{"id":39,"date":"2016-01-06T13:07:03","date_gmt":"2016-01-06T18:07:03","guid":{"rendered":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/?page_id=39"},"modified":"2016-01-06T13:07:03","modified_gmt":"2016-01-06T18:07:03","slug":"colloquia-fall-2000","status":"publish","type":"page","link":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/colloquia-fall-2000\/","title":{"rendered":"Colloquia, Fall 2000"},"content":{"rendered":"\n\n
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<\/font>Vanderbilt Mathematics<\/font><\/a><\/font>
\n<\/font>Colloquia<\/font>
\n<\/font>Fall 2000<\/font><\/h1>\n

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Colloquia are listed in reverse chronological order. The
\ntop of the list is subject to change, since more colloquia
\nare still being planned. Our colloquia, as well as our
\nseminars and other activities, feature
\nspeakers not only from our own department but also from other
\ndepartments all over the world. You may also want to consult our
\nweekly
\ncalendar<\/a> and past calendars<\/a>.
\nSome additional information about this year’s colloquia may be available
\nat the
\nweb page maintained by this year’s Colloquium Chairperson<\/a>.<\/p>\n

\n

   <\/p>\n

December 7.
\nMarek
\nKimmel<\/a>, of Rice University<\/a>.
\nTBA.<\/b>\n<\/p>\n

   <\/p>\n

November 30.
\nVan Vu, of
\nMicrosoft<\/a>.
\nOn a refinement of Waring’s assertion.<\/b>
\nIn 1770 Waring asserted (without proof) that for every fixed k, there is
\na number s such that every natural number can be represented as sum of s
\nkth<\/sup> powers. For instance, every natural number can be written
\nas sum of four squares, 9 cubes and so on. This has become the famous
\nWaring’s conjecture in number theory.
\n¶
\nWaring’s conjecture was proved in its full generality by several famous
\nmathematicians
\n(including Hilbert, Hardy-Littlewood, Vinogradov, Hua etc)
\nin the beginning of the 20th century.
\n¶
\nIn this talk, we consider a more “economical” version of Waring’s
\nassertion and
\ninvestigate the following question: Given k and s, do we really need
\nall kth<\/sup> powers
\nto guarantee that Waring’s assertion holds? Is it possible
\nthat
\nonly a small fraction still suffices? If yes, then how small?
\n¶
\nThe answer will be given during the talk.
\n\n<\/p>\n

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November 13.
\nAvner Friedman<\/a>,
\nof the
\nUniversity of Minnesota<\/a>.
\nA simple model of tumor growth.<\/b>
\nWe shall describe a phenomenological model of tumor growth with
\ninhibitors. Mathematically the model consists of a system of partial
\ndifferential equations in a region with free boundary – the moving
\nboundary of the tumor. We shall prove that the model produces dormant
\nstates that have non-radially symmetric solutions.
\n\n<\/p>\n

   <\/p>\n

November 9.
\nChris Godsil<\/a>,
\nof the University of Waterloo<\/a>.
\nSpin Models and Knots.<\/b>
\nJones introduced two ways to get knot invariants –
\nalgebraically, using a class of traces on braid groups and
\ncombinatorially, using partition functions on planar graphs. He
\nobserved that these methods were closely related. Subsequent work has
\nextended the combinatorial view by introducing so-called “four-weight
\nspin models”. However this theory is complicated and not obviously
\nrelated to braid groups. I will describe a new approach which shows
\nthat these spin models are related to representations of the braid
\ngroups and have a surprisingly rich structure.
\n\n<\/p>\n

   <\/p>\n

November 2. Larry
\nTaylor<\/a>, of the University of Notre Dame<\/a>.
\nMultivariable Calculus Reprised.<\/b>
\nI will begin by explaining the strong sense in which the calculus of
\nn<\/i>-variables is unique if n<\/i> is not 4, a result known since
\nthe early 70’s. In the last 20 years we have begun to understand just
\nhow bizarre the 4-variable case really is. Using results of mine and
\nothers, I will explore some of the exciting results that have emerged,
\nmany of which should be accessible to anyone with an understanding of
\nmulti-variable calculus and the elementary theory of smooth manifolds.
\nThese results range from basic existence results from the 80’s to recent
\nconstraints on the differential geometry of exotic 4-variable calculus.
\n\n<\/p>\n

   <\/p>\n

October 26.
\nIgor Mineyev<\/a>,
\nof the
\nUniversity of South Alabama<\/a>.
\nHyperbolicity and cohomology.<\/b>
\nThis talk will be accessible to a general audience.
\nWe review basic properties of hyperbolic groups introduced
\nby Gromov. Following Gersten and Gromov, we discuss how geometric
\nproperties of hyperbolic groups (and more generally, of finitely
\npresented groups) can be described homologically. We also give
\nthe following characterization:<\/p>\n

\nTheorem.<\/b>
\nA finitely presentable group G is hyperbolic if and only if
\nthe map from bounded cohomology H2<\/sup>b<\/sub>(G,V)
\nto H2<\/sup>(G,V) is
\nsurjective for all bounded G-modules V.\n<\/p><\/blockquote>\n

This extends the Gromov’s claim of surjectivity for real coefficients,
\nand is analogous to the Gersten’s cohomological characterization of
\nhyperbolic groups and to the Johnson’s characterization of amenable
\ngroups. —
\nRelations to computations, geometry, and analysis will be discussed.
\n\n<\/p>\n

   <\/p>\n

October 5. Xingxing
\nYu<\/a>, of Georgia Tech<\/a>.
\nLong cycles in graphs.<\/b>
\nMotivated by the 4-color problem, Whitney (1931) proved
\nthat every 4-connected triangulation of the sphere contains a
\nHamilton cycle. Tutte (1956) generalized Whitney’s result to
\n4-connected planar graphs. However, 3-connected planar graphs
\nneed not contain Hamilton cycles. In this talk, I will briefly
\nsurvey some results on long cycles in graphs, and sketch the
\nproofs of two related conjectures.<\/p>\n

\n<\/p>\n

   <\/p>\n

September 28. Dr.
\nYuri Gurevich<\/a>, of Microsoft Research and
\nUniversity of Michigan<\/a>.
\nWhat is an algorithm?<\/b> One may think that the title problem was
\nsolved long ago by Church and Turing. It wasn’t: there is more to an
\nalgorithm than the function it computes. (Besides, what function does an
\noperating system compute?) The interest in the problem is not only
\ntheoretical: applications include modeling, specification, verification
\nand design of software and hardware systems. However, in this talk, we
\nwill concentrate on the theoretical aspects. We will explain the
\nsequential Abstract State machine thesis. (See http:\/\/www.acm.org\/tocl\/accepted.html<\/a>
\n in this connection.) If time permits, we will mention parallel,
\ndistributed and real-time abstract state machines.
\n\n<\/p>\n

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September 21. Jon
\nMcCammond<\/a>, of Texas A&M
\nUniversity<\/a>. Calculating curvatures in concrete complexes.<\/b>
\nGeometric group theorists have been borrowing techniques from
\ndifferential geometers for more then 15 years. As a result there is now
\na well developed theory of nonpositively curved metrical simplicial
\ncomplexes which can be used to study of the “geometry of a group”.
\nUnfortunately for the working group theorist, there are very few tests
\nfor determining whether a given finite complex will fit into this
\nframework. … Specifically, given an explicit, finite, piecewise
\nEuclidean (PE) complex, there does not at present exist an algorithm for
\ndetermining whether this particular complex is nonpositively curved. …
\nAfter a quick introduction to\/review of the theory of metric simplicial
\ncomplexes, I will present a recent result (joint with Murray Elder) in
\nwhich we provide an algorithm to solve this problem if the PE complex
\nhas dimension less than four.
\n\n<\/p>\n

   <\/p>\n

\n

Previous semesters:<\/b> <\/p>\n