Math Department
https://wp0.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 22 Feb 2019 22:10:26 +0000en-UShourly1https://wordpress.org/?v=4.9.8146940276Covariant Derivations
https://wp0.vanderbilt.edu/math/2019/02/talk-title-tba-288/
Fri, 22 Feb 2019 22:10:26 +0000https://wp0.vanderbilt.edu/math/?p=8738Let M be a von Neumann algebra equipped with an action of a locally compact group G. In this talk I will discuss how derivations on M can be lifted to derivations on the crossed product of M by G, and what properties are inherited by this new derivation. As an application, I will show how a result of Davies and Lindsay (about the closure of a derivation on a tracial von Neumann algebra) can be extended to the non-tracial case by way of the continuous core.
]]>8738Long Time Behavior of 2d Water Waves with Point Vortice
https://wp0.vanderbilt.edu/math/2019/02/long-time-behavior-of-2d-water-waves-with-point-vortice/
Fri, 22 Feb 2019 22:10:06 +0000https://wp0.vanderbilt.edu/math/?p=8732We study the motion of the two dimensional inviscid incompressible, infinite depth water waves with point vortices in the fluid. We show that Taylor sign condition ∂P/∂n >= 0 can fail if the point vortices are sufficient close to the free boundary, so the water waves could be subject to the Taylor instability. Assuming the Taylor sign condition, we prove that the water wave system is locally wellposed in Sobolev spaces. Moreover, we show that if the water waves is symmetric with a certain symmetric vortex pair traveling downward initially, then the free interface remains smooth for a long time, and for initial data of size ε << 1, the lifespan is at least O(ε^-2).
]]>8732Euclidean Distance Degree of Algebraic Varieties
https://wp0.vanderbilt.edu/math/2019/02/euclidean-distance-degree-of-algebraic-varieties/
Fri, 22 Feb 2019 21:10:47 +0000https://wp0.vanderbilt.edu/math/?p=8728The Euclidean distance degree of an algebraic variety is a well-studied topic in applied algebra and geometry.
It has direct applications in geometric modeling, computer vision, and statistics. I will first describe a new topological interpretation of the Euclidean distance degree of an affine variety in terms of Euler characteristics. As a concrete application, I will present a solution to the open problem in computer vision of determining the Euclidean distance degree of the affine multiview variety. Secondly, I will present a solution to a conjecture of Aluffi-Harris concerning the Euclidean distance degree of projective varieties (Joint work with J. Rodriguez and B. Wang.)
]]>8728Constructing Quantum Modular Forms of Depth Two
https://wp0.vanderbilt.edu/math/2019/02/talk-title-tba-286/
Thu, 21 Feb 2019 21:10:24 +0000https://wp0.vanderbilt.edu/math/?p=8726Work of Bringmann, Kaszian and Milas in 2017 introduced the concept of higher depth quantum modular forms
(qmfs), and therein provided one such example of a qmf of depth two, related to characters of vertex algebras. In this talk we see how to generalise their work to obtain an infinite family of non-trivial qmfs of depth two. To show this, we relate our constructed function F asymptotically to double Eichler integrals on the lower half plane, and further to non-holomorphic theta functions with coefficients given by double error functions.
]]>8726Asymptotics of Cheeger Constants and Unitarisability of Groups
https://wp0.vanderbilt.edu/math/2019/02/talk-title-tba-291/
Wed, 20 Feb 2019 22:10:30 +0000https://wp0.vanderbilt.edu/math/?p=8744 Let \(\Gamma\) be a discrete group. A group \(\Gamma\) is called unitarisable if for any Hilbert space \(H\) and any uniformly bounded representation \(\pi: \Gamma \to B(H)\) of \(\Gamma\) on \(H\) there exists an operator \(S: H\to H\) such that \(S^{-1}\pi(g)S\) is a unitary representation for every \(g \in \Gamma\). It is well known that amenable groups are unitarisable. It has been open ever since whether amenability characterises unitarisability of groups. Dixmier: Are all unitarisable groups amenable? One of the approaches to study unitarisability is related to the space of Littlewood functions \(T_1(\Gamma)\). We define the Littlewood exponent \({\rm Lit}(\Gamma)\) of a group \(\Gamma\): \({\rm Lit}(\Gamma)=\inf\big\{ p : T_1(\Gamma) \subseteq \ell^p(\Gamma) \big\}.\) On the one hand, \({\rm Lit}(\Gamma)\) is related to unitarisability and amenability and, on the other hand, it is related to some geometry of \(\Gamma\) or, more precisely, to the behavior of Cayley graphs when one increases the generating sets of \(\Gamma\). We will also discuss some corollaries of this connection, for example, about the number of colours one\ need to colour the Cayley graphs of some groups.
]]>8744Sharks in the Shallows
https://wp0.vanderbilt.edu/math/2019/02/8782/
Wed, 20 Feb 2019 00:00:58 +0000https://wp0.vanderbilt.edu/math/?p=8782On a fine sunny day, little Sri decided to go fishing because he really enjoyed it and was passionate about it and wanted to become a great fisher. Since he wasn’t an expert nor an experienced fisher, he decided to go to the local shallow waters thinking that he’ll find only small fish, and it’ll be a tractable job for him to catch them. Everything was going well until he discovered that several unconquerable beasts and sharks have been hiding there for centuries tricking several young and amateur fishers into catching them. Sometimes, even great expert fishermen go back to these shallows to try and catch them, and indeed a very few of them have succeeded and received immense glory and fame… Do you want to know more about the mystery of these legendary beasts??? Come to my talk.
]]>8782Generalizing Orthomodularity
https://wp0.vanderbilt.edu/math/2019/02/generalizing-orthomodularity/
Mon, 18 Feb 2019 22:10:15 +0000https://wp0.vanderbilt.edu/math/?p=8808Quantum logics represent a long standing research tradition, and a vaste literature on algebraic structures related to quantum logics — sharp quantum structures (Orthomodular Posets, Orthomodular Lattices, Orthoalgebras) and unsharp quantum structures (Effect Algebras) — has appeared over the last fifty years. In this talk, we consider a generalization of the notion of orthomodularity for posets to the concept of the generalized orthomodularity property (GO-property) by considering the LU-operators. This seemingly mild generalization of orthomodular posets and its order theoretical analysis yield rather strong application to effect algebras, orthomodular posets (lattices), and Boolean posets (algebras). In presence of the GO-property, it will turn out that pastings of Boolean algebras are in fact orthomodular posets, thus establishing a novel connection between Greechie’s Theorems, orthomodular posets and the coherence law for effect algebras. It will turn out that an order theoretical notion of orthomodularity make sense, also in a weaker form, only in the framework of orthomodular posets. It would suggest that sharp and unsharp quantum logics are capable of a neat separation in terms of their order structure.
]]>8808Qualifying Examination: Commuting Squares and Hyperfinite Subfactors
https://wp0.vanderbilt.edu/math/2019/02/qualifying-examination-commuting-squares-and-hyperfinite-subfactors/
Mon, 18 Feb 2019 22:00:13 +0000https://wp0.vanderbilt.edu/math/?p=88048804Eulerian Hypergraphs
https://wp0.vanderbilt.edu/math/2019/02/eulerian-hypergraphs/
Mon, 18 Feb 2019 21:10:51 +0000https://wp0.vanderbilt.edu/math/?p=8811Let $H$ be a $k$-uniform hypergraph, $k\ge 3$. An Euler tour in $H$ is an alternating sequence $v_0, e_1, v_1, e_2, v_2, \cdots, v_{m-1}, e_m, v_m=v_0$ of vertices and edges in $H$ such that each edge of $H$ appears in this sequence exactly once and $v_{i-1}, v_i\in e_i$,$v_{i-1}\ne v_i$, for every $i= 1,2, . . . , m$. This is an obvious generalization of the graph theoretic concept of an Euler tour. A straight forward necessary condition for existence of an Euler tour in a $k$-uniform hypergraph $H$ is $|V_{\text{odd}}(H)|\le (k-2)|E(H)|$, where $V_{\text{odd}}(H)$ is the set of vertices of odd degrees in $H$ and $E(H)$ is the set of edges in $H$. Lonc and Naroski showed that for $k\ge 3$, the problem of determining if a given $k$-uniform hypergraph has an Euler tour is NP-complete. For $1\le \ell \le k$, the minimum $\ell$-degree of $H=(V,E)$ is $\delta_{\ell}(H)=\min_{S\subseteq V, |S|=\ell}|\{ e \,|\, S\subseteq e, e\in E\}|$. \v{S}ajna and Wagner showed that every 3-uniform hypergraph $H=(V,E)$ with $\delta_2(H)\ge 1$ and with $|E|\ge 2$admits an Euler tour. As a consequence, every $k$-uniform hypergraph $H=(V,E)$ with $\delta_{k-1}(H)\ge 1$ and with $|E|\ge 2$ has an Euler tour. In this talk, we investigate existence of Euler tour in $k$-uniform hypergraphs for $k\ge 4$ under $\ell$-degree conditions with $1\le \ell \le k-2$. In particular, for $k\ge 4$, we show that every $k$-uniform hypergraph $H=(V,E)$ with $\delta_2(H)\ge k$ or $\delta_{k-2}(H)\ge 4$ and with $|V|\ge \min\{\frac{k^2}{2}+\frac{k}{2}\}$ admits an Euler tour.
]]>8811A Consensus Framework for Quantifying Drug Synergy
https://wp0.vanderbilt.edu/math/2019/02/a-consensus-framework-for-quantifying-drug-synergy/
Fri, 15 Feb 2019 21:00:21 +0000https://wp0.vanderbilt.edu/math/?p=87998799