{"id":45,"date":"2016-01-06T13:35:53","date_gmt":"2016-01-06T18:35:53","guid":{"rendered":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/?page_id=45"},"modified":"2016-01-06T13:35:53","modified_gmt":"2016-01-06T18:35:53","slug":"colloguia-spring-2002","status":"publish","type":"page","link":"https:\/\/my.vanderbilt.edu\/pastcolloquia\/colloguia-spring-2002\/","title":{"rendered":"Colloguia, Spring 2002"},"content":{"rendered":"\n\n
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<\/font>Vanderbilt Mathematics<\/font><\/a><\/font>
\n<\/font>Colloquia<\/font>
\n<\/font>Spring 2002<\/font><\/h1>\n

<\/center>\n<\/td>\n<\/tr>\n<\/table>\n<\/td>\n<\/tr>\n<\/table>\n

<\/p>\n

Colloquia are listed in reverse chronological order. The
\ntop of the list is subject to change, since more colloquia
\nare still being planned.
\nAll colloquia are held at 4:10p.m.
\nin 1431 Stevenson Center unless otherwise noted.<\/p>\n

\nOur 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.
\nFor further information on activities in the department,
\nyou may also consult our
\nweekly
\ncalendar<\/a> and
\npast
\ncalendars<\/a>.<\/p>\n

\n

   
\nFriday, May 24th, 12:45p.m., SC 1320.<\/i>
\nA. M. W. Glass<\/b>, of the University of Cambridge.
\nCatalan’s Conjecture.<\/b>
\nIn 1844, Catalan asked if the only two consecutive whole numbers that
\nare proper powers are 8 & 9. In 1976, using the Fields Medal work of
\nAlan Baker, Tijdeman proved that the number of solutions to
\nxp<\/sup>-yq<\/sup><\/i>=1 was indeed finite, and subsequent
\nwork has put severe restrictions on their size. In 1996, using
\nStickleberger elements, Preda Mihailescu improved prior algebraic work
\nof Inkeri et al to show that two strong congruences must hold, whence
\ntranscendence theory gives that m<\/i>=min{p,q<\/i>} >
\n105<\/sup>
\nand M<\/i>=max{p,q<\/i>} < m2<\/sup> (so M<\/i>≠ 1
\n(mod m<\/i>)). By considering subgroups and quotients of groups of
\nunits in the cyclotomic field for Mth<\/sup><\/i> roots of unity,
\nMihailescu has a most ingeniously scheme to show that
\nM>m<\/i>2<\/sup>.
\nI will sketch the background and Mihailescu’s anticipated solution.
\n(Hosts: Matt Gould and Peter Jipsen.)<\/i><\/p>\n

   
\nThursday, April 25th.<\/i>
\nRaul Curto<\/b>, of the
\nUniversity of Iowa<\/a>.
\nTruncated moment problems: A survey of recent results.<\/b>
\nLet g<\/font>(2n)<\/sup>:
\ng<\/font>00<\/sub>,
\ng<\/font>01<\/sub>,
\ng<\/font>10<\/sub>,…,
\ng<\/font>0,2n<\/sub>,…,
\ng<\/font>2n,0<\/sub>
\nbe a given set of complex numbers, with
\ng<\/font>00<\/sub> > 0
\nand g<\/font>ji<\/sub>
\n=`<\/font>
\ng<\/font>ij<\/sub>
\nfor all i,j. The truncated complex moment problem entails
\nfinding necessary and sufficient conditions for the existence
\nof a positive Borel measure m<\/font>,
\nsupported in the complex plane, such that
\ng<\/font>ij<\/sub>=
\n\ufffd<\/font>
\n`<\/font>zi<\/sup>zj<\/sup>
\n  dm<\/font>  (0
\n\ufffd<\/font> i+j
\n\ufffd<\/font> 2n). We first
\ndescribe briefly some classical approaches to (full and
\ntruncated) moment problems, in one or several variables.
\nWe discuss the Hamburger, Stieltjes, Hausdorff,
\nand Toeplitz MP, and the work of Riesz, Haviland, Fuglede,
\nand others. We then present a new operator-theoretic approach,
\nbased on matrix positivity and extension, centered around
\nSmul’jan’s criterion for positivity of 2×2-operator
\nmatrices. In this approach, the structure of an associated
\nmoment matrix M(n) \ufffd<\/font>
\nM(n)[g<\/font>] plays a
\nfundamental role. For instance, when M(n) is flat
\n(meaning that rank M(n) = rank M(n
\n–<\/font>1)), then the truncated moment problem always admits
\na unique representing measure m<\/font>
\n[g<\/font>], which has precisely rank M(n)
\natoms. Our techniques allow for a concrete description of
\nthe support and densities of m<\/font>
\n[g<\/font>].  
\n—
\nThere is a close connection between the existence of
\nrepresenting measures supported in a prescribed algebraic
\nvariety and the presence of corresponding dependence relations
\nin the columns of the moment matrix M(n). If
\ng<\/font> \ufffd<\/font>
\ng<\/font>(2n)<\/sup> admits a
\nrepresenting measure m<\/font>,
\nthen M(n) is positive, recursively generated, and
\ncard V(g<\/font>)
\n\ufffd<\/font> rankM(n), where
\nV(g<\/font>) is the variety associated
\nto g<\/font>. We show how to solve the
\nmoment problem for Z`<\/font>Z=A1+BZ+C
\n`<\/font>Z+DZ2<\/sup>, D
\n\ufffd<\/font> 0, where positivity and
\nrecursiveness are not sufficient for a representing measure.
\n—
\nFor the quadratic MP (n=1) and singular quartic MP
\n(n=2 and detM(2)=0), a complete description of necessary
\nand sufficient conditions for the existence of representing
\nmeasures can be formulated concretely in terms of the
\ninitial data. For the singular quartic MP, we show that rank
\nM(2)-atomic measures exist in case the moment problem is
\nsubordinate to an ellipse, a parabola, or a non-degenerate
\nhyperbola, but the minimal measures for certain degenerate
\nhyperbola problems may require more than rank M(2) atoms.
\n Finally, we present applications to the classical
\nQuadrature Problem.
\n(Host: Dechao Zheng.)<\/i><\/p>\n

   
\nMonday, April 22th.<\/i>
\nGennadi Kasparov<\/b>, of the
\n
\nUniversité d’Aix Marseille II<\/a> and
\nVanderbilt University<\/a>.
\nOn the Baum-Connes conjecture.<\/b>
\nAn important object of study in representation theory of locally compact
\ngroups is the reduced C*-algebra of a group. It contains all information
\nabout the irreducible unitary representations weakly contained in the
\nregular representation.
\n
   
\nThe Baum-Connes conjecture, first stated about twenty years ago,
\nproposes a way to calculate the K-theory of the reduced C*-algebra of
\na group G by mainly topological methods. More precisely, the conjecture
\nasserts that this K-theory group is isomorphic to the K-homology of the
\nclassifying space for proper actions of G.
\n
   
\nBy now the conjecture has already been proved for large classes of
\ngroups: Lie groups, reductive p-adic groups, amenable groups, etc.
\nI will discuss in this talk the statement of the conjecture and also
\nexamples and methods of proof in some known cases of the conjecture.
\n(Host: Guoliang Yu.)<\/i><\/p>\n

   
\nFriday, April 19th.<\/i>
\nJerry Kaminker<\/b>, of the
\nIUPUI<\/a>.
\nNoncommutative geometry and solid state physics.<\/b>
\nNoncommutative geometry, as introduced by Alain Connes, has
\nfound several applications in physics. One of the most successful has
\nbeen in solid state physics. This has largely been due to the program
\ndeveloped by Jean Bellissard. Recently one of the main conjectures in
\nthe area, the “gap labeling conjecture”, was resolved by three different
\ngroups of workers. This talk will start with an introduction to some
\naspects of noncommutative geometry, explaining its motivation and basic
\napplications. The way it fits very well with the physics of solids will
\nthen be discussed and a sketch of the proof of the gap labeling
\nconjecture presented. The latter is joint work with Ian Putnam.
\n(Host: Guoliang Yu.)<\/i><\/p>\n

   
\nThursday, April 18th.<\/i>
\nM. Zuhair Nashed<\/b>, of the
\nUniversity of Delaware<\/a>.
\nVariational inequalities, nonsmooth calculus, and Newton-like
\nmethods for ill-posed problems: Un ménage à trois.<\/b>
\nNewton’s method is one of the most widely used algorithms for finding
\napproximate solutions of nonlinear operator equations F(x) = 0<\/i>.
\nThe method and the (Kantorovich) theory for its convergence require
\nthe existence and bounded invertibility of the Fréchet derivative
\nof the operator F<\/i>. The goal of this talk is to describe a theory
\nfor Newton-like methods when the derivative does not exist or when
\nthe derivative has no bounded inverse or bounded generalized inverse.
\nAlong the way we visit variational inequalities and discuss their
\nrole in minimization of nonsmooth functionals. We also introduce a
\nnew concept of “differentiability” for nonsmooth operators and use it
\nto formulate a new Newton-like method. Finally, we give applications
\nto bounded-variation regularization and nonsmooth ill-posed problems.
\n(Host: Akram Aldroubi.)<\/i><\/p>\n

   
\nTuesday, April 16th.<\/i>
\nYuri Muranov<\/b> of Vitebsk State Technological University,
\nBelarus.
\nSplitting of homotopy equivalence along submanifolds.<\/b>
\nLet X<\/i> be a submanifold of a manifold Y<\/i>. A homotopy
\nequivalence f:M –> Y<\/i> splits along the submanifold X<\/i>
\nif f<\/i> is homotopic to a map g<\/i> which is transversal to
\nX<\/i>, and the restrictions of g<\/i> to a transversal preimage
\nof X<\/i> and to its complement are homotopy equivalences. If a
\nmap f<\/i> is homotopic to a homeomorphism then, obviously,
\nthis map splits along any submanifold. The corresponding obstruction
\ngroups were introduced by Wall.
\nThe splitting methods are effectively applied for computation of maps
\nin surgery exact sequence and for solution of the oozing problem
\n(the problem of realizing elements of Wall groups by normal maps of
\nclosed manifolds). We describe geometrical and algebraic aspects of the
\nsplitting problem and relations of this problem with surgery theory.
\n(Host: John Ratcliffe.)<\/i><\/p>\n

   
\nTuesday, April 9th.<\/i>
\nStephen Smale<\/a><\/b>, of
\n
\nUniversity of California at Berkeley<\/a>.
\nEvolution of language.<\/b>
\n
\n   
\nA mathematical model is presented which helps to understand how
\nlanguages are formed. A theorem in this setting is the convergence
\nto a common language under a hypothesis on linguistic encounters.
\n(Host: Akram Aldroubi.)<\/i><\/p>\n

   
\nThursday, April 4th.<\/i>
\n
\nLaurent Pujo-Menjouet<\/a><\/b>, of
\nMcGill University<\/a>
\nand the Université de Pau<\/a>.
\nAsymptotic behavior of a singular transport equation modeling
\ncell division.<\/b>
\nThis paper analyses the behavior of the solutions of a model of cells
\nthat are capable of simultaneous proliferation and maturation.
\nThis model is described by a first-order singular partial differential
\nsystem with a retardation of the maturation and a time delay. Both
\ndelays are due to cell replication. We prove that uniqueness and
\nasymptotic behavior of solutions depend only on cells with small
\nmaturity (stem cells).
\n(Host: Glenn Webb.)<\/i><\/p>\n

   
\nThursday, March 21st.<\/i>
\nMartin Kochol<\/b>, of the Slovak Academy of Sciences, Bratislava,
\nand the Georgia Institute of
\nTechnology<\/a>.
\n

\nSuperposition — a method for constructing graphs without
\nnowhere-zero flows.<\/b>
\nNowhere-zero flow problems in graphs are dual to graph coloring
\nbecause, by Tutte, a planar graph is k-colorable iff its dual has a
\nnowhere-zero k-flow (its edges can be oriented and assigned values
\n1,…,k so that for every vertex, the sum of the values of the incoming
\nedges equals the sum of the outcoming ones). Graphs without
\nnowhere-zero k-flows are called k-snarks. In particular, snarks are
\nnontrivial cubic 4-snarks (by nontrivial we mean cyclically
\n4-edge-connected and with girth at least 5). Snarks present an
\nimportant family of graphs, because many conjectures about graphs can
\nbe reduced on them. Among the most interesting belong the 5-flow
\nconjecture of Tutte (every bridgeless graph has a nowhere-zero 5-flow)
\nand the cycle double cover conjecture (every bridgeless graph has a
\nfamily of circuits containing each edge twice).
\n

\n   
\nWe present a method for constructing graphs without nowhere-zero
\nk-flows. Using this method we obtain several results regarding
\nnowhere-zero flows. Primarily we construct new families of snarks. The
\nmost interesting is the construction of snarks with arbitrary large
\ngirth, which disproves a conjecture of Jaeger and Swart that every
\nsnark has girth at most 6. (Note that if this conjecture would be true,
\nit would imply the 5-flow and cycle double cover conjectures). We also
\npresent new results about the 3-flow conjecture of Tutte (every graph
\nwithout 1- and 3-edge cuts has a nowhere-zero 3-flow) and show that
\nthis is equivalent with seemingly stronger or weaker statements.
\n(Host: Mark Ellingham.)<\/i> <\/p>\n

   
\nMonday, March 18th.<\/i>
\n
\nDietmar Bisch<\/a><\/b>, of
\nthe
\n
\nUniversity of California, Santa Barbara<\/a>.
\nEntanglement – the spooky action at a distance.<\/b>
\n
\n   
\nEntanglement is a feature of quantum mechanics, which does
\nnot exist in classical physics. It expresses a correlation of two
\nsubsystems of a quantum physical system which appears naturally as
\nsoon as the commutative algebras of functions in classical physics
\nare replaced by non-commutative algebras of operators (matrices) in
\nquantum physics. Einstein called this phenomenum “spooky action at
\na distance”. Entanglement is believed to be related to what speeds
\nup a quantum computer and is currently the subject of intense study in
\nquantum information science.
\n   
\nI will discuss entanglement and show how it can be used to transmit
\nquantum information on a classical channel (“quantum teleportation”).
\nIf time permits I will present some of the proposals of how to quantify
\nentanglement, most of which have the flavor of entropy-like quantities
\nin operator algebras.
\n(Host: Guoliang Yu.)<\/i><\/p>\n

   
\nFriday, March 15th.<\/i>
\nAlexei Myasnikov<\/b> of CUNY, New York.
\nThe Andrews-Curtis conjecture and black box groups.<\/b>
\nIf G<\/i> is a group and V_k(G)<\/i> is the set of all
\nk<\/i>tuples of elements in G<\/i> which generate G<\/i>
\nas a normal subgroup, then the Andrews-Curtis graph
\nDelta_k(G)<\/i> of G<\/i> is the set V_k(G)<\/i> (as
\nthe set of its vertices) in which two elements are connected by
\nan edge iff one of them can be obtained from another by an
\nelementary transformation (Nielsen transformations and conjugation
\nof one of the components). These objects appear naturally in the
\nAndrews-Curtis conjecture in algebraic topology and in the theory
\nof black box groups in probabilistic group theory.
\n
   
\nIn my talk I am going to discuss some recent results on these
\nsubjects based on study of Andrews-Curtis graphs of various
\ngroups.
\n(Host: Mark Sapir.)<\/i><\/p>\n

   
\nThursday, March 14th. (Biomathematics Colloquium)<\/i>
\n
\nPrahlad Ram<\/b>, of the
\nMount Sinai School of Medicine<\/a>.
\nComputational analysis of a biological signaling network.<\/b>
\nSignaling networks receive and process information to control the
\nfunction of cellular machines. The MAP-kinase 1,2\/protein kinase C
\nsystem is one such network that regulates many cellular machines,
\nincluding the cell cycle machinery and autocrine\/paracrine factor
\nsynthesizing machinery. We used a combination of computational analysis
\nand experiments in NIH-3T3 fibroblasts to understand some of the design
\nprinciples of this controller network. We find that the growth factor
\nstimulated MAP-kinase 1,2\/protein kinase C network can operate as both a
\nmonostable as well as a bistable system. At low concentrations of
\nMAP-kinase phosphatase the system exhibits bistable behavior, such that
\nbrief stimulus results in sustained MAP-kinase activation. The
\nMAP-kinase induced increase in the levels of MAP-kinase phosphatase
\nmoves the network to a monostable state, where it behaves as
\nproportional response system responding acutely to stimulus, but
\nincapable of sustained responses. Thus the MAP-kinase1, 2\/protein
\nkinase C controller network is flexibly designed and MAP-kinase
\nphosphatase is the locus of flexibility.
\n(Host: Emmanuele DiBenedetto.)<\/i><\/p>\n

   
\nTuesday, March 12th.<\/i>
\n
\nLaurent Baratchart<\/a><\/b> of
\nINRIA Sophia-Antipolis<\/a>.
\n
   
\nSome extremal problems arising in frequency identification and
\ndeconvolution for recovering Hardy functions from incomplete
\nboundary values.<\/b>
\nWe consider the problem of L2<\/sub><\/i> or
\nL¥<\/font><\/sup><\/i>
\napproximating a function on a subarc of the unit circle (or on a
\nsubinterval of the imaginary axis) by the trace of a Hardy function
\nsatisfying pointwise or norm constraints on the remaining of the
\ncircle (or the imaginary axis). This generalization of Carleman’s
\nrecovery problem exhibits connections with Hankel and Toeplitz
\noperators, whose spectral theory allows one to derive error
\nestimates as well as explicit computational
\nschemes. These can be used in several practical situations of
\nengineering science.
\n(Host: Ed Saff.)<\/i><\/p>\n

   
\nThursday, March 7th.<\/i>
\nJoachim Cuntz<\/b> of the
\nMathematisches Institut, Universitaet Muenster.
\n
   
\nK-homology for the Heisenberg commutation relations.<\/b>
\nA very prominent quantum space is represented by the so called
\nWeyl algebra which is generated by two elements satisfying the
\nHeisenberg commutation relations. Until recently it was not
\nknown how to define and compute some of the standard invariants
\nof noncommutative geometry for this space. We describe a new
\ntheory which does exactly that.
\n(Host: Guoliang Yu.)<\/i><\/p>\n

   
\nFriday, March 1st,
\n3:10-4:00p.m.,
\n1307 Stevenson Center<\/blink><\/font><\/i>
\nKonstantin Rybnikov<\/b> of
\nCornell University.
\n
   
\nGain graphs and their applications to convex polyhedra and splines.<\/b>
\nA gain graph (G,h,H)<\/i> is a homomorphism h<\/i> from the free
\ngroup on the edges of a graph G<\/i> to some group H<\/i>; it
\nis called balanced if all closed walks of G<\/i> lie in
\nthe kernel of h<\/i>. I’ll explain a test, formulated in terms
\nof the binary cycle space of G<\/i>, that can be used to
\ndetect if (G,h,H)<\/i> is balanced for some choices of H<\/i>,
\ne.g. the abelian case.
\n
   
\nI am going to show a few examples of gain graphs
\narising from discrete geometry. In the 1860s Maxwell
\ndescribed a relationship between equilibrium stresses
\nin a plane framework and polyhedral surfaces projected
\non this framework. In the most simple case, Maxwell’s
\ncorrespondence can also be interpreted in terms of
\nlifting a tiling of the plane to a spatial surface.
\nLifting a tiling of Rd<\/sup><\/i> to a convex surface, tangent
\nto a paraboloid, appears to be a powerful technique
\nin geometry of numbers (Voronoi, 1908) and
\ncomputational geometry (Brown 1978, Edelsbrunner
\n1986).
\n
   
\nI’ll present various criteria for a tiling of Rd<\/sup><\/i> or,
\nmore generally, a PL-manifold in Rd<\/sup><\/i>, to be the vertical
\nprojection of a convex d-surface. These criteria lead
\nto improvements of algorithms determining whether a
\ngiven tiling can be regarded as the projection of a PL-surface.
\nGain graphs can also be used to obtain some topological
\nresults on the dimension of the space of splines over
\na non-simplicial tiling of a domain in Rd<\/sup><\/i>.
\n
   
\nSome of the discussed results are
\njoint with S. Ryshkov and T. Zaslavsky.
\n(Host: Paul Edelman.)<\/i><\/p>\n

   
\nThursday, February 28th.<\/i>
\nZlil Sela<\/a><\/b>, of the
\n
\nHebrew University of Jerusalem<\/a>.
\nDiophantine geometry over groups and the elementary theory of
\na free group.<\/b>
\nWe study sets of solutions to equations over a free group,
\nprojections of such sets, and the structure of elementary sets defined
\nover a free group. The structure theory we obtain enable us to answer
\nsome questions of A. Tarski’s, and classify those finitely generated
\ngroups that are elementary equivalent to a free group. Connections
\nwith low dimensional topology, and a generalization to general
\nhyperbolic groups and their elementary classification will also be
\ndiscussed.
\n(Host: Mark Sapir.)<\/i><\/p>\n

   
\nThursday, February 21st.<\/i>
\n
\nMichael E. Adams<\/a><\/b>, of
\n
\nSUNY-New Paltz<\/a>.
\nUniversal varieties of algebras.<\/b>
\n
   
\nFor a variety of algebras (equational class), the notions
\nof universal in the categorical sense and universal in
\nthe quasivariety sense will be considered.
\n
   
\nThe two notions will be compared and their relationship
\nillustrated by different examples, including varieties of bounded
\nlattices.
\n(Host: Matthew Gould.)<\/i><\/p>\n

   
\nWednesday, February 20th.<\/i>
\n
\nAmos Ron<\/a><\/b>, of the
\n
\nUniversity of Wisconsin, Madison<\/a>.
\nWavelet frames: The power of redundant representation.<\/b>
\nOne of the hallmarks of the IT revolution is the rapid increase in
\nconnectivity and data acquisition capabilities. Questions concerning the
\neffective processing of this avalanche of data is becoming a top
\nnational priority, fueled even further by the recent increase in
\nmilitary and security needs.
\n
   
\nWavelets are widely considered to be among the most successful
\ncontributions of the mathematical community to the theory and
\napplications of data processing. Most of the progress during the 1990s
\nwas confined to non-redundant wavelet systems, primarily because their
\ntheory was developed first.
\n
   
\nIn the last 5-6 years, a theory for wavelet frames (which are a major
\ntype of redundant wavelet systems) was established. The theory leads to
\neffective constructions of finely-tuned wavelet frames, together with
\nfast implementation algorithms, opening thereby the door to a range of
\npossible applications.
\n
   
\nThe talk will be devoted to a review of this exciting development.
\nAfter highlighting the main ingredients of the theory, I will show
\nexamples of an on-going research on applications, and will conclude
\nwith a demo of the Wavelet IDR Framenet<\/i>, a web-based
\ninteractive software, currently under development, too.
\n(Host: Larry Schumaker.)<\/i><\/p>\n

   
\nTuesday, February 19th.<\/i>
\n
\nLászló Lipták<\/a><\/b>, of the
\n
\nUniversity of Waterloo<\/a>.
\nStable set problem and the lift-and-project ranks of graphs.<\/b>
\n
\n   
\nWe study the lift-and-project procedures for solving combinatorial
\noptimization problems, as described by Lovász and Schrijver, in the
\ncontext of the stable set problem on graphs. We investigate how the
\nprocedures’ performance changes as we apply fundamental graph
\noperations. We give examples showing that adding, deleting, or
\nsubdividing an edge can increase the N0<\/sub><\/i>– and
\nN<\/i>-rank of a graph, and define two classes of
\ngraphs when these and the subdivision of a star operation does not
\nincrease the N0<\/sub><\/i>– and N<\/i>-rank of the
\nunderlying graph. We present a graph-minor based characterizations
\nof the rank of subdivisions of the complete graph Kn<\/sub><\/i>,
\nand define a class of graphs with large rank that
\ncan be obtained from Kn<\/sub><\/i> using just the
\nstretching of a vertex operation. Finally, we provide improved bounds
\nfor the N+<\/sub><\/i>-rank of graphs in terms of the number of
\nvertices in the graph and prove that the
\nsubdivision of an edge or cloning a vertex operations can increase the
\nN+<\/sub><\/i>-rank of a graph.
\n
\n   
\nThis is joint work with Levent Tuncel.
\n(Host: Paul Edelman.)<\/i><\/p>\n

   
\nThursday, February 14th,
\n3:10-4:00p.m.,
\n1312 Stevenson Center<\/blink><\/font><\/i>
\n
\nDietmar Bisch<\/a><\/b>, of the
\n
\nUniversity of California, Santa Barbara<\/a>.
\nSubfactors and symmetry.<\/b>
\n
\n   
\nJohn von Neumann discovered in the 30’s that certain algebras
\nof bounded operators on a Hilbert space are the natural algebras of
\nsymmetries of quantum physical systems. These
\nvon Neumann algebras<\/i>
\nas they are now called can be viewed as non-commutative measure spaces
\nwhich feature many astonishing mathematical structures and have led to
\nrich theories, largely due to Connes, Jones and Voiculescu.
\n
\n   
\nVaughan Jones initiated in the early 80’s the
\ntheory of subfactors<\/i>,
\na theory which deals with certain highly non-commutative, infinite
\ndimensional probability spaces. These subfactors turn out to display
\na surprising rigidity phenomenon, which ultimately led Jones to the
\ndiscovery of his famous knot invariant, the Jones polynomial. A
\nsubfactor is a functional analytical object that captures what one
\nmight call the
\ngeneralized symmetries<\/i> of the mathematical or physical situation
\nfrom which it was constructed. Analytical techniques can then be used
\nto decode this information and to compute with it. Surprising connections
\nto statistical mechanics and knot theory appear naturally.
\n
\n   
\nI will present in my talk some of the basic ideas and concepts in
\nsubfactor theory and will discuss some applications if time allows.
\nNo prior knowledge of operator algebras is required for this talk.
\n(Host: Glenn Webb.)<\/i><\/p>\n

   
\nMonday, February 11th,
\n2:10-3:00p.m.,
\n1214 Stevenson Center<\/blink><\/font><\/i>
\n
\n
\nFlorian Pfender<\/a><\/b>, of
\n
\nEmory University<\/a>.
\nPancyclicity of 3-connected graphs: Pairs of forbidden subgraphs.<\/b>
\n
\n   
\nWe say that G<\/i> is {H1<\/sub>,…Hl<\/sub>}<\/i>-free,
\nif it contains no induced copies of any of the graphs
\nH1<\/sub>,…Hl<\/sub><\/i>. The problem of characterizing all
\nfamilies of H1<\/sub>,…Hl<\/sub><\/i> such that each “sufficiently
\nconnected” {H1<\/sub>,…Hl<\/sub>}<\/i>-free graph has some
\nHamiltonian property has been studied by a number of authors.
\nIn particular, the family of all pairs of graphs X, Y,<\/i>
\nsuch that each 2-connected {X,Y}<\/i>-free graph G\\neq Cn<\/sub><\/i>
\non n\\geq 10 vertices is pancyclic, has been characterized by Faudree
\nand Gould. In this talk, I will characterize all graphs
\nX, Y,<\/i> such that each 3-connected {X, Y}<\/i>-free
\ngraph is pancyclic.
\n
\n   
\nThis is joint work with R. Gould and T. Luczak.
\n(Host: Mark Ellingham.)<\/i><\/p>\n

   
\nThursday, February 7th.<\/i>
\n
\nGil Strang<\/a><\/b>, of the
\n
\nMassachusetts Institute of Technology<\/a>.
\nFiltering and signal processing.<\/b>
\n
\n   
\nWe discuss two filters that are frequently used to smooth data.
\nOne is the (nonlinear) median filter, that chooses the median
\nof the sample values in the sliding window. This deals effectively
\nwith “outliers” that are beyond the correct sample range, and will
\nnever be chosen as the median. A straightforward implementation of
\nthe filter is expensive, particularly in two dimensions (for images).
\n
   
\nThe second filter is linear, and known as “Savitzky-Golay”. It is
\nfrequently used in spectroscopy, to locate positions and peaks and
\nwidths of spectral lines. This filter is based on a least-squares fit
\nof the samples in the sliding window to a polynomial of relatively
\nlow degree. The filter coefficients are unlike the equiripple filter
\nthat is optimal in the maximum norm, and the “maxflat” filters that
\nare central in wavelet constructions.
\n
   
\nWe will discuss the analysis and the implementation of both filters.
\n(Host: Doug Hardin.)<\/i><\/p>\n

   
\nThursday, January 31st.<\/i>
\n
\n
\nGui-Qiang Chen<\/a><\/b>, of
\nNorthwestern University<\/a>.
\nOn nonlinear degenerate parabolic-hyperbolic equations.<\/b>
\n
   
\nIn this talk we will discuss a well-posedness theory for solutions
\nin L1<\/sup><\/i> to the Cauchy problem of general degenerate
\nparabolic-hyperbolic equations with non-isotropic nonlinearity. A
\nnotion of kinetic solutions and a corresponding kinetic formulation
\nwill be introduced. The notion of kinetic solutions applies to more
\ngeneral situations than that of entropy solutions; and its advantage
\nis that the kinetic equations in the kinetic formulation are well
\ndefined even when the macroscopic fluxes are not locally integrable,
\nso that L1<\/sup><\/i> is a natural space on which the kinetic
\nsolutions are posed. It includes a new ingredient, a chain rule type
\ncondition, which makes it different from the isotropic case. Based
\non this notion, we will present an effective approach to prove the
\ncontraction property of kinetic solutions in L1<\/sup><\/i>,
\nespecially including entropy solutions, among others.
\n(Host: Glenn Webb.)<\/i><\/p>\n

   
\nWednesday, January 23rd.<\/i>
\nEric Schechter<\/b>, of
\n
\nVanderbilt University<\/a>.
\nNonclassical logics for undergraduates.<\/b>
\n
\n   
\n“If it is raining right now then the moon is round.” That’s logical
\nto a mathematician, but nonsense to everyone else. Classical logic,
\nwhen presented by itself, lacks the relevance, causality,
\nconstructivism, quantitativeness and other features that people
\n(even mathematicians!) are accustomed to using in their everyday
\nnonmathematical reasoning. Also, a single example (i.e., classical
\nlogic) does not provide adequate illustrations for an abstract idea
\n(e.g., completeness). These may be some of the reasons that the
\ntraditional classical-only approach has fared badly in our
\nundergraduate introduction to mathematical logic. Consequently, I am
\nexperimenting with a new curriculum that introduces several nonclassical
\nlogics alongside the classical.
\n
\n   
\nThe math in this talk is not mine, nor is it new — it can be found in
\nresearch-level monographs and journals from the last few decades. What
\nis new is the attempt to convey this material at a much more elementary
\nlevel.
\n
\n   
\nThis talk is intended not just for logicians, but for general
\nmathematicians (including graduate students), plus a few folks from the
\nphilosophy department who have expressed interest. Basics of logic
\nare sketched in the talk, so they’re not a prerequisite for attending.
\nThe nonclassical examples may be amusing if you haven’t seen them before.<\/p>\n

   
\nMonday, January 21st.<\/i>
\n
\nVilmos Totik<\/a><\/b>, of the
\nUniversity of Szeged<\/a> and the
\nUniversity of South Florida<\/a>.
\n
\nPolynomial inverse images and how to transfer results from one
\ninterval to more general sets.<\/b>
\n
\n   
\nI proposed the following problem on the 1991 Schweitzer contest:
\nTo divide an inheritance, n brothers hire an impartial judge.
\nSecretly however, each brother bribes the judge. The value of the
\ninheritance that a given brother gets strictly (and continuously)
\nincreases in his own bribe and strictly decreases in everybody
\nelses bribe. Show that if the eldest brother does not give too
\nmuch to the judge, then the others can give so that the decision
\nwill be fair.
\n
\n   
\nIn the talk a few other systems with similar characteristics will be
\nmentioned, and some basic properties of such monotone systems will
\nbe discussed. In particular, equilibrium measures on sets of
\nfinitely many intervals form such systems. A small modification of
\nthe problem, in which each brother gets
\na rational fraction of the inheritance no matter what the
\ninitial judgement of the judge is, turns out to be the same problem
\n– when translated in the language of equilibrium measures –
\nas the density of polynomial inverse images of intervals among
\nsets consisting of finitely many intervals. This was proved in order
\nto transfer some polynomial inequalities from one interval to general
\ncompact sets on R. In the talk some further applications of the density
\ntheorem will be described that are of similar nature, namely they are
\nthe extensions of results on compact sets that
\nhave been known only on intervals.
\n(Host: Ed Saff.)<\/i><\/p>\n

\n

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