Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 13 Jan 2017 22:10:06 +0000enhourly1http://wordpress.org/?v=3.1.4On Convergence almost everywhere of Spectral Resolutions of Elliptic Differential Operators and The Multiple Fourier Integrals
http://as.vanderbilt.edu/math/2017/01/pde-seminar-on-convergence-almost-everywhere-of-spectral-resolutions-of-elliptic-differential-operators-and-the-multiple-fourier-integrals/
http://as.vanderbilt.edu/math/2017/01/pde-seminar-on-convergence-almost-everywhere-of-spectral-resolutions-of-elliptic-differential-operators-and-the-multiple-fourier-integrals/#commentsFri, 13 Jan 2017 22:10:06 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4282The question of the validity of the Luzin conjecture for the spherical partial sums of the multiple Fourier integrals is open so far. But if we consider the Riess means of the multiple Fourier integrals or in addition if we let f=0 on an open set G, and investigate the convergence to zero a.e. on G (i.e. generalized localization principle), then there are many possitive results. We first remind some of these results, and then study generalized localization principle for compactly supported distributions and present sharp conditions for its fullfilment.
]]>http://as.vanderbilt.edu/math/2017/01/pde-seminar-on-convergence-almost-everywhere-of-spectral-resolutions-of-elliptic-differential-operators-and-the-multiple-fourier-integrals/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2017/01/graduate-student-tea-63/
http://as.vanderbilt.edu/math/2017/01/graduate-student-tea-63/#commentsWed, 11 Jan 2017 20:30:48 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4287http://as.vanderbilt.edu/math/2017/01/graduate-student-tea-63/feed/0Special Colloquium: Towards a Higher Dimensional Analog of Uniform Boundedness of Torsion on Elliptic Curves
http://as.vanderbilt.edu/math/2017/01/special-colloquium-towards-a-higher-dimensional-analog-of-uniform-boundedness-of-torsion-on-elliptic-curves/
http://as.vanderbilt.edu/math/2017/01/special-colloquium-towards-a-higher-dimensional-analog-of-uniform-boundedness-of-torsion-on-elliptic-curves/#commentsTue, 10 Jan 2017 22:10:24 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=4266Let E be an elliptic curve over a number field K. Mordell proved that E(K), the set of K-rational points on E, forms a finitely generated abelian group. So one can ask how this abelian group varies as E and K vary, in particular one can ask whether the torsion subgroup can be arbitrarily large. In 1996, Merel gave a definitive answer to this question, showing that the size of the torsion subgroup of E(K) can be bounded by a constant that depends only on the degree of K over Q. K3 surfaces are in many ways similar to elliptic curves, although there is no group structure on a K3 surface. Despite this key difference, we explain how one can formulate an analog of Merel’s theorem for K3 surfaces and state some results in this direction. This talk will be suitable for a general audience. Tea at 3:30 pm in SC 1425. (Contact Person: John Ratcliffe)
]]>http://as.vanderbilt.edu/math/2017/01/special-colloquium-towards-a-higher-dimensional-analog-of-uniform-boundedness-of-torsion-on-elliptic-curves/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-151/
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-151/#commentsWed, 14 Dec 2016 22:10:25 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4182http://as.vanderbilt.edu/math/2016/12/talk-title-tba-151/feed/0Roots, Schottky Semigroups, and a Proof of Bandt’s Conjecture
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-137/
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-137/#commentsThu, 08 Dec 2016 22:10:26 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=3896In 1985, Barnsley and Harrington defined a “Mandelbrot Set” M for pairs of similarities — this is the set of complex numbers z with norm less than 1 for which the limit set of the semigroup generated by the similarities x -> zx and x -> z(x-1)+1 is connected. Equivalently, M is the closure of the set of roots of polynomials with coefficients in {-1,0,1}. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small “holes” in M, and conjectured that these holes were genuine. These holes are very interesting, since they are “exotic” components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, but his methods were not strong enough to show the existence of infinitely many holes; one difficulty with his approach was that he was not able to understand the interior points of M, and on the basis of numerical evidence he conjectured that the interior points are dense away from the real axis. We introduce the technique of *traps* to construct and certify interior points of M, and use them to prove Bandt’s Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in M. This is joint work with Sarah Koch and Alden Walker. Tea at 3:30 pm in SC 1425. (Contact Person: Mark Sapir)
]]>http://as.vanderbilt.edu/math/2016/12/talk-title-tba-137/feed/0Invariable Generation of Thompson Groups
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-150/
http://as.vanderbilt.edu/math/2016/12/talk-title-tba-150/#commentsWed, 07 Dec 2016 22:10:35 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=4158 A subset S of a group G invariably generates G if for every choice of g(s) ∈ G,s ∈ S the set {s^{g(s)}:s ∈ S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.
]]>http://as.vanderbilt.edu/math/2016/12/talk-title-tba-150/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2016/12/graduate-student-tea-62/
http://as.vanderbilt.edu/math/2016/12/graduate-student-tea-62/#commentsWed, 07 Dec 2016 20:30:23 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4227http://as.vanderbilt.edu/math/2016/12/graduate-student-tea-62/feed/0Special Colloquium: Fast Alternating Direction Algorithms for Nonsmooth, Convex/Nonconvex Optimization with Imaging Applications
http://as.vanderbilt.edu/math/2016/12/special-colloquiumfast-alternating-direction-algorithms-for-nonsmooth-convexnonconvex-optimization-with-imaging-applications/
http://as.vanderbilt.edu/math/2016/12/special-colloquiumfast-alternating-direction-algorithms-for-nonsmooth-convexnonconvex-optimization-with-imaging-applications/#commentsTue, 06 Dec 2016 22:10:38 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4211In this talk, I propose several efficient, reliable, and practical computational algorithms to solve challenging optimization problems arising in medical imaging and image processing. These problems are non-differentiable and can be ill-conditioned, non-convex, and/or highly nonlinear such that traditional sub-gradient based methods converge very slowly. To tackle the computational complexities, I use relaxation and approximation techniques. In addition, I exploit splitting variables and alternating direction method of multipliers to decouple the original challenging problems into sub-problems which are easier to solve. To obtain fast results, I develop innovative line search strategies and solve the sub-problems using Fourier transforms and shrinkage operators. I present the analytical properties of these algorithms as well as various numerical experiments on parallel Magnetic Resonance Imaging, image inpainting, and image colorization. The comparison with other methods are given to show the efficiency and the effectiveness of the proposed methods. Tea at 3:30 pm in SC 1425. (Contact Person: Alex Powell).
]]>http://as.vanderbilt.edu/math/2016/12/special-colloquiumfast-alternating-direction-algorithms-for-nonsmooth-convexnonconvex-optimization-with-imaging-applications/feed/0Circuit Cover of Signed Graphs
http://as.vanderbilt.edu/math/2016/12/circuit-cover-of-signed-graphs/
http://as.vanderbilt.edu/math/2016/12/circuit-cover-of-signed-graphs/#commentsMon, 05 Dec 2016 21:10:30 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4236A signed graph is a graph G associated with a mapping σ : E(G) → {−1, +1}, denoted by (G, σ). A cycle of (G, σ) is a connected 2-regular subgraph. A cycle C is positive if it has an even number of negative edges, and negative otherwise. A circuit of a signed graph (G, σ) is a positive cycle or a barbell consisting of two edge-disjoint negative cycles joined by a path. The definition of a circuit of a signed graph comes from the signed-graphic matroid. A circuit cover of (G, σ) is a family of circuits covering all edges of (G, σ). A circuit cover with the smallest total length is called a shortest circuit cover of (G, σ) and its length is denoted by scc(G, σ). Bouchet proved that a signed graph has a circuit cover if and only if it is flow-admissible (i.e., has a nowhere-zero integer flow). We show that a flow-admissible signed graph may not have a circuit double cover, but it is not known whether every flow-admissible signed graph has a circuit cover with length at most 2|E(G)|. We show that every 2-connected cubic signed graph has scc(G, σ) ≤ 26|E(G)|/9 if it is flow-admissible, and scc(G, σ) ≤ 23|E(G)|/9 if it has even negativeness, which improves previous results of Máčajová et. al and Chen et. al. This is joint work with Yezhou Wu.
]]>http://as.vanderbilt.edu/math/2016/12/circuit-cover-of-signed-graphs/feed/0Departmental Holiday Party
http://as.vanderbilt.edu/math/2016/12/departmental-holiday-party/
http://as.vanderbilt.edu/math/2016/12/departmental-holiday-party/#commentsSat, 03 Dec 2016 00:00:20 +0000hannah.grayhttp://as.vanderbilt.edu/math/?p=4194For tickets see K.T. Griffis, SC 1426C
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