Distinguished Visitor Lecture Series Archives
Academic Year 2005-2006
| February 27, 28, March 1, 2, 2005 | |
| Speaker | Pere Ara, Universitat Aut'omoma de Barcelona, Spain |
| Titles | Leavitt Path Algebras and Related Rings I, II, III, IV |
| Host | Victor Camillo |
| Abstracts | I: Introduction. The separativity problem for exchange rings, von Neumann regular rings and C*-algebras of real rank zero.The realization problem for von Neumann regular rings. II: Algebras associated to a quiver. The Leavitt path algebra and the von Neumann regular envelope(I) III: Algebras associated to a quiver. The Leavitt path algebra and the von Neumann regular envelope(II) IV: Modules over the Leavitt algebra. Algebraic K-theory. |
| March 23-24, 2005 | |
| Speaker | Ken Ono, University of Wisconsin |
| Titles | Modular Forms, Infinite Products and Singular Moduli |
| Host | Department of Mathematics |
| Abstracts | Modular forms play many roles in mathematics. In numbertheory, modular forms often arise as generating functions for interestingquantities such as representation numbers of integers by quadratic forms,partition functions, values of L-functions, and also degrees of characters ofsporadic simple groups like the Monster. In his 1994 ICM lecture, Borcherdsfound a striking new phenomenon. He proved that certain modular forms ofhalf-integral weight serve as generating functions for the infinite productexponents of other modular forms, thereby greatly generalizing some of theprettiest q-series dating back to work of Euler and Jacobi on classical thetafunctions. His work pertained to an exceptionally rich family of modular forms,those with a `Heegner divisor'. Zagier later found a beautiful number theoreticexplanation for Borcherds' phenomenon, one involving singular moduli,complex multiplication, and elliptic curves. In this lecture we provide ageneral framework which includes Zagier's reformulation of Borcherds' theory asa special case. We show that all of these results follow from beautifulproperties of a delightfully rich sequence of modular forms, the weakMaass-Poincare series of half-integral weight. |
| May 5, 8, 2005 | |
| Speaker | Andrzej Stasiak, University de Lausanne |
| Titles | Natural Classification of Knots; Scaling Behavior, Equilibrium Lengths and Probabilities of Knotted Polymers |
| Host | David Manderscheid |
| Abstracts | Natural Classification of Knots: (joint work with Alessandro Flammini, School ofInformatics, Indiana University).The principal objective of knot theory is toprovide a simple way of classifying and orderingall the knot types. The talk will present anatural classification of knots based on theirintrinsic position in the knot space that isdefined by the set of knots to which a given knotcan be converted by individual intersegmentalpassages. In addition various knots will becharacterized using a set of simple quantumnumbers that can be determined upon inspection ofminimal crossing diagram of a knot. These numbersinclude: crossing number, average 3-D writhe,number of topological domains and the averagerelaxation value. Scaling Behavior, Equilibrium Lengths and Probabilities of Knotted Polymers: (joint work with: Akos Dobay, Ludwig-Maximilians-Universitat, Munich, Germany,Kenneth C. Millett, University of California, Santa Barbara, USA,Michael Piatek, University of Washington, Seattle, USA andEric Rawdon, DuquesneUniversity, Pittsburgh, USA)Previous works on radius of gyration and averagecrossing number have demonstrated that polymerswith fixed topology show a different scalingbehavior with respect to these characteristicsthan polymers with unrestricted topology. Thetalk will discuss the results showing that thedifference in the scaling behavior betweenpolymers with restricted and unrestrictedtopology also applies to various other spatialcharacteristics such as total curvature and totaltorsion. In the finite-length range, polymersforming different knot types show distinctscaling profiles with respect to the giveninvestigated characteristics. For each knot type,the equilibrium length with respect to a givenspatial characteristic is the number of edges atwhich the value of the characteristic is the sameas the average for all polygons. This numberappears to be correlated to physical propertiesof macromolecules. The extent to which theequilibrium length is universal for all spatialcharacteristics versus being associated with agiven characteristic or class of characteristics,will be discussed. |
