Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 27 Nov 2015 16:14:11 +0000enhourly1http://wordpress.org/?v=3.1.4Product rigidity for the von Neumann algebras of hyperbolic groups
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-56/
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-56/#commentsFri, 20 Nov 2015 21:10:42 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2427Suppose $\Gamma_1,\ldots, \Gamma_n $ are each hyperbolic i.c.c. groups and $L(\Gamma_1\times\cdots\times \Gamma_n) \cong L(\Lambda) $ for an arbitrary group $\Lambda $. Then we show $\Lambda $ decomposes as an $n $-fold product, $\Lambda= \Lambda_1\times\cdots\times \Lambda_n $ such that for each $i=1,\ldots, n $, we have \$L(\Gamma_i)\cong L(\Lambda_i) $ up to amplification. This strengthens Ozawa and Popa’s unique prime decomposition results.
]]>http://as.vanderbilt.edu/math/2015/11/talk-title-tba-56/feed/0Hyperbolicity of surface group extensions, and convex cocompact subgroups of mapping class groups
http://as.vanderbilt.edu/math/2015/11/hyperbolicity-of-surface-group-extensions-and-convex-cocompact-subgroups-of-mapping-class-groups/
http://as.vanderbilt.edu/math/2015/11/hyperbolicity-of-surface-group-extensions-and-convex-cocompact-subgroups-of-mapping-class-groups/#commentsFri, 20 Nov 2015 21:10:49 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2847Convex cocompact subgroups of mapping class groups, as introduced by Farb and Mosher, are subgroups whose action on Teichmuller space is analogous to that of convex cocompact Kleinian groups acting on hyperbolic 3-space. Moreover, it is exactly the convex cocompact subgroups that give rise to Gromov hyperbolic surface bundles and to hyperbolic extensions of free groups. In this talk I will describe a setting, arising from hyperbolic fibered 3-manifolds, in which there is a concrete connection between these two notions of convex cocompactness and explain how one may use this connection to prove certain subgroups of mapping class groups are convex cocompact. This is joint work with Richard Kent and Christopher Leininger.
]]>http://as.vanderbilt.edu/math/2015/11/hyperbolicity-of-surface-group-extensions-and-convex-cocompact-subgroups-of-mapping-class-groups/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-80/
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-80/#commentsWed, 18 Nov 2015 21:10:35 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2754http://as.vanderbilt.edu/math/2015/11/talk-title-tba-80/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2015/11/graduate-student-tea-32/
http://as.vanderbilt.edu/math/2015/11/graduate-student-tea-32/#commentsWed, 18 Nov 2015 20:30:04 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2849http://as.vanderbilt.edu/math/2015/11/graduate-student-tea-32/feed/0Symmetric graphs via odd automorphisms
http://as.vanderbilt.edu/math/2015/11/symmetric-graphs-via-odd-automorphisms/
http://as.vanderbilt.edu/math/2015/11/symmetric-graphs-via-odd-automorphisms/#commentsMon, 16 Nov 2015 21:10:43 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2853An automorphism (or symmetry) of a combinatorial graph may be called *even* or *odd* according to whether it acts as an even or odd permutation on the vertices of the graph. In this talk, we will present a partial result about the following question: When does the existence of even automorphisms of a graph force existence of odd automorphisms? In particular, we will present complete information the on existence of odd automorphisms for cubic symmetric graphs.
]]>http://as.vanderbilt.edu/math/2015/11/symmetric-graphs-via-odd-automorphisms/feed/0C*-simplicity for discrete groups
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-55/
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-55/#commentsFri, 13 Nov 2015 21:10:12 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2425A discrete group is said to be C*-simple if its reduced C*-algebra is simple. It is not difficult to see that a group with this property does not have any non-trivial normal amenable subgroups, however it was an open question for many years to determine whether the converse holds. Recent examples constructed by Le Boudec show that the answer to this question is negative, but raise the question of whether there is an intrinsic algebraic characterization of C*-simplicity. In this talk, after a brief review of some background material, I will discuss recent work providing such a characterization.
]]>http://as.vanderbilt.edu/math/2015/11/talk-title-tba-55/feed/0Surface bundles, Teichmuller space, and mapping class groups
http://as.vanderbilt.edu/math/2015/11/surface-bundles-teichmuller-space-and-mapping-class-groups/
http://as.vanderbilt.edu/math/2015/11/surface-bundles-teichmuller-space-and-mapping-class-groups/#commentsFri, 13 Nov 2015 21:10:27 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2805This talk will introduce the mapping class group and Teichmuller space of a surface with a focus on how these objects are related to the theory of surface bundles. We’ll take the perspective of Teichmuller space as a (sort of) classifying space for surface bundles and explain how each surface bundle gives rise to a monodromy representation into the mapping class group. I’ll then describe how the geometry of Teichmuller space is related to the metric properties of surface bundles, which will lead us to the notion of convex cocompact subgroups of mapping class groups. The talk will be introductory (and I hope accessible!) in nature with the aim of setting the stage for a follow-up talk discussing some of my work in this area.
]]>http://as.vanderbilt.edu/math/2015/11/surface-bundles-teichmuller-space-and-mapping-class-groups/feed/0Far beyond the infinite
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-64/
http://as.vanderbilt.edu/math/2015/11/talk-title-tba-64/#commentsThu, 12 Nov 2015 21:10:01 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2468The modern mathematical story of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key ZFC axioms for Set Theory were in place and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory, that of Cantor’s Continuum Hypothesis, was not solvable on the basis of the ZFC axioms. The 50 years since Cohen’s work has seen a vast development of Cohen’s method and the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based. Thus a fundamental dilemma has emerged. On the one hand, the discovery, also over the last 50 years, of a rich hierarchy axioms of infinity seems to argue that Cantor’s conception is fundamentally sound. But on the other hand, the developments of Cohen’s method over this same period seem to strongly suggest there can be no preferred extension of the ZFC axioms to a system of axioms that can escape the ramifications of Cohen’s method. But this dilemma was itself based on a misconception and recent discoveries suggest there is a resolution (maybe). Tea at 3:30 pm in SC 1425. (Contact Person: Vaughan Jones)
]]>http://as.vanderbilt.edu/math/2015/11/talk-title-tba-64/feed/0Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation
http://as.vanderbilt.edu/math/2015/11/mathematical-analysis-of-a-clonal-evolution-model-of-tumour-cell-proliferation/
http://as.vanderbilt.edu/math/2015/11/mathematical-analysis-of-a-clonal-evolution-model-of-tumour-cell-proliferation/#commentsThu, 12 Nov 2015 21:10:21 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2545We introduce and analyse a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which corresponds to the clonal model of tumour cell growth. This assumption leads to a model with a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. In particular, we study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population typically dies out. On the other hand, with telomere restoration, we may observe exponential growth in the linear model. We also study the effects of crowding induced mortality on the qualitative behaviour, and study the existence and stability of steady states of a nonlinear model with crowding effect.
]]>http://as.vanderbilt.edu/math/2015/11/mathematical-analysis-of-a-clonal-evolution-model-of-tumour-cell-proliferation/feed/0Not all acylindrically hyperbolic groups have universal acylindrical actions
http://as.vanderbilt.edu/math/2015/11/not-all-acylindrically-hyperbolic-groups-have-universal-acylindrical-actions/
http://as.vanderbilt.edu/math/2015/11/not-all-acylindrically-hyperbolic-groups-have-universal-acylindrical-actions/#commentsWed, 11 Nov 2015 21:10:15 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2779The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(F_n) for n > 1. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated acylindrically hyperbolic group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. In this talk, I will answer this question in the negative, using Dunwoodys example of an inaccessible group as a counterexample.
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