University of Michigan Geometry Seminar

Schedule of Talks

January 23
Jake Rasmussen (Princeton University)
Torus Knots, HOMFLY homology, and the Hecke algebra

Abstract: I'll discuss a homological generalization of the HOMFLY polynomial due to Khovanov and Rozansky. This theory exhibits some remarkable similarities to certain "gauge theoretic" invariants of knots defined using Heegaard Floer homology. Via work of Khovanov, it is also known to be related to the Kazhdan-Lusztig basis for the Hecke algebra. I'll describe how this latter point of view can be used to compute the stable homology of torus knots, and explain how the result relates to some conjectured properties of the KR-homology.

January 30
Natasa Sesum (Columbia University)
Convergence of the Ricci flow

Abstract: In this talk I will mention a few results regarding the convergence of the Ricci flow. The first part of the talk will refer to compact manifolds. We will consider the flow $\frac{\partial}{\partial t}g = -2Ric + \frac{1}{\tau}g$ and give the sufficient conditions for the convergence of the flow to a unique limiting Ricci soliton metric (the metric that moves under the flow just by diffeomorphisms). We will mention a few results in the direction of the convergence of the K\"ahler Ricci flow as well. In the second part of the talk we will focus on a complete Ricci flow of a conformally flat metric on R^2 that blows up at finite time $T < \infty$. Studying the behaviour of this flow is equivalent to studying the behaviour of a solution to the logarithmic fast diffusion equation of the conformal factor $\frac{\partial}{\partial t}u = \Delta log u$. We will describe the extinction profile of the conformal factor by giving two regions of behaviour, that is, our solution after the right rescalings converges to a cigar in the interior and to a logarithmic cusp in the outer region. If the time allows we will say what happens in higher dimensions.

References (Check the title link above):

1. Type II extinction profile of maximal solutions to the Ricci flow in R^2
2. On the extinction profile of solutions to fast diffusion
3. Linear and dynamical stability of Ricci flat metrics
4. Convergence of the Ricci flow towards a unique soliton

February 6
Ron Fintushel (Michigan State University)
Surgery on nullhomologous tori and small 4-manifolds

Note Special Location: 1360 East Hall. This is the Departmental Colloquium as well this week.

Abstract: In the past few years there has been significant progress in constructing exotic smooth simply connected 4-manifolds with small 2nd betti number. I will review these results and show how they can be obtained by a unified technique involving surgery on nullhomologous tori.

Background reading : Six Lectures on Four 4-Manifolds, R. Fintushel and R. Stern, IAS Park City, 2006.

February 13
No meeting today, but check the Colloquium at 4PM, 1360 East Hall,and the specially scheduled seminar Thursday (below):

Yann Bernard (ETH Zurich)
The coupling of gravity to Yang-Mills fields and fermions in
static spherically symmetric spacetimes

Abstract: The coupling of gravity to other classical force fields has given rise to many unexpected phenomena. The resulting solutions of Einstein's field equations have in turn provided interesting insights into the intricate nature of the nonlinear interactions. I will be concerned with coupling SU(n) and U(1) x SU(n) Yang-Mills fields to Dirac fermions over static spherically symmetric Lorentzian spacetimes. I will first review the various ingredients which will be used throughout the talk. Based on plausible first principles, I will then explain how to couple altogether gravity to fermions and SU(n) Yang-Mills fields. Finally, I will study the special class of "black-hole solutions" for the equations resulting from the coupling. It is my hope this talk will be at least comprehensible to most.

February 15
Note change of date; time and place the same (3-4PM, 4088 East Hall).
Joel Fish (Courant Institute, NYU)
Think global, act local - a new approach to Gromov compactness

Abstract: Abstract: Since their introduction by Gromov, pseudo-holomorphic curves have been studied as maps from closed Riemann surfaces into almost complex manifolds with a taming symplectic form. This parameterized view has lead to a number of versions of Gromov compactness which are quite global in nature. For instance, in order to obtain convergence of a sequence of pseudo-holomorphic curves mapping into a family of symplectic manifolds, typically one must first assume the family has uniform bounds on geometric quantities like curvature, injectivity radius, energy threshold, etc. This talk will focus on a new approach to Gromov's compactness theorem, in which the curves are treated as generalized (unparameterized) surfaces. In particular, we prove a local compactness theorem which is useful when considering a family of target manifolds which develop unbounded geometry. This result recovers for instance compactness in the standard "stretching the neck" construction. Furthermore we will also provide applications of the local result to families of connected sums of contact manifolds in which the connecting handle degenerates to a point.

February 20
Anna Wienhard (University of Chicago)
The Teichmueller component for SL(4,R) and foliated projective structures

Abstract: The Teichmüller space of a Riemann surface S can be realized as the moduli space of hyperbolic structures on S. Associating to a hyperbolic structure its holonomy representation, it is naturally embedded into the space of representation of the fundamental group of S into PSL(2,R). There are higher Teichmüller components defined by Hitchin (and in a different way by Fock and Goncharov) as connected components of the space of representations of the fundamental group of S into PSL(n,R). For PSL(3,R) Goldman and Choi showed many years ago that this Teichmüller component is the moduli space of convex projective structures on S. In my talk I will explain joint work with O. Guichard showing that the Teichmüller component for PSL(4,R) is the moduli space of certain convex foliated projective structures on the unit tangent bundle of S.

February 27
No Speaker
"Spring" Break

March 6
Hiroshi Matsuzoe (Mathematics, Nagoya Institute of Technology)
A Generalization of Statistical Manifolds and Weyl Manifolds

Abstract: A statistical manifold is a generalization of a Riemannian manifold with the Levi-Civita connection. Geometirc structures of statistical manifolds naturally arise in geometry of a set of probability density functions, and this geometric structure is useful for the theory of statistical inferences. In this talk, a generalization of statistical manifolds is given.
This generalization turns out to be a generalization of Weyl manifolds, as well.

March 13
Dave Constantine (UM)
Rank Rigidity via Ergodic 2-Frame Flow

Abstract: The rank rigidity theorem of Ballmann and Burns-Spatzier states that a non-positively curved space with higher rank is locally symmetric. Analogous notions of higher rank in strict negative and positive curvature have been developed and similar theorems proven in those curvature settings. In this talk I'll present a recent result in this vein for negatively curved spaces, namely if a compact, negatively curved manifold has what's called higher hyperbolic rank then (subject to a curvature pinching condition in even dimension) it has constant curvature. This provides a new proof of Hammenstadt's hyperbolic rank rigidity theorem (subject to the pinching condition) and also adresses some previously untouched curvature settings. The proof uses a nice geometric description of the dynamics of the frame flow given by Brin.

March 19
Note extra date, time and place: 4-5 PM; 3866 East Hall.
Yael Karshon (University of Toronto)
A compact symplectic four-manifold admits only finitely many inequivalent toric actions

Abstract: A compact symplectic four-manifold admits only finitely many inequivalent toric actions.
This is joint work with Martin Pinsonnault and Liat Kessler.

March 20
C.S. Rajan (IAS, Princeton and Tata Institute, Mumbai)
On Spectrum and Arithmetic

Abstract: We will expound on the theme that the spectrum and arithmetic of compact, locally symmetric spaces arising from congruent arithmetic lattices should mutually determine each other. One side of these expectations can be considered as an archimedean analogue of Tate's conjecture. We verify that the arithmetic is determined in the examples constructed by Sunada. On the other hand for the spaces arising from quaternion division algebras we give examples of isospectral but non-isometric spaces generalizing the examples constructed by Vigneras. Such examples can be interpreted as saying that `Galois conjugation' preserves the spectrum of such spaces.

March 27
Alex Furman (University of Illinois, Chicgo)
Superrigidity via Weyl groups

Abstract: The celebrated superrigidity theorems of Margulis and Zimmer inspired a lot of research on possible generalizations and analogues of the superrigidity phenomena beyond the framework of Lie groups. In this talk I will describe a new approach which gives a unified proof for many of the known results and leads to new ones. The key new concept is a generalization of the classical Weyl group; we view these generalized Weyl groups as the cause of "higher rank" superrigidity phenomena. Based on joint works with Uri Bader, and with Bader and Shaker.

April 3
Yves Guivarch (Universite de Rennes)
On the spectrum of a Zariski-dense subgroup, and some dynamical consequences

Abstract: We consider a semi-simple R-algebraic group G, and a Zariski-dense subsemigroup D of G. We define the spectrum s(g) of a proximal element g, and we consider the closed subgroup <s(D)> generated by the the elements s(g) where g belongs to D. We show that <s(D)> is large, and we describe some dynamical consequences, related to cohomological equations,
central limit theorem for products of random matrices, minimal range of the cocycle associated with the above product,
and D-minimal subsets of compact group extensions of boundaries of G.

April 10
Tadeusz Januszkiewicz (Ohio State University)
Simplicial non-positive curvature and filling invariants

Abstract: "Simplicial non-positive curvature" (SNPC) is a combinatorial condition on links in a simplicial complex which implies many of the consequences of metric non-positive curvature. Many SNPC spaces exists but they -- especially high dimensional ones -- have properties very different from classical examples. Probably the most striking difference is the behaviour of filling invariants. I will try to describe basic concepts, constructions and, time permitting, some applications of wider interest.

The talk is based mainly on joint work with Jacek Swaitkowski.

April 12
Note extra date, time and place: 1-2 PM; room TBA.
Corinna Ulcigrai (Princeton University)
Ergodic properties of flows over interval exchange maps

Abstract: We are interested in a class of area-preserving flows on surfaces, flows given by multi-valued Hamiltonians, which was first introduced by the russian school of Novikov. We will first explain how such flows can be described through suspension flows over interval exchange transformatins (IETs) with logarithmic singularities.

We prove that if the roof function has an asymmetric logarithmic singularity, the suspension flow is mixing for a full measure set of IETs. If the singularity is symmetric, we show weak mixing for a full measure set of IETs and absence of mixing for a special class of IETs. The key ingredients in both cases are renormalization techniques for interval exchange maps estimates on the growth rate of Birkhoff sums of a non integrable function.

April 12-14
Special Event!
RTG Research Workshhop

ASPECTS OF TEICHMUELLER THEORY

This workshop provides an introduction to Teichmueller space and its geometry via lecture series by the speakers below.

Speakers:

Nikolai Ivanov, Michigan State University
Feng Luo, Rutgers University
Michael Wolf, Rice University
Scott Wolpert, University of Maryland

Funding: Limited funding is available for graduate students and post-docs. Please contact an organizer.

Register for the Workshop!

Sponsors: Sponsored by our RTG Grant in Geometry, Topology and Dynamics from the National Science Foundation and the Department of Mathematics at the University of Michigan.

April 17
Speaker: TBA
Title: TBA

Abstract:

May 3-5
Special Event!
Michigan Center for Theoretical Physics Research Workshhop:

There is increasing interest in and progress with G2 Compactifications.

On the one hand the Standard Model of particle physics, and grand unified extensions, have been embedded in manifolds with G2 holonomy. On the other hand it is possible to simultaneously stabilize all moduli, introduce supersymmetry breaking, and have the apparently observed unification of the gauge couplings. Achieving the embedding and these goals simultaneously seems to be easier than in other limits of M theory. However, as you know, there is still relatively little known about G2 manifolds. In particular the construction of compact examples with conical singularities is an important mathematical question with important physics applications.

We are organizing this workshop to try to greatly increase communication between mathematicians, string theorists, and particle theorists in this area.

There will be talks designed to familiarize all the participants about the status, issues, and progress with G2 compactifications, and considerable time left for discussion.

ORGANIZING COMMITTEE:

Bobby Acharya
Konstantin Bobkov
Sergei Gukov
Dominic Joyce
Gordon Kane
Piyush Kumar
Finn Larsen
Jim Liu
Joe Lykken

Register for the Workshop!

Sponsors: Sponsored by MCTP.