Department of Mathematics
http://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 24 Oct 2014 21:10:28 +0000enhourly1http://wordpress.org/?v=3.1.4Operator Algebras When Every Set is Lebesgue Measurable
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-19/
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-19/#commentsFri, 24 Oct 2014 21:10:28 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1327The assumption that every set of real numbers is Lebesgue measurable is particularly convenient in functional analysis. I will describe the theory of operator algebras in one model of set theory satisfying this assumption. I will then discuss the V*-algebras, which are $*$-algebras of bounded operators closed in a continuum analog of the ultraweak topology. Every unital separable C*-algebra has an enveloping V*-algebra, which may be identified with the space of strongly affine functions on the state space.
]]>http://as.vanderbilt.edu/math/2014/10/talk-title-tba-19/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-5/
http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-5/#commentsWed, 22 Oct 2014 19:30:56 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1405http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-5/feed/0Big Problems
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-6/
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-6/#commentsTue, 21 Oct 2014 21:10:51 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1225Have you ever wondered what math research is? This talk will present some of the major unsolved problems that have guided mathematics research throughout history. Some of them have been solved, some of them are very close to being solved, and some of them are still wide open. The talk will require no advanced mathematics and all audiences are welcome.
]]>http://as.vanderbilt.edu/math/2014/10/talk-title-tba-6/feed/0Contractible Edges and Some Applications
http://as.vanderbilt.edu/math/2014/10/contractible-edges-and-some-applications/
http://as.vanderbilt.edu/math/2014/10/contractible-edges-and-some-applications/#commentsMon, 20 Oct 2014 20:10:50 +0000rongiolhttp://as.vanderbilt.edu/math/?p=1403A *k-contractible edge* is an edge in a k-connected graph which when contracted leaves the resulting graph k-connected. We will present an introduction to the theory of 3- and 4-contractible edges, including some interesting known results and applications to the characterization of 4-connected planar K_2,5-minor-free graphs.
]]>http://as.vanderbilt.edu/math/2014/10/contractible-edges-and-some-applications/feed/0Nonlocal Phenomena in Partial Differential Equations
http://as.vanderbilt.edu/math/2014/10/nonlocal-phenomena-in-partial-differential-equations/
http://as.vanderbilt.edu/math/2014/10/nonlocal-phenomena-in-partial-differential-equations/#commentsFri, 17 Oct 2014 21:10:50 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1221Four examples of nonlocal phenomena in partial differential equations will be presented: (1) partial differential equations with time delay (non-local in time – the future depends not only on the present time, but on a history before the present time; (2) age structured population models (nonlocal in the boundary condition – offspring are born at age 0 from a mother with age in a specified age range); (3) cell-cell adhesion models (nonlocal in the transport term – cells have a spatial sensing radius, on the order of several cell diameters, that modulates their adhesion to other cells within their sensing radius); (4) interference phenomena in quantum mechanics (nonlocal in the probability density of spatial position – the detection of a quantum particle is determined only probabilistically).
]]>http://as.vanderbilt.edu/math/2014/10/nonlocal-phenomena-in-partial-differential-equations/feed/0Some Right-Angled Artin Groups Whose Automorphism Groups Do Not Have Property (T)
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-5/
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-5/#commentsWed, 15 Oct 2014 21:10:05 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1218Grunewald and Lubotzky described a family of representations of finite index subgroups of the automorphism group of a non-abelian free group, Aut(Fn). In particular, they showed that the image of these representations are arithmetic groups, and furthermore one particular subfamily of these representations have image SL(n-1,Z). A consequence of this is that Aut(F3) does not have Kazhdan’s Property (T). I will describe how these representations can be generalised to automorphism groups of right-angled Artin groups and hence describe a condition sufficient to imply when they do not have Property (T).
]]>http://as.vanderbilt.edu/math/2014/10/talk-title-tba-5/feed/0Graduate Student Tea
http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-4/
http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-4/#commentsWed, 15 Oct 2014 19:30:47 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1375http://as.vanderbilt.edu/math/2014/10/graduate-student-tea-4/feed/0Elementary Notion of Curvature
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-15/
http://as.vanderbilt.edu/math/2014/10/talk-title-tba-15/#commentsTue, 14 Oct 2014 23:00:50 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1281We shall discuss how to make intuitive notions of curvature (such as a plane is “flat” but a sphere is “curved”) mathematically precise.
]]>http://as.vanderbilt.edu/math/2014/10/talk-title-tba-15/feed/0Large Antichains in G-Independent Hypercubes
http://as.vanderbilt.edu/math/2014/10/large-antichains-in-g-independent-hypercubes/
http://as.vanderbilt.edu/math/2014/10/large-antichains-in-g-independent-hypercubes/#commentsMon, 13 Oct 2014 20:10:05 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1373Let F be a family of subsets of {1,2, … , n}. We say that F is an antichain if for every pair of distinct elements A, B in F, A is not a subset of B and B is not a subset of A. How large can an antichain be? This question was answered by a classical theorem of Sperner. In this talk we shall consider how that answer might change if F is restricted to picking its elements from the independent sets of a graph G on {1,2, …, n}.
]]>http://as.vanderbilt.edu/math/2014/10/large-antichains-in-g-independent-hypercubes/feed/0Universal Single Qubit and Qutrit Gates in the Kauffman-Jones Version of SU(2) Chern-Simons Theory at Level 4
http://as.vanderbilt.edu/math/2014/10/subfactor-seminar/
http://as.vanderbilt.edu/math/2014/10/subfactor-seminar/#commentsFri, 10 Oct 2014 21:10:58 +0000pam.joneshttp://as.vanderbilt.edu/math/?p=1148This is a recent development regarding universal topological quantum computation in a specific anyonic system, as appearing in the title, and joint work with Michael Freedman and Station Q. The anyonic system we use is hoped to become physically realizable. Our starting point are two Jones unitary representations of the braid group on four strands. One representation arises from braiding four anyons of respective topological charges 1,2,2,1 and the second representation occurs when braiding four anyons of identical topological charge 2. Both representations have a finite image and this image yields a finite subgroup of SU(2) and SU(3) respectively whose elements are called quantum gates. By protocols involving both braids and measurements, we show how to make in each case an additional quantum gate. In the qubit case, this new gate generates an infinite subgroup of SU(2) and in the qutrit case, the new gate enlarges the size of the finite SU(3) group issued from braiding only. Our method uses ancilla preparation with adequate norms and interesting relative phases and fusion of the ancilla into the input in order to form the gate.
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