Math Department
https://wp0.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityWed, 19 Sep 2018 22:10:49 +0000en-UShourly1https://wordpress.org/?v=4.9.4146940276Properly Proximal Groups and Their von Neumann Algebras
https://wp0.vanderbilt.edu/math/2018/09/properly-proximal-groups-and-their-von-neumann-algebras/
Wed, 19 Sep 2018 22:10:49 +0000https://wp0.vanderbilt.edu/math/?p=8120We will introduce a class of groups, which we call properly proximal, which includes all nonelementary hyperbolic groups, all nonelementary bi-exact groups, all convergence groups, all lattices in semisimple Lie groups, and is closed under commensurability and taking direct products, but excludes all amenable and even all inner-amenable groups. We will then discuss rigidity results for von Neumann algebras associated to measure-preserving actions of these groups.
]]>8120An Upper Bound for the Green Energy on SO(3)
https://wp0.vanderbilt.edu/math/2018/09/an-upper-bound-for-the-green-energy-on-so3/
Wed, 19 Sep 2018 22:10:47 +0000https://wp0.vanderbilt.edu/math/?p=8159In this short talk, we compute a simple expression for Green’s function on the Lie group SO(3) using Wigner D-functions, the eigenfunctions of the corresponding Laplace-Beltrami Operator, and use a result of Macchi-Soshnikov to calculate an upper bound for the Green energy there on. To this end, we give a small summary of Determinantal Point Processes, and if time permits, we will prove asymptotics of the L2-norm of Gegenbauer (hyperspherical) polynomials. This is a joint work with Carlos Beltrán from Universidad de Cantabria.
]]>8159Polygons and Polyhedrons in Space and Beyond
https://wp0.vanderbilt.edu/math/2018/09/8138/
Wed, 19 Sep 2018 00:00:15 +0000https://wp0.vanderbilt.edu/math/?p=8138How many regular polygons are there? Equilateral triangles, squares … as many as you like. Surprisingly, the answer to how many regular solid polyhedrons (3D version of polygons) there are, is a mere 5! I shall prove why there are just five of these Platonic Solids and attempt to argue that there ought to be more, hidden in hyperbolic space.
]]>8138The Relationship of Supernilpotence to Nilpotence (3)
https://wp0.vanderbilt.edu/math/2018/09/the-relationship-of-supernilpotence-to-nilpotence-3/
Mon, 17 Sep 2018 22:10:44 +0000https://wp0.vanderbilt.edu/math/?p=8134Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent.
]]>8134A result of Furman on boundary actions
https://wp0.vanderbilt.edu/math/2018/09/a-result-of-furman-on-boundary-actions/
Mon, 17 Sep 2018 22:10:14 +0000https://wp0.vanderbilt.edu/math/?p=8168After giving the definition and examples of boundary actions, we will sketch Furman’s proof that if a boundary action of a locally compact group H satisfies certain hypotheses (related to the “no small subgroups” property), then we may conclude that either H is discrete infinite countable, or H is a connected semi-simple real Lie group with trivial center.
]]>8168Shanks Workshop: Free Probability and Applications, Sept. 15 &16
https://wp0.vanderbilt.edu/math/2018/09/shanks-workshop-free-probability-and-applications-sept-15-16/
Sat, 15 Sep 2018 15:00:16 +0000https://wp0.vanderbilt.edu/math/?p=8069The aim of this workshop is to bring together experts in free probability theory and adjacent fields, including random matrix theory, probability, and subfactor theory to discuss possible applications of free probability in these fields. More information is available on the workshop website.
]]>8069Pinsker Algebras for 1-bounded Entropy
https://wp0.vanderbilt.edu/math/2018/09/pinsker-algebras-for-1-bounded-entropy/
Fri, 14 Sep 2018 22:10:21 +0000https://wp0.vanderbilt.edu/math/?p=8157I will discuss the notion of a Pinkser algebra for 1-bounded entropy (a modification of free entropy dimension for strongly 1-bounded algebras in the sense of Jung). Given a tracial von Neumann algebra M, a Pinsker algebra in M is a subalgebra P of M which is maximal with respect to the property that the 1-bounded entropy of P in M is zero. Such algebras always exist. I will discuss properties of Pinkser algebras, as well as give at least one interesting example of such an algebra, and discuss the difficulties involved in producing more examples.
]]>8157Least Dilatation of Pure Surface Braids
https://wp0.vanderbilt.edu/math/2018/09/least-dilatation-of-pure-surface-braids/
Wed, 12 Sep 2018 22:10:22 +0000https://wp0.vanderbilt.edu/math/?p=8117 The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.
]]>8117Annual Town Hall Meeting of Chair and DUG with Undergraduate Students
https://wp0.vanderbilt.edu/math/2018/09/annual-town-hall-meeting-of-chair-with-undergraduate-students/
Tue, 11 Sep 2018 00:00:52 +0000https://wp0.vanderbilt.edu/math/?p=8004The annual town hall meeting is an informal gathering where Department Chair Mike Neamtu along with Director of Undergraduate Studies John Rafter meet with undergraduate students to answer questions and discuss updates about the undergraduate program.
]]>8004The Relationship of Supernilpotence to Nilpotence (2)
https://wp0.vanderbilt.edu/math/2018/09/the-relationship-of-supernilpotence-to-nilpotence-2/
Mon, 10 Sep 2018 22:10:41 +0000https://wp0.vanderbilt.edu/math/?p=8131Supernilpotence is a condition on an algebra that is definable with a higher arity commutator that generalizes the classical binary commutator for general algebras. Supernilpotent algebras have received attention lately because of theorems of the flavor ‘nice property true satisfied by finite nilpotent groups’ is satisfied by finite supernilpotent Mal’cev algebras of finite type. For example, it is now know that there is a polynomial time algorithm to solve the equation equation satisfiability problem for such algebras. The exact relationship between supernilpotence and nilpotence had been unclear. We will discuss how supernilpotence implies nilpotence for algebras with a Taylor term, but that in general the two notions are independent.
]]>8131