Math Department
https://wp0.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 20 Sep 2019 21:10:12 +0000en-UShourly1https://wordpress.org/?v=4.9.8146940276A Plethora of Embeddings into an Ultraproduct of II_1 Factors
https://wp0.vanderbilt.edu/math/2019/09/talk-title-tba-308/
Fri, 20 Sep 2019 21:10:12 +0000https://wp0.vanderbilt.edu/math/?p=9160We will report on some results (joint with S. Atkinson) regarding new characterizations of amenability for Connes embeddable II_1 factors. First we introduce the notion of self-tracial stability and discuss immediate connections with amenability. Secondly we will discuss an interesting strengthening of the well known Jung’s tubularity result. The core of this is a technical argument of Kishimoto. There are also some interesting model theoretic questions that come out of this. Finally we build on work of N. Brown and N. Ozawa to answer a question of Popa concerning the “vastness” of the space of embeddings of a non-amenable II_1 factor into the ultraproduct of a given collection of II_1 factors. We will end with interesting questions in the group theory setting. See arxiv: 1907.03359 for submitted paper.
]]>9160The Strong Cosmic Censorship Conjecture in General Relativity
https://wp0.vanderbilt.edu/math/2019/09/talk-title-tba-305/
Thu, 19 Sep 2019 21:10:40 +0000https://wp0.vanderbilt.edu/math/?p=9101The Einstein equations in general relativity admit explicit black hole solutions which have the disturbing property that global uniqueness fails. As a way out, Penrose proposed the strong cosmic censorship conjecture, which says that this phenomenon of global non-uniqueness is non-generic. We will discuss this conjecture and some recent mathematical progress. This talk is based on joint works with Mihalis Dafermos, Sung-Jin Oh, Jan Sbierski and Yakov Shlapentokh-Rothman. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Jared Speck)
]]>9101Bounds on Multiplicities of Zeros of a Family of Zeta Functions
https://wp0.vanderbilt.edu/math/2019/09/talk-title-tba-322/
Thu, 19 Sep 2019 18:00:48 +0000https://wp0.vanderbilt.edu/math/?p=9281In “The Pair Correlation of Zeros of the Zeta Function”, Montgomery finds the asymptotics of the pair correlation function in order to give a lower bound on the proportion of zeros that are simple (assuming the Riemann Hypothesis). We will discuss some of the necessary tools to extend his proof to pair correlation for zeros of Dedekind zeta functions of abelian extensions, and as in the Riemann zeta case, we can then use this to obtain results on multiplicities of zeros for these zeta functions. However, we also are able to relate the counts of multiplicities to Cohn-Elkies sphere-packing type bounds, allowing us to use semi-definite programming techniques to obtain better results in lower degree extensions than could be found from a direct analysis. In particular, we are able to conclude that more than 45% of the zeros are distinct for Dedekind zeta functions of quadratic number fields. This is based on joint work with M. Alsharif, D. de Laat, M. Milinovich, L. Rolen, and I. Wagner.
]]>9281Hyperbolicity is Preserved under Elementary Equivalence
https://wp0.vanderbilt.edu/math/2019/09/hyperbolicity-is-preserved-under-elementary-equivalence/
Wed, 18 Sep 2019 21:10:55 +0000https://wp0.vanderbilt.edu/math/?p=9289Zlil Sela proved that a finitely generated group that satisfies the same first-order properties as a torsion-free hyperbolic group is torsion-free hyperbolic. I will explain that this result remains true for hyperbolic groups with torsion, as well as for subgroups of hyperbolic groups, and for hyperbolic and cubulable groups.
]]>9289Graduate Student Tea
https://wp0.vanderbilt.edu/math/2019/09/graduate-student-tea-78/
Wed, 18 Sep 2019 20:33:42 +0000https://wp0.vanderbilt.edu/math/?p=93109310Braids
https://wp0.vanderbilt.edu/math/2019/09/talk-title-tba-321/
Tue, 17 Sep 2019 23:00:11 +0000https://wp0.vanderbilt.edu/math/?p=9262Groups are found throughout mathematics to capture symmetries of the world around us. There is a famous group that you have interacted with if you have ever braided a friend’s hair. We will explore this incredible group and see what it can tell us about other mathematical objects.
]]>9262Glivenko Theorems and Semisimple Companions
https://wp0.vanderbilt.edu/math/2019/09/glivenko-theorems-and-semisimple-companions/
Mon, 16 Sep 2019 21:10:01 +0000https://wp0.vanderbilt.edu/math/?p=9295The Glivenko theorem connecting intuitionistic and classical logic by means of a double negation translation, as well as its analogues connecting the modal logic S4 with S5 and the fuzzy logic BL with Lukasiewicz logic, will be shown to be instances of a general phenomenon where a logic with a well-behaved negation displays a Glivenko-like connection to what we call its semisimple companion.
]]>9295Minimal Index and Dimension for Inclusions of von Neumann Algebras with Finite-Dimensional Centers
https://wp0.vanderbilt.edu/math/2019/09/minimal-index-and-dimension-for-inclusions-of-von-neumann-algebras-with-finite-dimensional-centers/
Fri, 13 Sep 2019 21:10:21 +0000https://wp0.vanderbilt.edu/math/?p=9190The notion of index for inclusions of von Neumann algebras goes back to the seminal work of Jones on subfactors of type II1. More generally, one can define the index of a conditional expectation associated with a subfactor and look for expectations that minimize the index. This minimal value is a number and it is called the minimal index of the subfactor. We report on our analysis of the minimal index for inclusions of arbitrary von Neumann algebras (not necessarily factorial, nor finite) with finite-dimensional centers (multi-factor inclusions). The theory is controlled by a matrix associated with the inclusion, that we call matrix dimension, whose squared L^2 norm equals the minimal index and which determines a further invariant, that we call spherical state of the inclusion, via Perron-Frobenius theory. We mention the properties of finite multi-factor inclusions, especially multi-matrices, for which the spherical state coincides on the relative commutant with the Markov trace (super-extremal inclusions). We also mention how the matrix dimension can be purely algebraically defined for 1-arrows in rigid 2-C*-categories and how it determines the so-called standard solutions of the conjugate equations, and we address some open questions. Based on joint work with R. Longo, arXiv:1805.09234.
]]>9190Attractors of the Einstein-Klein-Gordon System
https://wp0.vanderbilt.edu/math/2019/09/attractors-of-the-einstein-klein-gordon-system/
Fri, 13 Sep 2019 21:10:18 +0000https://wp0.vanderbilt.edu/math/?p=9180A key question in general relativity is whether solutions to the Einstein equations, viewed as an initial value problem, are stable to small perturbations of the initial data. For example, previous results have shown that the Milne spacetime, which represents an expanding universe emanating from a big bang singularity with a linear scale factor, is a stable solution to the Einstein equations. With such a slow expansion rate, particularly compared to related models with accelerated expansion (such as the exponentially expanding de Sitter spacetime modelling our universe), there are interesting questions one can ask about stability of this spacetime. Previous results have shown that the Milne model is a stable solution to the vacuum Einstein, Einstein-Klein-Gordon and Einstein-Vlasov systems. Motivated by techniques from the last result, I will present a new proof of the stability of the Milne model to the Einstein-Klein-Gordon system and compare our method to a recent result of J. Wang. This is joint work with David Fajman (Vienna).
]]>9180Waddington’s Epigenetic Landscape Does Not Exist
https://wp0.vanderbilt.edu/math/2019/09/waddingtons-epigenetic-landscape-does-not-exist/
Fri, 13 Sep 2019 20:00:55 +0000https://wp0.vanderbilt.edu/math/?p=92669266