Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 21 Nov 2014 21:10:55 +0000enhourly1http://wordpress.org/?v=3.1.4Formation 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/0Talk Title TBA
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-18/
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-18/#commentsTue, 18 Nov 2014 23:00:07 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1288http://as.vanderbilt.edu/math/2014/11/talk-title-tba-18/feed/0Cycles in 5-Connected Planar Triangulations
http://as.vanderbilt.edu/math/2014/11/cycles-in-5-connected-planar-triangulations/
http://as.vanderbilt.edu/math/2014/11/cycles-in-5-connected-planar-triangulations/#commentsMon, 17 Nov 2014 20:10:09 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1532We look at the natural question of how many cycles must be present in a graph with p vertices and q edges. After examining the cubic graph case, we will switch attention to 5-connected planar triangulations. Without going into too many details we will look at showing that these graphs have exponentially many Hamiltonian cycles.
]]>http://as.vanderbilt.edu/math/2014/11/cycles-in-5-connected-planar-triangulations/feed/0Talk Title TBA
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-23/
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-23/#commentsFri, 14 Nov 2014 21:10:46 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1364http://as.vanderbilt.edu/math/2014/11/talk-title-tba-23/feed/0Velocity of Free Boundaries for Obstacle Problems with Time Dependent Data
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-11/
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-11/#commentsFri, 14 Nov 2014 21:10:13 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1270I will present recent results (joint work with Ivan Blank) on the motion induced on free boundaries of the obstacle problem when either boundary data or the shape of the obstacle are allowed to vary in a time-dependent fashion. Starting with a basic introduction and motivation for the obstacle problem, I will motivate measure-theoretic bounds for the change in contact region under small perturbations and show how this measure-theoretic information leads to precise velocity formulas around regular portions of the free boundary. Application to the Hele-Shaw problem will also be discussed as time allows.
]]>http://as.vanderbilt.edu/math/2014/11/talk-title-tba-11/feed/0Random Walks on Groups and the Kaimanovich-Vershik Conjecture
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-13/
http://as.vanderbilt.edu/math/2014/11/talk-title-tba-13/#commentsThu, 13 Nov 2014 21:10:56 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1276Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if y is in xS. Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derriennic and Avez, showed that G admits non-constant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik also presented the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; in dimensions 5 and above, their conjecture was proved by Erschler (2011). I will discuss the background and present a simple proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Yuval Peres.
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