Difference between revisions of "Geometry and Topology Seminar"
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===Alex Suciu (Northeastern)=== | ===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)=== | ===Alvaro Pelayo (IAS)=== |
Revision as of 14:27, 18 April 2011
Contents
- 1 Spring 2011
- 2 Abstracts
- 2.1 Mohammed Abouzaid (Clay Institute & MIT)
- 2.2 Laurentiu Maxim (UW-Madison)
- 2.3 Richard Kent (UW-Madison)
- 2.4 Jeff Viaclovsky (UW-Madison)
- 2.5 David Massey (Northeastern)
- 2.6 Danny Calegari (Cal Tech)
- 2.7 Joerg Schuermann (Muenster)
- 2.8 Ishai Dan-Cohen (U. Hannover)
- 2.9 Graeme Wilkin (U of Colorado-Boulder)
- 2.10 Yong-Geun Oh (UW-Madison)
- 2.11 Steven Simon (Courant Institute, NYU)
- 2.12 Alex Suciu (Northeastern)
- 2.13 Alvaro Pelayo (IAS)
Spring 2011
The seminar will be held in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
date | speaker | title | host(s) |
---|---|---|---|
January 21 | Mohammed Abouzaid (Clay Institute & MIT) | Yong-Geun | |
February 4 | Laurentiu Maxim (UW-Madison) | local | |
February 11 | Richard Kent (UW-Madison) | local | |
February 18 | Jeff Viaclovsky (UW-Madison) |
Rigidity and stability of Einstein metrics for quadratic curvature functionals |
local |
March 4 | David Massey (Northeastern) |
Lê Numbers and the Topology of Non-isolated Hypersurface Singularities |
Maxim |
March 11 | Danny Calegari (Cal Tech) | Yong-Geun | |
March 23, Wed | Joerg Schuermann (University of Muenster, Germany) | Maxim | |
April 8 | Ishai Dan-Cohen (U. Hannover) | Jordan | |
April 15 | Graeme Wilkin (U of Colorado-Boulder) | Sukhendu | |
April 22 | Yong-Geun Oh (UW-Madison) | local | |
April 29 | Steven Simon (Courant Institute, NYU) | Maxim | |
May 6 | Alex Suciu (Northeastern) | Maxim | |
May 13 | Alvaro Pelayo (IAS) | Yong-Geun |
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)
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 done.
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.
Steven Simon (Courant Institute, NYU)
Equivariant and Orthogonal Ham Sandwich Theorems
This talk will present two generalizations of the Ham Sandwich Theorem, which states that under very broad conditions, any n finite measures on R^n can be bisected by a single hyperplane. Giving the theorem a S^0 interpretation, we provide equivariant analogues for the finite subgroups of the spheres S^1 and S^3. Secondly, we ask for the maximum number of pairwise orthogonal hyperplanes which can bisect a generic set of m measures on R^n, m<n. An ``orthogonal" Ham Sandwich Theorem is found in the form of a lower bound for this number which is defined in terms of the span of real projective space. The two generalizations are unified by finding orthogonal versions of the equivariant results.
Alex Suciu (Northeastern)
Betti numbers of abelian covers
The regular covers of a connected, finite cell complex X, with group of deck transformations a fixed abelian group A admit a natural parameter space, which in the case of free abelian covers of rank r is simply the Grassmannian of r-planes in H^{1}(X, Q). The Betti numbers of such covers are determined by the jump loci for homology with coefficients in rank 1 local systems on X, and the way these loci intersect with certain algebraic subgroups in the group of characters of the fundamental group of X. Under favorable circumstances, the finiteness of those Betti numbers is controlled by the jump loci of the cohomology ring of X. 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.