{"id":41,"date":"2016-01-06T13:32:49","date_gmt":"2016-01-06T18:32:49","guid":{"rendered":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/?page_id=41"},"modified":"2016-01-06T13:32:49","modified_gmt":"2016-01-06T18:32:49","slug":"colloquia-spring-2001","status":"publish","type":"page","link":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/colloquia-spring-2001\/","title":{"rendered":"Colloquia, Spring 2001"},"content":{"rendered":"\n\n
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<\/font>Vanderbilt Mathematics<\/font><\/a><\/font>
\n<\/font>Colloquia<\/font>
\n<\/font>Spring 2001<\/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

June 11.
\nSuhrit K. Dey, of
\nEastern Illinois University<\/a>.
\nBiomechanics of Lymphocytes with Applications for Prevention \/
\nCure of Breast Cancer.<\/b>
\nLymphocytes are white cells, which fight infections and cancer. T cells
\nare lymphocytes, which attack all antigens very aggressively. Thymus is
\na lymphoid organ which produces a hormone called thymosin which help T –
\ncells to proliferate and while staying in the thymus, T – cells become
\ntrained soldiers to defend the body against the invasion of cancer.
\nThymus is located inside the chest cavity above the breast and in the
\nspace between the lungs. —
\nThree mathematical models have been developed to represent Surveillance
\nof lymphocytes, Rate of mobilization of lymphocytes and the defense
\nmechanism of lymphocytes. The third model deals with activator-inhibitor
\nresponse system. Cancer cells are activators. They activate the
\nlymphocytes to respond as inhibitors. This model consists of two coupled
\nnonlinear partial differential equations, which were solved numerically.
\nIf the activator prevails, cancer spreads and if the inhibitor prevails
\nthe immune system overpowers cancer. The models are all one-dimensional
\nbased on the assumption that the cancer site is very near to at least
\none lympnode. Physiologically this scenario is applicable to breast
\ncancer.\n<\/p>\n

   <\/p>\n

June 7.
\nJoachim Escher,
\nof the
\nUniversity of Hannover<\/a>.
\nAnalytic Solutions for a Stefan Problem with Gibbs-Thomson Correction.<\/b>
\nStefan problems are widely used to model the freezing\/melting process of
\nwater\/ice. In this talk a general existence and uniqueness result of
\nclassical solutions for a class of Stefan problems with Gibbs-Thomson
\ncorrection in arbitrary space dimensions is provided. In addition, it
\nwill show that the moving boundary depends analytically on the temporal
\nand spatial variables. — Of crucial importance for the analysis is the
\nproperty of maximal Lp<\/sup>-regularity for the linearized problem,
\nwhich is based on the Dore-Venni theorem.\n<\/p>\n

   <\/p>\n

April 19.
\nPaul Baum<\/a>, of
\nPennsylvania State University<\/a>.
\nK theory for group C* algebras.<\/b>
\nSeveral issues in representation theory and geometry-topology
\ncan be unified by studying the K-theory of group C* algebras.
\nP.Baum and A.Connes have conjectured a formula for this K-theory.
\nThis talk states the conjecture and indicates how it is related
\nto various questions. The talk is intended for a general mathematical
\naudience, and basic definitions (C* algebra , K theory) will
\nbe carefully stated.\n<\/p>\n

   <\/p>\n

April 18.
\nRoger Horn, of the
\nUniversity of Utah<\/a>.
\nEqualities and Inequalities for Matrix Eigenvalues and Singular Values.<\/b>
\nMany classical inequalities for matrix eigenvalues and singular values can
\nbe understood in the context of simple counting arguments involving
\nsubspace intersections. These arguments often have the added benefit of
\nidentifying cases of equality. We illustrate these ideas by discussing
\nthe Weyl inequalities for the eigenvalues of a sum of two Hermitian
\nmatrices, and the Cauchy interlacing inequalities for a bordered Hermitian
\nmatrix.\n<\/p>\n

   <\/p>\n

April 13.
\nRichard Laver<\/a>,
\nof the
\nUniversity of Colorado<\/a>.
\nLarge cardinals and their implications in classical mathematical
\nareas.<\/b>
\nThis will be an expository talk. We will review some of the basic
\nconcepts of set theory, and state some “large cardinal” axioms—axioms which
\nassert the existence of infinite cardinals having properties which make them,
\nroughly speaking, so large that their existence cannot be proved by ordinary
\nmathematical methods. The question arises as to what applications the
\nexistence of such cardinals might have to classical mathematics; we’ll
\ndiscuss a couple of examples, one about projections of Borel sets and
\none about finite algebras in an area related to knot theory.
\n[In this last example, the large cardinal assumption is not known to
\nbe necessary.]\n<\/p>\n

   <\/p>\n

April 12.
\nDaniel E. Gonsor,
\nof
\nThe Boeing Company<\/a>,
\nSeattle, Washington.
\nThree Problems from Industrial Mathematics.<\/b>
\nMost talks on mathematics in industry present difficult problems
\nfor which mathematics played an important role in deriving an effective
\nsolution. In this talk we will take a different approach and look at
\nthree routine problems that arose in the context of everyday work at The
\nBoeing Company. In each case the (then) current solution method
\nproduced unsatisfactory results. The first problem involves data
\nfitting, the second calculating a geodesic, and the third calculating
\narc length. We will analyze each solution method, determine the
\nsource(s) of the deficiency, and propose a better solution. The reason
\nfor choosing these particular problems is threefold. First, each problem
\nis representative of an approach or mentality that one often encounters
\nin industrial mathematics. In the case of data fitting it is the
\ninsistence on interpolation, for the geodesic example it is the reliance
\non intuition and visual verification, and for the arc length example it
\nis the disregard of fundamental hypothesis. The second reason for
\nchoosing these examples is that the mathematics involved is fairly
\nelementary, and therefore will be accessible to undergraduate math
\nmajors. The last reason is that the fundamental deficiency in each
\nexample can be traced to a lack of mathematical expertise.\n<\/p>\n

   <\/p>\n

April 11.
\nKarin Goosen,
\nof the
\nUniversity of Stellenbosch<\/a>,
\nRepublic of South Africa.
\nInterpolatory Subdivision and Wavelets on an interval.<\/b>
\nWe consider a method of adapting the Dubuc-Deslauriers subdivision
\nscheme to accommodate sequences of finite length, in a way which ensures
\nconvergence of the adapted scheme and the existence of an associated
\nrefinable function. Then with an appropriate definition of a
\ninterpolation wavelet, we obtain decomposition and reconstruction
\nalgorithms. Illustrations of the theory are provided.\n<\/p>\n

   <\/p>\n

March 29.
\nEric Weber<\/a>,
\nof
\nTexas A&M<\/a>.
\nTranslation Invariant Wavelets.<\/b>
\nAll wavelets can be associated to a multiresolution like
\nstructure, i.e. an increasing sequence of subspaces of L2<\/sup>. We
\nconsider the interaction of a wavelet and the translation operator in
\nterms of which of the subspaces in this multiresolution like structure are
\ninvariant under the translation operator. This action defines the notion
\nof the translation invariance property of order n. In this talk we
\nshall characterize such wavelets and show that they exist. We shall
\nalso discuss how these special types of wavelets might lead to techniques
\nfor edge detection in images.\n<\/p>\n

   <\/p>\n

March 27.
\nMisha Kapovich<\/a>,
\nof the University of Utah<\/a>.
\nSingularities of representation varieties of finitely generated
\ngroups.<\/b>
\nGiven a finitely-generated group G<\/font> and an
\nalgebraic Lie group G (for instance, SL(2)) the space of representations
\nHom(G<\/font>,G) itself has structure of an algebraic
\nvariety. In this talk I will outline several “universality” theorems for
\nthe singularities of Hom(G<\/font>,G), whose main
\nmessage is that these singularities could be as bad as one wishes. This
\napplies to such classes of finitely generated groups as Coxeter groups,
\nArtin groups, fundamental groups of 3-manifolds and discrete groups of
\nisometries of the hyperbolic 3-space. This is a joint work with John
\nMillson.\n<\/p>\n

   <\/p>\n

March 19.
\nKirby Baker<\/a>, of
\nUCLA<\/a>.
\nUnavoidable patterns in long strings of symbols.<\/b>
\nThue showed that using an alphabet of three symbols it is
\npossible to construct an infinite string with no block that is
\nimmediately repeated. We say that such a string avoids the
\npattern xx . On the other hand some patterns, such as
\nxyxzxyx, are unavoidable no matter how many symbols are
\nin the alphabet; in other words, there are always blocks
\nX, Y, Z so that XYXZXYX occur consecutively. There are
\nintriguing questions as to which patterns can be avoided
\nusing what sizes of alphabets.
\n\n<\/p>\n

   <\/p>\n

March 15.
\nWai Shing Tang<\/a>,
\nof the
\nNational University of Singapore<\/a>.
\nA Hilbert space approach to wavelets.<\/b>
\nIn this talk, we first review the concept of multiresolutions of
\nL2<\/sup>(R), as introduced by Meyer and Mallat in the mid 1980’s,
\nand show how an orthonormal wavelet can be obtained from a
\nmultiresolution. Next we describe a connection between the existence of
\nwavelets and Robertson’s result on wandering subspaces for unitary
\noperators on Hilbert spaces. Finally, we give a brief summary of some
\nrecent work of the speaker and his collaborators on the approach of
\nwavelets in Hilbert spaces.\n<\/p>\n

   <\/p>\n

March 13.
\nAlexander Kostochka,
\nUniversity of Illinois<\/a>.
\nEquitable colorings of graphs.<\/b> In many applications of graph
\ncolorings color classes should not be large. A good model for such
\napplications is the notion of equitable coloring<\/i> — a proper
\ncoloring where the difference between the sizes of any two color classes
\nis at most one. Hajnal and Szemerédi proved that for every
\nD³<\/font>1
\nand
\nk³D<\/font>+1,
\neach graph with maximum degree at
\nmost
\nD<\/font>
\nadmits equitable coloring with k colors. The aim of the
\ntalk is to survey recent progress in studying equitable colorings and to
\nprove a conjecture on equitable colorings of outerplanar graphs. We also
\nwill discuss an analog of equitable coloring for list colorings.\n<\/p>\n

   <\/p>\n

March 5.
\nDmitry Kozlov<\/a>, of the
\nRoyal
\nInstitute of Technology, Stockholm, Sweden<\/a>.
\nStratifications indexed by partitions and combinatorial models for
\nhomology.<\/b>
\nIn this talk I will discuss several connections between various
\ncombinatorial objects (partitions, partially ordered sets, labeled forests,
\nmatchings) and algebraic invariants (homology groups, Betti numbers) of
\ncertain
\ntopological spaces.
\n—
\nMore specifically, I shall consider several topological spaces equipped with
\nstratifications indexed by integer partitions. In each case I consider the
\nproblem of studying homology groups of strata. I shall describe how to
\nconstruct various models for computing these groups and present the following
\napplications:<\/p>\n

    \n
  1. determining the homology of resonance-free orbit arrangements (with
    \nthe help of general lexicographic shellability), thereby settling a
    \nconjecture of Bjorner for this special case;\n<\/li>\n
  2. a combinatorial reproof of Arnol’d theorem regarding the rational
    \nhomology of the space of monic complex polynomials with at least q roots
    \nof multiplicity k;\n<\/li>\n
  3. a counterexample to a conjecture by Sundaram and Welker;\n<\/li>\n
  4. a computation of the homology groups of the space of hyperbolic
    \npolynomials with at least q roots of multiplicity k.\n<\/li>\n<\/ol>\n

       <\/p>\n

    March 1.
    \n    Bjarne Toft<\/a>,
    \n    University of Southern Denmark<\/a>
    \n& Vanderbilt University.
    \nJulius Petersen – the man, the myth, the legend.<\/b>
    \nThe emergence of Danish mathematics at the end of the nineteenth century and
    \nthe dominant role of geometry in Danish research is closely linked to the work
    \nof two mathematicians, Zeuthen and Petersen. Both are still remembered for
    \ntheir geometry, but Zeuthen also as a historian of mathematics, and Petersen
    \nfor his graph theory.
    \n—
    \nPetersen did pioneering work in a number of fields, including cryptography and
    \neconomics, but both his graph theory and some of the other brilliant pieces
    \nwent unnoticed or met with outright rejection in his own time. This is not to
    \nimply that his life was one of disappointment – far from it! He was the
    \nembodiment of the best sense of humor and the most vigorous joy in life.
    \n—
    \nA biography of Petersen by Jesper Lutzen, Gert Sabidussi and Bjarne Toft has
    \nbeen published in Discrete Mathematics<\/i> Volume 100. The talk is based on that
    \npaper.
    \n—
    \n(Related
    \n    web page<\/a>)\n<\/p>\n

       <\/p>\n

    February 22.
    \n    Nigel Higson<\/a>,
    \nof     Penn State<\/a>.
    \nGroup C*-Algebras and Topology.<\/b>
    \nI will give a survey of some current work at the interface
    \nof C*-algebra theory and the topology of manifolds. The central
    \nproblem here is the Baum-Connes conjecture, which has
    \nimplications not only for topology but also for harmonic analysis.
    \nThe conjecture will be the focus of the lecture.
    \n\n<\/p>\n

       <\/p>\n

    February 21.
    \n    Don Hong<\/a>,
    \nof
    \n    East Tennessee State University<\/a>.
    \nMultilevel structure of bivariate spline spaces over
    \ntriangulations.<\/b>
    \nIn this talk, we’ll investigate multilevel structure of bivariate spline
    \nspaces. Wavelet decomposition method, multigrid technique in finite
    \nelement, and subdivision scheme in splines and approximation theory
    \nactually reach the same goal by different routes. Wavelet theory
    \nprovides very efficient algorithms in decomposition and reconstruction
    \nby using the so-called wavelets, which are actually locally supported
    \nbasis functions like “little waves”. We’ll present some recent results
    \non multivariate splines and wavelet-type basis construction for
    \nbivariate spline spaces over triangulations.\n<\/p>\n

       <\/p>\n

    February 20.
    \n    Anthony
    \nHilton<\/a>, of     University
    \nof Reading<\/a>, UK.
    \nSome coloring theorems and conjectures.<\/b>
    \nThere is a well known conjecture that if a regular graph of even order has
    \ndegree greater than half the order then it is the union of edge-disjoint
    \n1-factors.
    \nWith David Cariolaro, I have shown that this is true if 3\/4 is substituted
    \nfor 1\/2.
    \nI shall sketch the proof of this result, and also discuss its relationship
    \nwith two other well known conjectures, the
    \nOverfull Conjecture<\/i> and
    \nthe Conformability Conjecture<\/i>.
    \n—
    \nLet us call the conjecture above about regular graphs the
    \nRegular Graph Conjecture<\/i>.
    \nLet us now explain what the other two conjectures say. —
    \nThe chromatic index of a graph is the least number of colours needed to
    \ncolour the edges of the graph so that no two edges with the same colour
    \nare incident with the same vertex.
    \nBy Vizing’s Theorem the chromatic index equals either the maximum degree
    \n(in which case the graph is called Class 1) or the maximum degree plus 1
    \n(in which case the graph is Class 2).
    \nA graph is Overfull if it is of odd order and the number of edges is
    \ngreater than half the product of the maximum degree and
    \n(the order lass one).
    \nThe Overfull Conjecture<\/i> is that if the maximum degree of a graph
    \nis greater than one third the order, then the graph is Class 2 if and only
    \nif it contains an overfull subgraph of the same maximum degree.
    \nThe Regular Graph Conjecture follows from the Overfull Conjecture.
    \n—
    \nThe total chromatic number of a graph is the least number of colours needed
    \nto colour the vertices and the edges of the graph so that no two incident
    \nor adjacent elements get the same colour.
    \nAn old conjecture of Behzad and Vizing (independently) is that the total
    \nchromatic number of a graph equals either (1 or 2) plus the maximum
    \ndegree.
    \nIn the first case the graph is called Type 1 and in the second Type 2.
    \nA vertex colouring of a graph with a number of colours equal to 1 plus the
    \nmaximum degree is called Conformable if the number of colour classes
    \nhaving parity different from that of the order is at most the deficiency
    \nof the graph, where the Deficiency is defined to be the product of
    \n(the maximum degree and the order) less twice the number of edges.
    \n(The deficiency measures the amount by which the graph fails to be
    \nregular.)
    \nA graph is called conformable if it has a conformable vertex colouring.
    \nThe Conformability Conjecture<\/i> is that a graph with the property that
    \nthe maximum degree is greater than twice the order is Type 2 if and only if
    \nit has a subgraph of the same maximum degree which is either nonconformable
    \nor is a complete graph of odd order with one edge subdivided.
    \n\n<\/p>\n

       <\/p>\n

    February 15.
    \n    Friedrich Wehrung<\/a>, of the
    \n    Université de Caen<\/a>, France.
    \nCongruence lattices of lattices: a survey.<\/b>
    \nThe Congruence Lattice Problem, that asks whether every distributive
    \nalgebraic lattice is isomorphic to the congruence lattice of a lattice,
    \nis, despite many attemps, still unsolved. I present the most recent
    \nresults about this problem, both negative and positive. The negative
    \nresults say essentially that one cannot solve the problem by using
    \nlattices with permutable congruences. The positive results imply that all
    \nknown representation theorems can be done with relatively complemented
    \nlattices with zero.
    \n\n<\/p>\n

       <\/p>\n

    February 8.
    \n    Ivo Dinov<\/a>, Neurology,
    \n    UCLA School of Medicine<\/a>.
    \nMathematical and Computational Challenges in Brain
    \nMapping and Neuroimaging.<\/b>
    \nThe incredibly complex (yet robust, efficient and
    \nelegant) functional, anatomical and bio-physiological
    \norganization of the brain provides a rich source for
    \ndeveloping interesting mathematical and computational
    \nmodels. Following an introduction to the goals of brain
    \nmapping research and the variety of brain-data acquisition
    \nmethods we will describe a number of problems and obstacles
    \nresearchers in this field encounter. Among the most needed
    \nalgorithms and data filters are models for: Stereotactic
    \ndata registration (alignment); Cortical surface modeling;
    \nTissue segmentation; Skull stripping and feature
    \nextraction; Construction of population specific brain
    \natlases; Measures of temporal\/developmental changes and
    \nvariability; Statistical assessment of structural or
    \nfunctional differences. We will devote most of our
    \nattention to the problems of signal representation and
    \nquantitative evaluation of different image registration
    \nmethods. MRI (magnetic resonance imaging) and PET (positron
    \nemission tomography) data from elderly normal controls and
    \ndementia patients will be used to illustrate the
    \nfunctionality and disadvantages of a variety of
    \nmathematical techniques.\n<\/p>\n

       <\/p>\n

    January 18.
    \n    Edward Saff<\/a>,
    \nof the     University of South Florida<\/a>.
    \nDistributing Many Points on a Sphere.<\/b>
    \nThe problem of distributing a large number of points
    \nuniformly over the surface of a sphere arises in
    \nmany practical and theoretical situations. We discuss
    \ngenerating such points by optimization with respect
    \nto a generalized energy criterion. Our interest is
    \nprimarily with the asymptotic behavior of these
    \noptimal spherical configurations of N points as
    \nN tends to infinity. Methods for generating “near
    \noptimal” points will also be discussed along with
    \nseveral challenging open problems.\n<\/p>\n

        <\/p>\n

    January 15.
    \n    Gui
    \nQuian Chen<\/a>, of     Northwestern University<\/a>.
    \nHyperbolic Conservation laws and Divergence-Measure Vector Fields.<\/b>
    \nIn this talk we first describe some connection between
    \nhyperbolic conservation laws and divergence-measure vector fields.
    \nThen we introduce a theory of divergence-measure vector fields,
    \nincluding the Gauss-Green formula and the normal traces, and discuss
    \nits applications to solving various nonlinear problems in partial
    \ndifferential equations.
    \n\n<\/p>\n

       <\/p>\n

       <\/p>\n

    \n

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