Difference between revisions of "Geometry and Topology Seminar"
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===Song Sun (Stony Brook)=== | ===Song Sun (Stony Brook)=== | ||
''Algebraic structure on Gromov-Hausdorff limits'' | ''Algebraic structure on Gromov-Hausdorff limits'' | ||
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+ | Given a sequence of compact Riemannian manifolds of fixed dimension, under fairly general assumptions we can obtain ``Gromov-Hausdorf limits" that are complete metric spaces. When the manifolds have bounded Ricci curvature and non-collapsing volume, the Anderson-Cheeger-Colding theory provides a regular-singular decomposition of a limit space. It is a central question in Riemannian geometry to understand these singularities. In the case when the manifolds are projective and the metrics are Kahler, we will discuss some recent progress towards an algebro-geometric understanding of the singularities of Gromov-Hausdorff limits. This talk is based on joint work with Simon Donaldson. | ||
== Fall 2014== | == Fall 2014== |
Revision as of 22:23, 28 April 2015
The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Richard Kent.
Contents
Spring 2015
date | speaker | title | host(s) | |
---|---|---|---|---|
January 23 | ||||
January 30 | ||||
February 6 | Balazs Strenner (Wisconsin) | Penner’s conjecture on pseudo-Anosov mapping classes. | local | |
Thursday, February 12, at 11AM in VV 901 | Ryan Budney (Victoria) | Operads and spaces of knots. | Kent | |
February 20 | Jenya Sapir (UIUC) | Counting non-simple closed curves on surfaces. | Dymarz | |
February 27 | ||||
March 6 | Botong Wang (Notre Dame) | Deformation theory with cohomology constraints. | Max | |
March 13 | Andrew Sale (Vanderbilt) | A geometric version of the conjugacy problem. | Dymarz | |
March 20 | ||||
March 27 | ||||
Spring Break | ||||
April 10 | Michael Hull (UIC) | Acylindrically hyperbolic groups | Dymarz | |
April 17 | Sean Li (UChicago) | Coarse differentiation of Lipschitz functions. | Dymarz | |
April 24 | ||||
May 1 | Song Sun (Stony Brook) | TBA | Wang | |
May 8 |
Spring Abstracts
Balazs Strenner (Wisconsin)
Penner’s conjecture on pseudo-Anosov mapping classes.
There are many constructions of pseudo-Anosov elements of mapping class groups of surfaces. Some of them are known to generate all pseudo-Anosov mapping classes, others are known not to. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. In this talk I prove that the conjecture is false for most surfaces. (This is joint work with Hyunshik Shin.)
Ryan Budney (Victoria)
Operads and spaces of knots.
I will describe a connection between the geometrization of 3-manifolds and a subject called operads. It manifests itself as a structure theorem for the space of smooth embeddings of the circle in the 3-sphere and points to a homotopy-theoretic approach to old questions about finite-type invariants.
Jenya Sapir (UIUC)
Counting non-simple closed curves on surfaces.
We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani has produced asymptotics for the growth of the number of simple closed curves, and this work has been extended to curves with at most 3 self-intersections with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result.
Botong Wang (Notre Dame)
Deformation theory with cohomology constraints.
Deformation theory is a powerful tool to study the local structure of moduli spaces. I will first give an introduction to the theory of Deligne-Goldman-Millson, which translates deformation theory problems to problems of differential graded Lie algebras. I will also talk about a generalization to deformation theory problems with cohomology constraints. This is used to study the local structure of cohomology jump loci in various moduli spaces.
Andrew Sale (Vanderbilt)
A geometric version of the conjugacy problem.
The classic conjugacy problem of Max Dehn asks whether, for a given group, there is an algorithm that decides whether pairs of elements are conjugate. Related to this is the following question: given two conjugate elements u,v, what is the shortest length element w such that uw=wv? The conjugacy length function (CLF) formalises this question. I will survey what is known for CLFs of groups, giving a sketch proof for a result in semisimple Lie groups. I will also discuss a new, closely related function, the permutation conjugacy length function (PCL). I will outline its potential application to studying the computational complexity of the conjugacy problem, and describe a result, joint with Y. Antolin, for the PCL of relatively hyperbolic groups.
Michael Hull (UIC)
Acylindrically hyperbolic groups
Hyperbolic and relatively hyperbolic groups have played an important role in the development of geometric group theory. However, there are many other groups which admit interesting and useful actions on hyperbolic metric spaces, including mapping class groups, Out(F_n), directly indecomposable RAAGs, and many 3-manifold groups. The class of acylindrically hyperbolic groups provides a framework for studying all of these groups (and many more) using many of the same techniques developed for hyperbolic and relatively hyperbolic groups. We will give a brief survey of examples and properties of acylindrically hyperbolic groups and show how the study of this class has yielded new results in a number of particular cases.
Sean Li (UChicago)
Coarse differentiation of Lipschitz functions.
Bates, Johnson, Lindenstrauss, Preiss, and Schechtman introduced a notion of large scale differentiation for Lipschitz functions between normed linear spaces. We discuss an extension of this result to the nonabelian setting of Carnot groups and use it to derive quantitative estimates for nonembeddability of such groups into certain classes of metric spaces.
Song Sun (Stony Brook)
Algebraic structure on Gromov-Hausdorff limits
Given a sequence of compact Riemannian manifolds of fixed dimension, under fairly general assumptions we can obtain ``Gromov-Hausdorf limits" that are complete metric spaces. When the manifolds have bounded Ricci curvature and non-collapsing volume, the Anderson-Cheeger-Colding theory provides a regular-singular decomposition of a limit space. It is a central question in Riemannian geometry to understand these singularities. In the case when the manifolds are projective and the metrics are Kahler, we will discuss some recent progress towards an algebro-geometric understanding of the singularities of Gromov-Hausdorff limits. This talk is based on joint work with Simon Donaldson.
Fall 2014
Fall Abstracts
Yuanqi Wang
Liouville theorem for complex Monge-Ampere equations with conic singularities.
Following Calabi, Pogorelov, Evans-Krylov-Safanov, and Trudinger's pioneer work on interior regularities and liouville theorems for Monge-Ampere equations, we prove the Liouville theorem for conic Kähler-Ricci flat metrics. We also discuss various applications of this Liouville theorem to conic Kähler geometry.
Chris Davis (UW-Eau Claire)
L^{2} signatures and an example of Cochran-Harvey-Leidy
Ben Knudsen (Northwestern)
Rational homology of configuration spaces via factorization homology
The study of configuration spaces is particularly tractable over a field of characteristic zero, and much effort has gone into producing chain complexes simple enough for explicit computations, formulas for Betti numbers, and homological stability results. I will discuss recent work identifying the homology of the configuration spaces of an arbitrary manifold M with the homology of a certain Lie algebra constructed from the compactly supported cohomology of M. The aforementioned results follow immediately from this identification, albeit with hypotheses removed; in particular, one obtains a new, elementary proof of homological stability for configuration spaces.
Kevin Whyte (UIC)
The rigidity theorems of Mostow and Margulis for lattices in semi simple Lie groups are some of the most celebrated in their field, and are motivation for much of geometric group theory. Mostow's result, which states that every isomorphism between lattices extends to an equivariant isometry between symmetric spaces, has been generalized by Kleiner and Leeb to say that any map between higher rank symmetric spaces which is quasi-isometric (a large scale version of bilipschitz) is actually a perturbation of an isometry. Margulis' superrigidity theorem, which says every homomorphism between lattices which has infinite image extends to an equivariant isometric embedding of symmetric spaces, has resisted a generalization of this sort. We will discuss one such result, which considers when quasi-isometric embeddings of symmetric spaces are near isometric embeddings. Our results show that the situation is complicated - in some cases one does have rigidity while in others there are exotic quasi-isometric embeddings. What geometric properties distinguish the two cases is only starting to be understood. This is joint work with David Fisher (Indiana).
Alden Walker (UChicago)
In 1985, Barnsley and Harrington defined a "Mandlebrot set" M for pairs of complex dilations. This is the set of complex numbers c such that the limit set generated by the pair of dilations x-> cx+1 and x-> cx-1 is connected. The set M is also the closure of the set of roots of polynomials with coefficients in {-1,0,1}. As with the usual Mandlebrot set, M has strong connections to dynamics and algebra, and it has been studied by Bousch, Bandt, Solomyak, Xu, Thurston, and Tiozzo. For a geometric group theorist, the study of M is qualitatively similar to the study of Kleinian groups acting on their limit sets or on universal circles.
Barnsley and Harrington noted the (numerically apparent) existence of infinitely many "holes" in M, which correspond to exotic components of the space of Schottky semigroups. Bandt rigorously confirmed a single hole in 2002 and conjectured that the interior of M is dense in M away from the real axis. We give the new technique of "traps" to certify an interior point of M, and we use these traps to prove Bandt's conjecture and certify the existence of infinitely many holes in M.
The only prerequisite for this talk is point-set topology. Fun pictures will be provided. This is joint work with Danny Calegari and Sarah Koch.
Jing Tao (Oklahoma)
Growth Tight Actions
Let G be a group equipped with a finite generating set S. G is called growth tight if its exponential growth rate relative to S is strictly greater than that of every quotient G/N with N infinite. This notion was first introduced by Grigorchuk and de la Harpe. Examples of groups that are growth tight include free groups relative to bases and, more generally, hyperbolic groups relative to any generating set. In this talk, I will provide some sufficient conditions for growth tightness which encompass all previous known examples.
Thomas Barthelmé (Penn State)
Counting orbits of Anosov flows in free homotopy classes
In 1972, Plante and Thurston asked the following question: If M is a manifold supporting an Anosov flow, does the number of conjugacy classes in the fundamental group grows exponentially fast with the length of the shortest orbit representative? This question can be solved by answering the following: Can one give an upper bound on the growth rate of the length of orbits inside a free homotopy class?
In this talk, I will explain how one can use the geometry and topology of Anosov flows to answer both questions in the 3-manifold case. This is joint work with Sergio Fenley.
Alexandra Kjuchukova (University of Pennsylvania)
On the classification of irregular branched covers of four-manifolds
It is a famous result of Hilden and Montesinos that every closed orientable three-manifold can be realized as an irregular three-fold cover of S^3 branched over a knot. In contrast, a smooth four-manifold which can be realized as a cover of S^4 branched over a smooth surface must have signature equal to zero. Given two simply-connected, closed, oriented four-manifolds X and Y and a surface B embedded in X with an isolated singularity, I will prove a necessary condition for the existence of an irregular dihedral branched covering map f: Y -> X with branching set B. Conversely, given a simply-connected oriented closed four-manifold X, I will outline a construction realizing as irregular dihedral covers of X infinitely many (and conjecturally all) of the manifolds Y afforded by the necessary condition.
Oyku Yurttas (Georgia Tech)
Dynnikov and train track transition matrices of pseudo-Anosov braids
In this talk we will compare a Dynnikov matrix with the train track transition matrix of a given pseudo-Anosov braid on the finitely punctured disk. Our main result is that these matrices are isospectral up to roots of unity and some zeros under particular conditions.
Archive of past Geometry seminars
2013-2014: Geometry_and_Topology_Seminar_2013-2014
2012-2013: Geometry_and_Topology_Seminar_2012-2013
2011-2012: Geometry_and_Topology_Seminar_2011-2012
2010: Fall-2010-Geometry-Topology