry and Topology Seminar"

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== Fall 2010 ==
 
 
The seminar will be held  in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
 
 
{| cellpadding="8"
 
!align="left" | date
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|-
 
|September 10
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (UW Madison)
 
|[[#Yong-Geun Oh (UW Madison)|
 
''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants'']]
 
|local
 
|-
 
|September 17
 
|Leva Buhovsky (U of Chicago)
 
|[[#Leva Buhovsky (U of Chicago)|
 
''On the uniqueness of Hofer's geometry'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
|-
 
|September 24
 
|[http://sites.google.com/site/polterov/home/ Leonid Polterovich] (Tel Aviv U and U of Chicago)
 
|[[#Leonid Polterovich (Tel Aviv U and U of Chicago)|
 
''Poisson brackets and symplectic invariants'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
|-
 
|October 8
 
|[http://www.math.wisc.edu/~stpaul/ Sean Paul] (UW Madison)
 
|[[#Sean Paul (UW Madison)|
 
''Canonical Kahler metrics and the stability of projective varieties'']]
 
|local
 
|-
 
|October 15
 
|Conan Leung (Chinese U. of Hong Kong)
 
|[[#Conan Leung (Chinese U. of Hong Kong)|
 
''SYZ mirror symmetry for toric manifolds'']]
 
|Honorary fellow, local
 
|-
 
|October 22
 
|[http://www.mathi.uni-heidelberg.de/~banagl/ Markus Banagl] (U. Heidelberg)
 
|[[# Markus Banagl (U. Heidelberg)|
 
''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry'']]
 
|[http://www.math.wisc.edu/~maxim/ Maxim]
 
|-
 
|October 29
 
|[http://www.math.umn.edu/~zhux0086/ Ke Zhu] (U of Minnesota)
 
|[[#Ke Zhu (U of Minnesota)|
 
''Thick-thin decomposition of Floer trajectories and adiabatic gluing'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
|-
 
|November 5
 
|[http://www.math.psu.edu/tabachni/ Sergei Tabachnikov]  (Penn State)
 
|[[#Sergei Tabachnikov (Penn State)|
 
''Algebra, geometry, and dynamics of the pentagram map'']]
 
|[http://www.math.wisc.edu/~maribeff/ Gloria]
 
|-
 
|November 19
 
|Ma Chit (Chinese U. of Hong Kong)
 
|[[#Ma Chit (Chinese U. of Hong Kong)|
 
''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics'']]
 
|Graduate student, local
 
|-
 
|December 3
 
|[http://www.math.northwestern.edu/~zaslow/ Eric Zaslow]  (Northwestern University)
 
|[[#Eric Zaslow (Northwestern University)|
 
''Ribbon Graphs and Mirror Symmetry'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun and Conan Leung]
 
|-
 
|December 10
 
|Wenxuan Lu  (MIT)
 
|[[#Wenxuan Lu (MIT)|
 
''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
 
Spaces'']]
 
|[http://www.math.wisc.edu/~oh/ Young-Geun and Conan Leung]
 
|-
 
|-
 
|-
 
|}
 
  
 
== Spring 2011 ==
 
== Spring 2011 ==
Line 117: Line 37:
  
 
== Abstracts ==
 
== Abstracts ==
 
==Fall 2010==
 
 
===Yong-Geun Oh (UW Madison)===
 
''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants''
 
 
Gopakumar-Vafa BPS invariant is some integer counting invariant of the cohomology
 
of D-brane moduli spaces in string theory. In relation to the Gromov-Witten theory,
 
it is expected that the invariant would coincide with the `number' of embedded
 
(pseudo)holomorphic curves (Gopakumar-Vafa conjecture). In this talk, we will explain the speaker's recent
 
result that the latter integer invariants can be defined for a generic choice of
 
compatible almost complex structures. We will also discuss the corresponding
 
wall-crossing phenomena and some open questions towards a complete solution to
 
the Gopakumar-Vafa conjecture.
 
 
===Leva Buhovsky (U of Chicago)===
 
''On the uniqueness of Hofer's geometry''
 
 
In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Yaron Ostrover.
 
 
===Leonid Polterovich (Tel Aviv U and U of Chicago)===
 
''Poisson brackets and symplectic invariants''
 
 
We discuss new invariants associated to collections of closed subsets
 
of a symplectic manifold. These invariants are defined
 
through an elementary variational problem involving Poisson brackets.
 
The proof of non-triviality of these invariants requires methods of modern
 
symplectic topology (Floer theory). We present applications
 
to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
 
The talk is based on a work in progress with Lev Buhovsky and Michael Entov.
 
 
===Sean Paul (UW Madison)===
 
''Canonical Kahler metrics and the stability of projective varieties"
 
 
I will give a survey of my own work on this problem, the basic reference is:
 
http://arxiv.org/pdf/0811.2548v3
 
 
===Conan Leung (Chinese U. of Hong Kong)===
 
''SYZ mirror symmetry for toric manifolds''
 
 
===Markus Banagl (U. Heidelberg)===
 
''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry.''
 
 
Using homotopy theoretic methods, we shall associate to certain classes of
 
singular spaces generalized geometric Poincaré complexes called intersection
 
spaces. Their cohomology is generally not isomorphic to intersection
 
cohomology.
 
In this talk, we shall concentrate on the applications of the new
 
cohomology theory to the equivariant real cohomology of isometric actions of
 
torsionfree discrete groups, to type II string theory and D-branes, and to
 
the relation of the new theory to classical intersection cohomology under
 
mirror symmetry.
 
 
===Ke Zhu (U of Minnesota)===
 
''Thick-thin decomposition of Floer trajectories and adiabatic gluing''
 
 
Let f be a generic Morse function on a symplectic manifold M.
 
For Floer trajectories of Hamiltonian \e f, as \e goes to 0 Oh proved that
 
they converge to “pearl complex” consisiting of J-holomorphic spheres
 
and joining gradient segments of f. The J-holomorphic spheres come from the
 
“thick” part of Floer trajectories and the gradient segments come from
 
the “thin” part. Similar “thick-thin” compactification result has
 
also been obtained by Mundet-Tian in twisted holomorphic map setting. In
 
this talk, we prove the reverse gluing result in the simplest setting: we
 
glue from disk-flow-dsik configurations to nearby Floer trajectories of
 
Hamitonians K_{\e} for sufficiently small \e and also show the
 
surjectivity. (Most part of the Hamiltonian K_{\e} is \ef). We will discuss
 
the application to PSS isomorphism. This is a joint work with Yong-Geun Oh.
 
 
===Sergei Tabachnikov (Penn State)===
 
''Algebra, geometry, and dynamics of the pentagram map''
 
 
Introduced by R. Schwartz almost 20  years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate  that the dynamics of the pentagram map  is completely integrable. I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. A surprising relation between the spaces of polygons and combinatorial objects called the 2-frieze patterns (generalizing the frieze patterns of Coxeter) will be described. Eight new(?) configuration theorems of projective geometry will be demonstrated. The talk is illustrated by computer animation.
 
 
===Ma Chit (Chinese U. of Hong Kong)===
 
''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics''
 
 
Using the existence of Einstein metrics on toric Kahler and Sasaki manifolds, a lower bound estimate on the growth of lattice points is obtained for Gorenstein cones. This talk is based on a joint work with Conan Leung. 
 
 
===Eric Zaslow (Northwestern University)===
 
''Ribbon Graphs and Mirror Symmetry''
 
 
I will define, for each ribbon graph, a dg category,
 
and explain the conjectural relation to mirror symmetry.
 
I will being by reviewing how T-duality relates
 
coherent sheaves on toric varieties to constructible sheaves
 
on a vector space, then use this relation to glue
 
toric varieties together.  In one-dimension, the
 
category of sheaves on such gluings has a
 
description in terms of ribbon graphs.
 
These categories are conjecturally
 
related to the Fukaya category of a noncompact
 
hypersurface mirror to the variety with toric
 
components.
 
 
I will use very basic examples.
 
This work is joint with Nicolo' Sibilla
 
and David Treumann.
 
 
 
===Wenxuan Lu (MIT)===
 
''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
 
Spaces''
 
 
We study two instanton correction problems of Hitchin's moduli spaces along with
 
their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space
 
can be put into an instanton-corrected form according to physicists Gaiotto,
 
Moore and Neitzke. The problem boils down to the construction of a set of
 
special coordinates which can be constructed as Fock-Goncharov coordinates
 
associated with foliations of quadratic differentials on a Riemann surface. A
 
wall crossing formula of Kontsevich and Soibelman arises both as a crucial
 
consistency condition and an effective computational tool. On the other hand
 
Gross and Siebert have succeeded in determining instanton corrections of
 
complex structures of Calabi-Yau varieties in the context of mirror symmetry
 
from a singular affine structure with additional data.  We will show that the
 
two instanton correction problems are equivalent in an appropriate sense. This
 
is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which
 
till now has been mostly studied in the framework of geometric Langlands
 
duality.  This result provides examples of Calabi-Yau varieties where the
 
instanton correction (in the sense of mirror symmetry) of  metrics and complex
 
structures can be determined.
 
 
==Spring 2011==
 
  
 
===Mohammed Abouzaid (Clay Institute & MIT)===
 
===Mohammed Abouzaid (Clay Institute & MIT)===
Line 262: Line 59:
 
===Alex Suciu (Northeastern)===
 
===Alex Suciu (Northeastern)===
 
''TBA''
 
''TBA''
 +
 +
 
[[Fall-2010-Geometry-Topology]]
 
[[Fall-2010-Geometry-Topology]]

Revision as of 13:35, 13 January 2011

Spring 2011

The seminar will be held in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date speaker title host(s)
January 21 Mohammed Abouzaid (Clay Institute & MIT)

A plethora of exotic Stein manifolds

Yong-Geun
March 4 David Massey (Northeastern)

TBA

Maxim
March 11 Danny Calegari (Cal Tech))

TBA

Yong-Geun
May 6 Alex Suciu (Northeastern)

TBA

Maxim

Abstracts

Mohammed Abouzaid (Clay Institute & MIT)

A plethora of exotic Stein manifolds

In real dimensions greater than 4, I will explain how a smooth manifold underlying an affine variety admits uncountably many distinct (Wein)stein structures, of which countably many have finite type, and which are distinguished by their symplectic cohomology groups. Starting with a Lefschetz fibration on such a variety, I shall per- form an explicit sequence of appropriate surgeries, keeping track of the changes to the Fukaya category and hence, by understanding open-closed maps, obtain descriptions of symplectic cohomology af- ter surgery. (joint work with P. Seidel)

David Massey (Northeastern)

TBA

Danny Calegari (Cal Tech)

TBA

Alex Suciu (Northeastern)

TBA


Fall-2010-Geometry-Topology