Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversitySat, 06 Feb 2016 14:00:47 +0000enhourly1http://wordpress.org/?v=3.1.4Shanks Workshop on Ordered Algebras and Logic, February 6 – 7, 2016
http://as.vanderbilt.edu/math/2016/02/shanks-workshop-on-ordered-algebras-and-logic-take-3/
http://as.vanderbilt.edu/math/2016/02/shanks-workshop-on-ordered-algebras-and-logic-take-3/#commentsSat, 06 Feb 2016 14:00:47 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2984http://as.vanderbilt.edu/math/2016/02/shanks-workshop-on-ordered-algebras-and-logic-take-3/feed/0Topological rigidity for closed aspherical manifolds fibering over the unit circle
http://as.vanderbilt.edu/math/2016/02/topological-rigidity-for-closed-aspherical-manifolds-fibering-over-the-unit-circle/
http://as.vanderbilt.edu/math/2016/02/topological-rigidity-for-closed-aspherical-manifolds-fibering-over-the-unit-circle/#commentsFri, 05 Feb 2016 21:10:19 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3065The Borel conjecture in manifold topology predicts that every closed aspherical manifold is topologically rigid, i.e. every homotopy equivalence between any two closed aspherical manifolds is homotopic to a homeomorphism. There are variants of the Borel conjecture, such as the simple Borel conjecture and the bordism Borel conjecture, corresponding to other types of topological rigidity. In this talk, I consider topological rigidity for closed aspherical manifolds that fiber over the unit circle. We show that, in dimensions greater than or equal to 5, both the simple Borel conjecture and the bordism Borel conjecture hold for such an aspherical manifold, provided the fundamental group of the fiber belongs to a large class of groups, including Gromov hyperbolic groups, CAT(0) groups, and lattices in virtually connected lie groups. The main ingredients in proving this rigidity result are some general results that we obtain in algebraic L-theory. These results also have some applications to the Novikov conjecture.
]]>http://as.vanderbilt.edu/math/2016/02/topological-rigidity-for-closed-aspherical-manifolds-fibering-over-the-unit-circle/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2016/02/talk-title-tba-90/
http://as.vanderbilt.edu/math/2016/02/talk-title-tba-90/#commentsFri, 05 Feb 2016 20:10:42 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3006http://as.vanderbilt.edu/math/2016/02/talk-title-tba-90/feed/0Gradings on Simple Algebras and Beyond
http://as.vanderbilt.edu/math/2016/02/talk-title-tba-89/
http://as.vanderbilt.edu/math/2016/02/talk-title-tba-89/#commentsThu, 04 Feb 2016 22:10:36 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2997Gradings by groups on simple algebras, like Lie, Jordan, etc., play an important role for the classification and representation theory of these algebras and in applications. At this time, we have a very good understanding of how the group gradings look like on finite-dimensional associative, Lie and Jordan algebras over algebraically closed fields of characteristic zero. There are a number of nice results in the case of algebraically closed fields of prime characteristic, very few when the field is not algebraically closed. Lie and Jordan superalgebras are another class where the work has just started. Even less explored are gradings on simple infinite-dimensional algebras, even locally finite. But we have good classification results for gradings on simple finitary Lie algebras. It should also be mentioned that gradings on not necessarily simple Lie algebras, even nilpotent algebras, are of interest in Differential Geometry. To deal with so different situations, various methods have been offered, from Classical Ring Theory to Algebraic Groups to Hopf Algebras to Polynomial identity algebras. In this talk I will try to explain some of the ideas and present some recent result of this now large area. Tea at 3:30 pm in SC 1425. (Contact Person: Alexander Olshanskiy)
]]>http://as.vanderbilt.edu/math/2016/02/talk-title-tba-89/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2016/02/graduate-student-tea-38/
http://as.vanderbilt.edu/math/2016/02/graduate-student-tea-38/#commentsWed, 03 Feb 2016 20:30:48 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3101http://as.vanderbilt.edu/math/2016/02/graduate-student-tea-38/feed/0Orientable quadrilateral embeddings of cartesian product graphs
http://as.vanderbilt.edu/math/2016/02/orientable-quadrilateral-embeddings-of-cartesian-product-graphs/
http://as.vanderbilt.edu/math/2016/02/orientable-quadrilateral-embeddings-of-cartesian-product-graphs/#commentsMon, 01 Feb 2016 21:10:42 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3103White, Pisanski and others proved a number of results on the existence of quadrilateral embeddings of cartesian products of graphs; in some cases these provide minimum genus embeddings. In a 1992 paper Pisanski posed three questions. First, if G and H are connected 1-factorable r-regular graphs with r >= 2, does the cartesian product of G and H have an orientable quadrilateral embedding? Second, if G is r-regular, does the cartesian product of G with sufficiently many even cycles have an orientable quadrilateral embedding? Third, if G is an arbitrary connected graph, does the cartesian product of G with a sufficient large cube Q_n have an orientable quadrilateral embedding? We answer all three questions. The answer to the first question is negative, as we show using 3-regular examples. The answers to the second and third questions are positive, as we show using a general theorem that answers both. This is joint work with Wenzhong Liu, Dong Ye, and Xiaoya Zha. Wenzhong Liu spoke about this last semester, but since then we have strengthened many of the results.
]]>http://as.vanderbilt.edu/math/2016/02/orientable-quadrilateral-embeddings-of-cartesian-product-graphs/feed/0The Local Perturbation Analysis: a non‐linear stability technique for detectingspatial responses of complex cell regulatory systems
http://as.vanderbilt.edu/math/2016/01/the-local-perturbation-analysis-a-non%e2%80%90linear-stability-technique-for-detectingspatial-responses-of-complex-cell-regulatory-systems/
http://as.vanderbilt.edu/math/2016/01/the-local-perturbation-analysis-a-non%e2%80%90linear-stability-technique-for-detectingspatial-responses-of-complex-cell-regulatory-systems/#commentsFri, 29 Jan 2016 22:10:54 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3077How cells make key decisions is an important, if mysterious question that has generated sustained investigation in mathematical modeling, biophysics, and molecular biology. In response to sufficiently large stimuli, cells can respond in a number of manners include “spreading” to form greater contact area with the substratum, “contracting” to avoid contact, “polarizing” to prepare for motility, or generating highly dynamic “waves” of intracellular activity. While understanding these cellular responses is a classical PDE pattern formation problem, the models representing the underlying regulatory systems can be highly complex and the questions of interest are inherently non‐linear in nature. To address this issue I will describe a new nonlinear perturbation technique, the “Local Perturbation Analysis”, that has proven immensely useful in investigating these responses. This method is capable of producing non‐linear stability information for a common class (in cell regulatory systems) of reaction diffusion systems of arbitrary dimension (i.e. number of variables), while being no more complex to implement than existing linear methods. I will use this technique to investigate a number of biological regulatory systems and I) provide a hypothesis for how a simple, evolutionary conserved biochemical regulatory system can give rise a diverse array of responsesand II) show that this function can be preserved as this system becomes increasingly more complex through evolution.
]]>http://as.vanderbilt.edu/math/2016/01/the-local-perturbation-analysis-a-non%e2%80%90linear-stability-technique-for-detectingspatial-responses-of-complex-cell-regulatory-systems/feed/0The Information Theory of Joinings
http://as.vanderbilt.edu/math/2016/01/the-information-theory-of-joinings-2/
http://as.vanderbilt.edu/math/2016/01/the-information-theory-of-joinings-2/#commentsFri, 29 Jan 2016 22:10:06 +0000rongiolhttp://as.vanderbilt.edu/math/?p=3072I will present ongoing research into an area I am developing based on the idea of treating joinings of quasi-invariant actions of groups on probability spaces along similar lines are treating random variables as representing information, in particular I consider the “mutual information” of two spaces in terms of their joinings. Furstenberg entropy is a numerical measure of how far a quasi-invariant action of a group on a probability space is from measure-preserving. The main new tool I introduce is a relative version of this entropy measuring how far a homomorphism between such spaces is from being relatively measure-preserving. I show that it enjoys the properties one would expect such as additivity over compositions and apply this notion to develop an “information theory” of joinings proving analogues of many of the key theorems in the information theory of random variables.
]]>http://as.vanderbilt.edu/math/2016/01/the-information-theory-of-joinings-2/feed/0Orthogonal bases and tiling: analysis, number theory and combinatorics
http://as.vanderbilt.edu/math/2016/01/talk-title-tba-77/
http://as.vanderbilt.edu/math/2016/01/talk-title-tba-77/#commentsThu, 28 Jan 2016 21:10:17 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2627In 1974 Bent Fuglede conjectured that if $\Omega$ is a bounded domain in ${\Bbb R}^d$, then $L^2(\Omega)$ has an orthogonal basis of exponentials if and only if $\Omega$ tiles ${\Bbb R}^d$ by translation. Even though this conjecture was disproved by Terrance Tao in 2004 in dimensions $5$ and higher, it is continuing to lead researchers to fascinating connections and ideas that involve a variety of areas of modern mathematics. In this talk we will present a sampling of these ideas and connections between them, as well as some recent developments in this fascinating field. Tea at 3:30 pm in SC 1425. (Contact Person: Akram Aldroubi)
]]>http://as.vanderbilt.edu/math/2016/01/talk-title-tba-77/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2016/01/talk-title-tba-75/
http://as.vanderbilt.edu/math/2016/01/talk-title-tba-75/#commentsWed, 27 Jan 2016 22:10:21 +0000rongiolhttp://as.vanderbilt.edu/math/?p=2912http://as.vanderbilt.edu/math/2016/01/talk-title-tba-75/feed/0