Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityThu, 26 Mar 2015 22:10:50 +0000enhourly1http://wordpress.org/?v=3.1.4Why the oracle may not exist: ergodic families of Jacobi matrices, absolute continuity without almost periodicity
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-28/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-28/#commentsThu, 26 Mar 2015 22:10:50 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1614We will explain the recent solution of Kotani’s problem pertinent to the existence/non-existence of “oracle” (almost periodicity) for the ergodic families of Jacobi matrices (discrete Schroedinger operators). Kotani suggested that such families are subject to the following implication: if family has a non-trivial absolutely continuous spectrum (this happens almost surely) then almost surely it consists of almost periodic matrices (hence the possibility to predict the future by the past). Kotani proved an important positive result of this sort. Recently independently Artur Avila and Peter Yuditskii–myself disproved this conjecture of Kotani (by two different approaches). We will show the hidden singularity that defines when such Kotani’s oracle exists or not.
]]>http://as.vanderbilt.edu/math/2015/03/talk-title-tba-28/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-34/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-34/#commentsWed, 25 Mar 2015 22:10:47 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1839http://as.vanderbilt.edu/math/2015/03/talk-title-tba-34/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2015/03/graduate-student-tea-18/
http://as.vanderbilt.edu/math/2015/03/graduate-student-tea-18/#commentsWed, 25 Mar 2015 20:30:17 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1931http://as.vanderbilt.edu/math/2015/03/graduate-student-tea-18/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-41/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-41/#commentsTue, 24 Mar 2015 23:00:50 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1707http://as.vanderbilt.edu/math/2015/03/talk-title-tba-41/feed/0Projectable l-groups and algebras of logic: Categorical and Algebraic Connections
http://as.vanderbilt.edu/math/2015/03/projectable-l-groups-and-algebras-of-logic-categorical-and-algebraic-connections/
http://as.vanderbilt.edu/math/2015/03/projectable-l-groups-and-algebras-of-logic-categorical-and-algebraic-connections/#commentsMon, 23 Mar 2015 21:10:33 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1935P. F. Conrad and other authors launched a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex l-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. In this broader perspective, we revisit the Galatos-Tsinakis categorical equivalence between integral generalized MV algebras and negative cones of l-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Upon recalling that projectable integral generalized MV algebras and negative cones of projectable l-groups can be endowed with a positive Gödel implication, and turned into varieties by including this implication in their signature, we prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications.
]]>http://as.vanderbilt.edu/math/2015/03/projectable-l-groups-and-algebras-of-logic-categorical-and-algebraic-connections/feed/0Inverses of Bipartite Graphs
http://as.vanderbilt.edu/math/2015/03/inverses-of-bipartite-graphs/
http://as.vanderbilt.edu/math/2015/03/inverses-of-bipartite-graphs/#commentsMon, 23 Mar 2015 21:10:02 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1933Let G be a bipartite graph. Then G has a unique perfect matching if and only if its vertices can be ordered so that its bipartite adjacency matrix is a lower triangular matrix with 1 on its diagonal entries. It is an open problem raised by Godsil to characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-zero matrices. Godsil’s problem has connection with the Möbius functions of geometric lattice. In this talk, we present a solution to Godsil’s problem. This is joint work with Yujun Yang.
]]>http://as.vanderbilt.edu/math/2015/03/inverses-of-bipartite-graphs/feed/0Dynamics and dimension
http://as.vanderbilt.edu/math/2015/03/dynamics-and-dimension/
http://as.vanderbilt.edu/math/2015/03/dynamics-and-dimension/#commentsFri, 20 Mar 2015 22:10:53 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1922The notion of dimension within the context of dynamics has become an important tool both in the classification theory of nuclear C*-algebras and in the study of the relation between K-theory and asymptotic geometry. I will present a combinatorial perspective on this dimension theory and speculate about its applications in ergodic theory and operator algebras.
]]>http://as.vanderbilt.edu/math/2015/03/dynamics-and-dimension/feed/0Blow-up phenomena in elliptic and parabolic models
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-45/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-45/#commentsFri, 20 Mar 2015 22:10:03 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1788Abstract available online at http://www.math.vanderbilt.edu/~disconmm/PDE-Seminar/Abstract_PDE_seminar_March_18_2015.pdf
]]>http://as.vanderbilt.edu/math/2015/03/talk-title-tba-45/feed/0Entropy inside out
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-29/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-29/#commentsThu, 19 Mar 2015 21:10:01 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1617In the late 1950s Kolmogorov introduced the concept of entropy into ergodic theory, and since that time entropy has become a pervasive presence in the theory of dynamical systems with applications to various areas including Riemannian geometry, analytic number theory, and Diophantine approximation. Kolmogorov’s approach is based on Shannon’s theory of information from the 1940s and is most generally applicable to actions of groups satisfying a kind of internal finite approximation property called amenability.
In the last few years a new approach to entropy in dynamics was pioneered by Lewis Bowen and further developed by Hanfeng Li and myself. Here one externalizes the finite approximation of the dynamics so that it occurs outside the acting group, and then counts these models in the spirit of Boltzmann’s work in statistical mechanics. This notion of entropy applies to the much larger class of acting groups satisfying the property of soficity, which includes free groups. In fact it is not known whether non-sofic groups exist. I will discuss all of these developments, and describe how the passage from single transformations to actions of general amenable and sofic groups marks a shift in applications away from geometry and smooth dynamics and more towards noncommutative harmonic analysis and operator algebras.
]]>http://as.vanderbilt.edu/math/2015/03/talk-title-tba-29/feed/0Mathematical Foundation of the Minimum Error Entropy Algorithm
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-44/
http://as.vanderbilt.edu/math/2015/03/talk-title-tba-44/#commentsWed, 18 Mar 2015 21:10:49 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1784Information theoretical learning (ITL) is an important research area in signal processing and machine learning. It uses concepts of entropies and divergences from information theory to substitute the conventional statistical descriptors of variances and covariances. The empirical minimum error entropy (MEE) algorithm is a typical approach falling into this this framework and has been successfully used in both regression and classification problems. In this talk, I will discuss the consistency analysis of the MEE algorithm. For this purpose, we introduce two types of consistency. The error entropy consistency, which requires the error entropy of the learned function to approximate the minimum error entropy, is proven when the bandwidth parameter tends to 0 at an appropriate rate. The regression consistency, which requires the learned function to approximate the regression function, however, is a complicated issue. We prove that the error entropy consistency implies the regression consistency for homoskedastic models where the noise is independent of the input variable. But for heteroskedastic models, a counterexample is constructed to show that the two types of consistency are not necessarily coincident. A surprising result is that the regression consistency holds when the bandwidth parameter is sufficiently large. Regression consistency of two classes of special models is shown to hold with fixed bandwidth parameter. These results illustrate the complication of the MEE algorithm.
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