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\nabstract.<\/a><\/p>\nThe fundamental group F(M)
\nof a type II1<\/sub> factor M was introduced by
\nMurray and von Neumann in 1943 in connection with their notion of
\ncontinuous dimension. It measures the extent to which “amplifications”
\nof M are isomorphic to M (e.g., if the algebra of
\n2×2 matrices over M is isomorhic to M then 2 is in F(M)}.
\nIt is a puzzling and still poorly understood invariant.<\/p>\nWe will present results providing the first examples of factors M
\nwith trivial fundamental group. Thus, if G is the arithmetic group
\nZ2<\/sup> \\rtimes SL(2, Z) and M=L(G) is the associated
\ngroup von Neumann algebra then F(M)={1}. The proof uses in a
\ncrucial way the “weak amenability” of \\Gamma=SL(2, Z)
\n(i.e. Haagerup’s approximation property or equivalently Gromov’s
\na-T-menability) and the relative property (T) of Kazhdan-Margulis
\nof the inclusion Z2<\/sup> \\subset Z2<\/sup> \\rtimes \\Gamma.
\nThe combination of these two properties makes it possible to prove
\na unique decomposition of M as a cross-product
\nM = L^\\infty(T2<\/sup>) \\rtimes \\Gamma, thus allowing us to define
\nl2<\/sup>-Betti number invariants bn<\/sub>(M) from the
\nl2<\/sup>-Betti numbers bn<\/sub>(R), defined by Gaboriau
\nin 2001, of the equivalence relation R induced by \\Gamma on
\nT2<\/sup>.<\/p>\nContact persons<\/u>: Dietmar Bisch and Gennadi
\nKasparov<\/font><\/p>\n<\/td>\n<\/tr>\n\n| March 20, 2003<\/b> <\/th>\n | \nRalph McKenzie, Vanderbilt University<\/b><\/center>
\n
\n
\nDefining and Recognizing Structure in General Algebras;
\nCongruence Lattices are the Key to Deep Results
\n<\/font><\/strong><\/center>
\n
\nAbstract<\/u>:
\nIn the last three decades of the twentieth century,
\nuniversal algebra began to realize many of the lofty goals Garrett
\nBirkhoff had envisioned for it in 1933. Especially notable is the
\nability to
\nformulate and proof deep results about all finite and locally finite
\nalgebraic sysems. Tame congruence theory is an analysis of the possible
\nways a clone of operations on a set may be organized relative to a
\ncovering pair of congruences that it admits. Applied to all the covering pairs
\nof congruences of a finite algebra A, and also those of finite
\nalgebras of functions derived from A, this theory reveals a
\nwealth of previously unrecognized structural features in finite
\nalgebras, and provides natural and useful new ways of classifying them.
\nThe task of working out the implications and extending the insights
\nof tame congruence theory has been the dominant theme of research
\nin general algebra for the past twenty years. Many of the results
\ndiscovered with its aid have since been extended by other means
\nto all algebraic systems (without local finiteness assumptions).<\/p>\nI originated this theory in 1981–84 (with the considerable help of my
\nthen graduate student David Hobby).
\nIn this talk, I will tell the story of how a long-running fascination
\nwith one little problem and several big problems, combined with
\nstubbornness and luck, led to some big results.<\/p>\n Contact person<\/u>: Dietmar Bisch <\/font><\/p>\n<\/td>\n<\/tr>\n\n| March 27, 2003<\/b> <\/th>\n | \nZhong-Jin Ruan, University of Illinois at Urbana-Champaign<\/b><\/center>
\n
\nOperator Spaces: A Natural
\nNon-commutative Quantization of Functional Analysis <\/font><\/strong><\/center>
\n
\nAbstract<\/u>:
\nAn operator space is a norm closed subspace of bounded operators on
\nsome Hilbert space together with a distinguished “matrix norm”.
\nMorphisms between operator spaces are “completely bounded linear maps”.<\/p>\nOperator space theory is a natural non-commutative quantization of
\nfunctional analysis (Banach space theory). In this talk, I will first
\n discuss some fundamental results in operator spaces, and then
\n discuss some interesting applications to operator algebras and
\n non-commutative harmonic analysis.<\/p>\n Contact person<\/u>: Guoliang Yu and Dechao
\nZheng <\/font><\/p>\n<\/td>\n<\/tr>\n\n| April 3, 2003<\/b> <\/th>\n | \nZhenghan Wang, Indiana University <\/b><\/center>
\n
\nTopological Quantum Computation<\/font><\/strong><\/center>
\n
\nAbstract<\/u>:
\nAn equivalent model of quantum computing based on topological quantum
\nfield theories has been proposed in the work of Freedman, Kitaev, Larsen
\nand Wang. This new way of looking at quantum computation provides efficient
\nquantum algorithms to approximately compute quantum invariants of links and
\n3-manifolds, and a possible way to realize a large scale quantum computer.
\nWe will start with a general introduction to quantum information science,
\nand then discuss the connection to topology, computer science and condensed
\nmatter physics.<\/p>\nContact persons<\/u>: Dietmar Bisch and Bruce
\nHughes<\/font><\/p>\n<\/td>\n<\/tr>\n\n| April 14, 2003<\/b> <\/th>\n | \nRostislav Grigorchuk, Texas A&M University <\/b><\/center>
\n
\nThe Ihara Zeta Function of Infinite
\nGraphs, the KNS Spectral Measure and Integrable Maps
\n <\/font><\/strong><\/center>
\n
\nNote: Special Colloquium, Monday, 4:10pm in SC 1431 <\/b><\/center>
\n(Part 1 of talk 4:10-5:00pm, 5 minutes break, Part 2 of talk
\n5:05-5:50pm)<\/b><\/center>
\n
\nAbstract<\/u>:
\nWe define the Ihara zeta function for Cayley graphs of infinite finitely
\ngenerated groups.
\nWe extend the definition of the Ihara zeta function to infinite
\ngraphs which are limits of sequences Xn<\/sub>
\nof finite k-regular graphs such that Xn+1<\/sub> covers
\nXn<\/sub>.
\nWe associate to such a graph a measure mu with support
\nin [-1,1] called the Kesten-von Neumann-Serre spectral measure.
\nWe present a few examples of computation of zeta function and a measure
\n\\mu for Schreier graphs of some fractal groups generated by finite automata.
\nThese computations are closely related to the integrability of
\nsome 2-dimensional mappings which are also in focus of our
\nconsiderations. (joint work with A. Zuk (University of Chicago))<\/p>\nContact person<\/u>: Mark Sapir<\/font><\/p>\n<\/td>\n<\/tr>\n\n| April 22, 2003<\/b> <\/th>\n | \nYuri Bahturin, visiting Vanderbilt University<\/b><\/center>
\n
\nNote: Tuesday, 4:10-5:00pm in SC 1431 <\/b><\/center>
\n
\n
\nBicharacters on Hopf Algebras
\n <\/font><\/strong><\/center><\/p>\nAbstract<\/u>:
\nThe notion of skew-symmetric bicharacter on a Hopf algebra is dual to
\nthat of
\nR-matrix widely used in mathematics and beyond. It appears in the theory of
\nquantum groups, Yang – Baxter equations, etc. While R-matrices work in the
\ncase of finite-dimensional Hopf algebras, bicharacters are more universal
\nand work fine in the case of infinite dimensions. One of the basic examples
\nof bicharacters are the commutation factors on abelian groups used, in
\nparticular, to define so called color Lie superalgebras, a notion generated
\nin physics few decades ago. In general, bicharacters on a commutative and
\ncocommutative Hopf algebra H allow one to define Lie structures on the
\nalgebras with the coaction H. On the other hand, Hopf algebras whose
\nstructure includes a fixed skew-symmetric bicharacter form an important
\nclass of cotriangular Hopf algebras, dual to the triangular ones
\nintroduced by Drinfeld. If we want to classify such Lie structures or such Hopf
\nalgebras, we may apply so called Scheunert’s trick, saying that any
\nskew-symmetric
\nbicharacter b(g,h) on a finitely generated abelian group G can be written in
\nthe form b(g,h)=a(g,h)[s(g,h)\/s(h,g)], where a(g,h) is either trivial or the
\nbicharacter defining ordinary Lie superalgebras and s(g,h) is a not
\nnecessarily skew-symmetric bicharacter. In particular, if we deform the
\nproduct in a G-graded algebra using s(g,h), then the color commutator defined
\nby b(g,h) becomes either an ordinary Lie bracket or an ordinary superbracket.
\nThe goal of this talk is to report on most recent result in this area,
\nincluding the extension of Scheunert’s trick to arbitrary cocommutative Hopf
\nalgebras of characteristic different from 2 and the classification of
\nfinite-dimensional algebras, which are commutative under a suitable
\ngeneralized Lie bracket.<\/p>\n Contact person<\/u>: Mark Sapir<\/font><\/p>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/table>\n
\n
\nColloquium Chair (Spring 2003): Dietmar Bisch<\/b>
\n<\/p>\n\nBack to the Vanderbilt Mathematics Department homepage<\/a>\n<\/li>\n<\/h2>\n","protected":false},"excerpt":{"rendered":"Mathematics Colloquia, Spring 2003 Thursdays 4:10 pm in 1431 Stevenson Center, unless otherwise noted Tea at 3:30 pm in 1425 Stevenson Center January 23, 2003 Glenn Webb, Vanderbilt University Mailborne Transmission of Anthrax: Modeling and Implications Abstract: A mathematical model is developed to analyze the transmission of inhalational anthrax through the postal system by cross-contamination…<\/p>\n","protected":false},"author":637,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-49","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/49","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/users\/637"}],"replies":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/comments?post=49"}],"version-history":[{"count":3,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/49\/revisions"}],"predecessor-version":[{"id":52,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/pages\/49\/revisions\/52"}],"wp:attachment":[{"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/media?parent=49"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/wp-json\/wp\/v2\/tags?post=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}} | | | | |