Difference between revisions of "Geometry and Topology Seminar"

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(Replacing page with '== Fall 2012 == The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm {| cellpadding="8" !align="left" | date !align="left" | speaker !align...')
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== Spring 2011 ==
== Fall 2012 ==
The seminar will be held  in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
The seminar will be held  in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
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!align="left" | host(s)
!align="left" | host(s)
|January 21
|October 7
|Mohammed Abouzaid (Clay Institute & MIT)
|David Fisher (Indiana University)
|[[#Mohammed Abouzaid (Clay Institute & MIT)|
|[[#David Fisher (Indiana University)|
''A plethora of exotic Stein manifolds'']]
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|[http://www.math.wisc.edu/~rkent/ Richard]
|February 4
|[http://www.math.wisc.edu/~maxim/ Laurentiu Maxim] (UW-Madison)
|[[#Laurentiu Maxim (UW-Madison)|
''Intersection Space Homology and Hypersurface Singularities'']]
|February 11
|[http://www.math.wisc.edu/~rkent/ Richard Kent] (UW-Madison)
|[[#Richard Kent (UW-Madison)|
''Mapping class groups through profinite spectacles'']]
|February 18
|[http://www.math.wisc.edu/~jeffv/ Jeff Viaclovsky] (UW-Madison)
|[[#Jeff Viaclovsky (UW-Madison)|
''Rigidity and stability of Einstein metrics for quadratic curvature functionals'']]
|March 4
|[http://www.massey.math.neu.edu/ David Massey] (Northeastern)
|[[#David Massey (Northeastern)|
''Lê Numbers and the Topology of Non-isolated Hypersurface Singularities'']]
|[http://www.math.wisc.edu/~maxim/ Maxim]
|March 11
|Danny Calegari (Cal Tech)
|[[#Danny Calegari (Cal Tech)|
''Random rigidity in the free group'']]
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|'''March 23, Wed'''
|Joerg Schuermann (University of Muenster, Germany)
|[[#Joerg Schuermann (University of Muenster, Germany)|
''Generating series for invariants of symmetric products'']]
|[http://www.math.wisc.edu/~maxim/ Maxim]
|April 8
|[http://www.iazd.uni-hannover.de/~dancohen/ Ishai Dan-Cohen] (U. Hannover)
|[[#Ishai Dan-Cohen (U. Hannover)|
''Moduli of unipotent representations'']]
|[http://www.math.wisc.edu/~ellenber/ Jordan]
|April 15
|[http://euclid.colorado.edu/~gwilkin/ Graeme Wilkin] (U of Colorado-Boulder)
|[[#Graeme Wilkin (U of Colorado-Boulder)|
''Moment map flows and the Hecke correspondence for quivers'']]
|[http://www.math.wisc.edu/~mehrotra/ Sukhendu]
|April 22
|[http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (UW-Madison)
|[[#Yong-Geun Oh (UW-Madison)|
''Floer homology and continuous Hamiltonian dynamics'']]
|May 6
|[http://www.math.neu.edu/~suciu/ Alex Suciu] (Northeastern)
|[[#Alex Suciu (Northeastern)|
''Betti numbers of abelian covers'']]
|[http://www.math.wisc.edu/~maxim/ Maxim]
|May 13
|[http://www.math.wustl.edu/~apelayo/ Alvaro Pelayo] (IAS)
|[[#Alvaro Pelayo (IAS)|
''Symplectic Dynamics of integrable Hamiltonian systems'']]
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
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== Abstracts ==
== Abstracts ==
===Mohammed Abouzaid (Clay Institute & MIT)===
===David Fisher (Indiana University)===
''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)
===Laurentiu Maxim (UW-Madison)===
''Intersection Space Homology and Hypersurface Singularities''
A recent homotopy-theoretic procedure due to Banagl assigns to a certain singular space a cell complex, its intersection space, whose rational cohomology possesses Poincare duality. This yields a new cohomology theory for singular spaces, which has a richer internal algebraic structure than intersection cohomology (e.g., it has cup products), and which addresses certain questions in type II string theory related to massless D-branes arising during a Calabi-Yau conifold transition.
While intersection cohomology is stable under small resolutions, in recent joint work with Markus Banagl we proved that the new theory is often stable under smooth deformations of hypersurface singularities. When this is the case, we showed that the rational cohomology of the intersection space can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface.
===Richard Kent (UW-Madison)===
''Mapping class groups through profinite spectacles''
It is a theorem of Bass, Lazard, and Serre, and, independently,
Mennicke, that the special linear group SL(n,Z) enjoys the congruence
subgroup property when n is at least 3.  This property is most quickly
described by saying that the profinite completion of the special
linear group injects into the special linear group of the profinite
completion of Z.  There is a natural analog of this property for
mapping class groups of surfaces.  Namely, one may ask if the
profinite completion of the mapping class group embeds in the outer
automorphism group of the profinite completion of the surface group.
M. Boggi has a program to establish this property for mapping class
groups.  I'll discuss some partial results, and what remains to be
===Jeff Viaclovsky (UW-Madison)===
''Rigidity and stability of Einstein metrics for quadratic curvature functionals''
===David Massey (Northeastern)===
''Lê Numbers and the Topology of Non-isolated Hypersurface Singularities''
The results of Milnor from his now-classic 1968 work "Singular Points of Complex Hypersurfaces" are particularly strong when the singular points are isolated. One of the most striking subsequent results in this area, was the 1976 result of Lê and Ramanujam, in which the h-Cobordism Theorem was used to prove that constant Milnor number implies constant topological-type, for families of isolated hypersurfaces.
In this talk, I will discuss the Lê cycles and Lê numbers of a singular hypersurface, and the results which seem to indicate that they are the "correct" generalization of the Milnor number for non-isolated hypersurface singularities.
===Danny Calegari (Cal Tech)===
''Random rigidity in the free group''
We prove a rigidity theorem for the geometry of the unit ball in the stable commutator length norm spanned by k random elements of the commutator subgroup of a free group of fixed big length n; such unit balls are C^0 close to regular octahedra. A heuristic argument suggests that the same is true in all hyperbolic groups. This is joint work with Alden Walker.
===Joerg Schuermann (Muenster)===
''Generating series for invariants of symmetric products"
We explain new formulae for the generating series of
Hodge theoretical invariants for symmetric products
of complex quasi-projective varieties and mixed Hodge module
complexes. These invariants include the corresponding Hodge
polynomial as well as Hirzebruch characteristic classes,
including those accociated to middle intersection cohomology.
This is joint work with L. Maxim, M. Saito, S. Cappell,
J. Shaneson and S. Yokura.
===Ishai Dan-Cohen (U. Hannover)===
''Moduli of unipotent representations''
Let $G$ be a unipotent group over a field of characteristic zero. The moduli problem posed by all representations of a fixed dimension $n$ is badly behaved. We set out to define an appropriate nondegenracy condition, and to construct a quasi-projective variety parametrinzing isomorphism classes of nondegenerate representations. In my thesis I defined an invariant $w$ of $G$, its \textit{width}, and a nondegeneracy condition appropriate for representations of dimension $n \le w+1$. Unfortunately, the width is bounded by the depth. But for groups $G$, unipotent of depth $\le 2$, a different nondegeneracy condition gives rise to a quasi projective moduli space for \textit{all} $n$.
This talk is based in part on my thesis, and in part on joint work with Anton Geraschenko, part of which was covered by his recent talk in the number theory seminar here in Madison.
===Graeme Wilkin (U of Colorado-Boulder)===
''Moment map flows and the Hecke correspondence for quivers''
Quiver varieties are a fundamental part of Nakajima's work in
Geometric Representation Theory, but some of their basic topological
invariants (such as the cohomology ring) are not yet well-understood. In
the first part of the talk I will give the definition of a quiver variety
and describe some examples, before giving an overview (again with
examples) of some of Nakajima's constructions, one of which is the Hecke
correspondence for quivers. In the second part of the talk I will explain
a new theorem that gives an analytic description of the Hecke
correspondence in terms of the gradient flow of an energy functional.
This is related to an ongoing program to use Morse theory to study the
cohomology of quiver varieties, and, if time permits, then I will state
some conjectures in this direction.
===Yong-Geun Oh (UW-Madison)===
''Floer homology and continuous Hamiltonian dynamics''
Alexander isotopy on the n-disc exists in almost all the known categories
of existing topology; e.g., diffeomorphism, homeomorphism, symplectic diffeomorphism
and symplectic homeomorphism, measure-preserving homeomorphism and others.
In this talk, we will explain our recent result  that Alexander isotopy exists in the category
of Hamiltonian homeomorphisms which  were introduced  by Mueller and the speaker a
few years ago. As a consequence, this implies that the group
of area preserving homeomorphisms of the 2-disc (compactly supported in the interior)
is not simple. The proof uses chain-level Floer homology theory in full throttle.
We will try to give some overview of the proof in this talk.
===Alex Suciu (Northeastern)===
''Betti numbers of abelian covers''
The regular covers of a connected, finite cell
complex <i>X</i>, with group of deck transformations
a fixed abelian group <i>A</i> admit a natural parameter
space, which in the case of free abelian covers of
rank <i>r</i> is simply the Grassmannian of <i>r</i>-planes
in <i>H</i><sup>1</sup>(<i>X</i>, <b>Q</b>). 
The Betti numbers of such covers are determined by the
jump loci for homology with coefficients in rank 1
local systems on <i>X</i>, and the way these loci intersect
with certain algebraic subgroups in the group of characters 
of the fundamental group of <i>X</i>.  Under favorable
circumstances, the finiteness of those Betti numbers is
controlled by the jump loci of the cohomology ring of <i>X</i>. 
In this talk, I will discuss this circle of ideas, and give
some examples from geometry, topology, and group theory
where such computations play a role.
===Alvaro Pelayo (IAS)===
''Symplectic Dynamics of integrable Hamiltonian systems''
I will start with a review the basic notions of Hamiltonian/symplectic vector field and of Hamiltonian/symplectic
group action, and the classical structure theorems of Kostant, Atiyah,
Guillemin-Sternberg and Delzant on Hamiltonian torus actions.
Then I will state a structure theorem for general symplectic torus
actions, and give an idea of its proof. In the second part of the talk
I will introduce new symplectic invariants of completely integrable
Hamiltonian systems in low dimensions, and explain how these invariants
determine, up to isomorphisms, the so called "semitoric systems".
Semitoric systems are Hamiltonian systems which lie somewhere between the more
rigid toric systems and the usually complicated general integrable
systems. Semitoric systems form a fundamental class of integrable systems,
commonly found in simple physical models such as the coupled
spin-oscillator, the Lagrange top and the spherical pendulum. Parts of
this talk are based on joint work with with Johannes J. Duistermaat and
San Vu Ngoc.

Revision as of 16:49, 29 August 2011

Fall 2012

The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date speaker title host(s)
October 7 David Fisher (Indiana University)




David Fisher (Indiana University)