Geometric Analysis and Topology Seminar

Spring 2016


The seminar's usual time is Friday at 11:00am in 517 Warren Weaver Hall (Directions). Special times and dates are marked in red. Click on the title of a talk for the abstract (if available).



Upcoming seminars:

Feb 26,  11am
517 WWH
Richard Bamler
(UC Berkeley)
Convergence of Ricci flows with bounded scalar curvature
Apr 15,  11am
517 WWH
Ian Agol
(UC Berkeley)
Certifying the Thurston norm via SL(2, C)-twisted homology
Apr 22,  11am
517 WWH
Nicola Gigli
(SISSA)
Notions of differential calculus on metric measure spaces
Apr 29,  11am
517 WWH
Balazs Strenner
(IAS)
Algebraic degrees of pseudo-Anosov stretch factors
May 6,  11am
517 WWH
Misha Hlushchanka
(Jacobs University Bremen)
On growth of iterated monodromy groups: a geometric point of view


Organizers: Sylvain Cappell, Jeff Cheeger, Bruce Kleiner, and Robert Young.

Previous semesters:

Abstracts:

Convergence of Ricci flows with bounded scalar curvature, Richard Bamler.  It is a basic fact that the Riemannian curvature becomes unbounded at every finite-time singularity of the Ricci flow. Sesum showed that the same is true for the Ricci curvature. It has since remained a conjecture whether also the scalar curvature becomes unbounded at any singular time. In this talk I will show that, given a uniform scalar curvature bound, the Ricci flow can only degenerate on a set of codimension bigger or equal to 4, if at all. This result is a consequence of a structure theory for such Ricci flows, which relies on and generalizes recent work of Cheeger and Naber.
Certifying the Thurston norm via SL(2, C)-twisted homology, Ian Agol.  We study when the Thurston norm is detected by twisted Alexander polynomials associated to representations of the 3-manifold group to SL(2,C). Specifically, we show that the hyperbolic torsion polynomial determines the genus for a large class of hyperbolic knots in the 3-sphere which includes all special arborescent knots and many knots whose ordinary Alexander polynomial is trivial. This theorem follows from results showing that the tautness of certain sutured manifolds can be certified by checking that they are a product from the point of view of homology with coefficients twisted by an SL(2, C)-representation.
Notions of differential calculus on metric measure spaces, Nicola Gigli.  I shall discuss in which sense generic metric measure spaces possess a weak first-order differential structure. Building on this, I will then turn to spaces with Ricci curvature bounded from below and illustrate how a second-order calculus can be built on them. In particular, concepts like Hessian, covariant derivative and Ricci curvature will all be well defined.
Algebraic degrees of pseudo-Anosov stretch factors, Balazs Strenner.  Consider a mapping of the torus that stretches and compresses it in two directions. (These are called Anosov maps.) The lift of such a map to the universal cover is the action of a matrix in SL(2,Z) on the plane and the stretch factor is an eigenvalue of the matrix. Therefore only quadratic algebraic integers can be stretch factors of the torus. For higher genus surfaces, the topology of the surface still imposes constraints on the possible algebraic degrees of the stretch factors, but now a wider variety of degrees may appear. In this talk, I will explain a construction that realizes stretch factors of all possible degrees.
On growth of iterated monodromy groups: a geometric point of view, Misha Hlushchanka.  Iterated monodromy group (IMG) is a self-similar group associated to every branched covering f of the 2-sphere (in particularly to every rational map). It was observed that even very simple maps generate groups with complicated structure and exotic properties which are hard to find among groups defined by more classical methods. For instance, IMG(z^2+i) is a group of intermediate growth and IMG(z^2-1) is an amenable group of exponential growth. Unfortunately, we still face a lack of general theory which would unify and explain these nice examples. In the talk I will first make a detour to the theory of growth of groups and overview the current state of studies of algebraic properties of IMGs. Then I will concentrate on a specific example of a rational map, whose Julia set is given by the whole Riemann sphere, and prove that its IMG has exponential growth. The proof is based on the Geometry of the tilings associated to the map. This is a joint project with Mario Bonk and Daniel Meyer.


Please email comments and corrections to bkleiner@cims.nyu.edu.