Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityThu, 18 Dec 2014 16:59:44 +0000enhourly1http://wordpress.org/?v=3.1.4Character Rigidity for Lattices in Higher-Rank Groups
http://as.vanderbilt.edu/math/2014/12/character-rigidity-for-lattices-in-higher-rank-groups/
http://as.vanderbilt.edu/math/2014/12/character-rigidity-for-lattices-in-higher-rank-groups/#commentsFri, 05 Dec 2014 21:10:49 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1412We show that lattices in a higher rank center-free simple Lie groups are operator algebraic superrigid, i.e., any unitary representation of the lattice which generates a II_1 factor extends to a homomorphism of its group von Neumann algebra. This generalizes results of Margulis, and Stuck and Zimmer, and answers in the affirmative a conjecture of Connes.
]]>http://as.vanderbilt.edu/math/2014/12/character-rigidity-for-lattices-in-higher-rank-groups/feed/0Computer-assisted mathematics
http://as.vanderbilt.edu/math/2014/12/talk-title-tba-14/
http://as.vanderbilt.edu/math/2014/12/talk-title-tba-14/#commentsThu, 04 Dec 2014 21:10:42 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1278The advent of cheap and sophisticated computing has often affected the way in which we do pure mathematics. We give some examples of this phenomenon, including (i) the determination of symmetry groups of knots and links, (ii) an extension of Lenstra’s approach to the class number problem in number theory, and (iii) the exact determination of representation varieties of 3-manifold fundamental groups.
]]>http://as.vanderbilt.edu/math/2014/12/talk-title-tba-14/feed/0Geometrically maximal knots
http://as.vanderbilt.edu/math/2014/12/geometrically-maximal-knots/
http://as.vanderbilt.edu/math/2014/12/geometrically-maximal-knots/#commentsWed, 03 Dec 2014 21:10:37 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1553The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. This motivates several questions, such as, for which knots is the ratio very near the upper bound? For fixed crossing number, which knots have largest volume or determinant? We show that many families of alternating knots and links simultaneously maximize both ratios, and investigate related questions for these families. This is joint work with Abhijit Champanerkar and Ilya Kofman.
]]>http://as.vanderbilt.edu/math/2014/12/geometrically-maximal-knots/feed/0Special colloquium
http://as.vanderbilt.edu/math/2014/12/special-colloquium/
http://as.vanderbilt.edu/math/2014/12/special-colloquium/#commentsTue, 02 Dec 2014 21:10:37 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1535http://as.vanderbilt.edu/math/2014/12/special-colloquium/feed/0The dancing bear – what is a combinatorist doing in a law school?
http://as.vanderbilt.edu/math/2014/12/the-dancing-bear-what-is-a-combinatorist-doing-in-a-law-school/
http://as.vanderbilt.edu/math/2014/12/the-dancing-bear-what-is-a-combinatorist-doing-in-a-law-school/#commentsMon, 01 Dec 2014 20:10:51 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1555Law may not be the first place to look for mathematical inspiration, but I will show in this talk that legal controversies can lead to natural combinatorial questions. Focusing on the Banzhaf measure of voting power in simple games, I will illustrate how questions of legal doctrine, such as what voting methods meet a “one person – one vote ” standard, give rise to combinatorial theorems.
]]>http://as.vanderbilt.edu/math/2014/12/the-dancing-bear-what-is-a-combinatorist-doing-in-a-law-school/feed/0Formation of Trapped Surfaces in General Relativity
http://as.vanderbilt.edu/math/2014/11/formation-of-trapped-surfaces-in-general-relativity/
http://as.vanderbilt.edu/math/2014/11/formation-of-trapped-surfaces-in-general-relativity/#commentsFri, 21 Nov 2014 21:10:55 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1262The first is a simplified approach to Christodoulou’s monumental result which showed that trapped surfaces can form dynamically by the focusing of gravitational radiation from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region. The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Luk.
]]>http://as.vanderbilt.edu/math/2014/11/formation-of-trapped-surfaces-in-general-relativity/feed/0Dynamic Sampling
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-24/
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-24/#commentsFri, 21 Nov 2014 21:10:27 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1366Let $Y=\{f(i), Af(i), \dots, A^{\li}f(i): i \in \Omega\}$, where $A$ is a bounded operator on $\ell^2(I)$. The problem under consideration is to find necessary and sufficient conditions on $A, \Omega, \{l_i:i\in\Omega\}$ in order to recover any $ f \in \ell^2(I)$ from the measurements $Y$. This is the so called dynamical sampling problem in which we seek to recover a function $f$ by combining coarse samples of $f$ and its futures states $A^lf$. For self adjoint operators in infinite dimensional spaces, the M\”untz-Sz\’asz Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that $Y$ can never be a Riesz basis when $\Omega$ is finite. Moreover, when $\Omega$ is finite, $Y=\{f(i), Af(i), \dots, A^{\li}f(i): i \in \Omega\}$ is not a frame except for some very special cases. The existence of these special cases is derived from Carleson’s Theorem for interpolating sequences in the Hardy space $H(D)$.
]]>http://as.vanderbilt.edu/math/2014/11/talk-title-tba-24/feed/0Partition Regular Equations
http://as.vanderbilt.edu/math/2014/11/partition-regular-equations/
http://as.vanderbilt.edu/math/2014/11/partition-regular-equations/#commentsThu, 20 Nov 2014 21:10:22 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1473A finite or infinite matrix M is called ‘partition regular’ if whenever the natural numbers are finitely coloured there exists a vector x, with all of its entries the same colour, such that Mx=0. Many of the classical results of Ramsey theory, such as van der Waerden’s theorem or Schur’s theorem, may be naturally rephrased as assertions that certain matrices are partition regular. While the structure of finite partition regular matrices is well understood, little is known in the infinite case. In this talk we will review some known results and then proceed to some recent developments. No knowledge of the subject will be assumed. Tea at 3:30 pm in SC 1425.
]]>http://as.vanderbilt.edu/math/2014/11/partition-regular-equations/feed/0An Infinite Rank Summand of Topologically Slice Knots
http://as.vanderbilt.edu/math/2014/11/an-infinite-rank-summand-of-topologically-slice-knots/
http://as.vanderbilt.edu/math/2014/11/an-infinite-rank-summand-of-topologically-slice-knots/#commentsWed, 19 Nov 2014 21:10:10 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1469The knot concordance group consists of knots in the 3-sphere, modulo the equivalence relation of smooth concordance. The group operation is induced by connected sum, and the identity element is generated by slice knots. We will consider the subgroup T generated by topologically slice knots. Endo showed that T contains an infinite rank subgroup, and Livingston and Manolescu-Owens showed that T contains a rank 3 summand. We will show that in fact T contains an infinite rank summand. The proof relies on knot Floer homology.
]]>http://as.vanderbilt.edu/math/2014/11/an-infinite-rank-summand-of-topologically-slice-knots/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2014/11/graduate-student-tea-9/
http://as.vanderbilt.edu/math/2014/11/graduate-student-tea-9/#commentsWed, 19 Nov 2014 19:30:30 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1538http://as.vanderbilt.edu/math/2014/11/graduate-student-tea-9/feed/0