Updated on Apr. 18, 2008
Robert Tuzun
University at Buffalo, State University of New York / SUNY BrockportComputational Search for Nontrivial Knots with Unit Jones Polynomial
Abstract: A famous and still unsolved conjecture in knot theory states that there are no non-trivial knots in S2 with unit Jones polynomial. A computational search for such a knot is underway using a technique similar to one previously used by Yamada. Much of the work in this talk focuses on dealing with the combinatorial explosion in computational effort with the number of crossings. The intricacies of enumerating knots will be discussed in some detail. Reductions in computational effort are accomplished in part by eliminating unnecessary cases: by exploiting algebraic symmetry in expressions for the Kauffman bracket, by not considering connected sums of knots, and by considering knots composed only of "trivializable" algebraic tangles. Other reductions in computational effort, and in some cases of memory usage, may be accomplished by reducing computational effort for each case: by computing Kauffman brackets rather than Jones polynomials, by using simple strategies such as Horner's rule, and by evaluating Kauffman brackets at a specific value of independent variable, thereby enabling the replacement of computations involving Laurent polynomials with much simpler floating point, but still exact, computations.
Amy Evans and Tracy Stepien
University at Buffalo, State University of New York
The presenters, currently seniors in the Mathematics Department, have just been announced as winners of the international Mathematical Contest in Modeling held in February 2008. The problem they tackled was to consider the effects on land from the melting of the North Polar Ice Cap due to the predicted increase in global temperatures. They will present their prize-winning report, and provide advice and encouragement to students considering participating in the Contest next year.
Dr. Alexander Gorokhovsky
University of Colorado, BoulderHeat equation, index theory and noncommutative geometry
Abstract: One of the approaches to Atiyah-Singer index theorems is based on the study of behavior of heat kernels. I will describe how this approach allows to prove index theorem for a single operator and for a family of operators. Then I will discuss how heat equation methods can be applied to compute Chern character of groupoid-equivariant operators in KK-theory and to give a (new) proof of A. Connes' index theorem for foliations.
Dr. Thomas Schroeder
Department of Learning and Instruction
University at Buffalo, State University of New YorkThe UB Master’s/Teacher Certification Program in Mathematics Education
All undergraduates interested in math teaching as a career are invited to attend. A presentation will be followed by a question and answer session.
Greg Schneider
University at Buffalo, State University of New YorkA Combinatorial Survey of Knot Floer Homology
Abstract: Knot Floer homology, developed by Ozsvath and Szabo, is a categorification of the Alexander polynomial of a knot. Their construction was found to have a connection with grid diagrams of knots, which in turn led to a purely combinatorial description of this homology. We will begin with a brief introduction to grid diagrams and the combinatorial construction of knot Floer homology, and then proceed to a survey of some of the recent developments to this theory, focusing primarily on results which can be proven entirely within the context of grid diagrams.
Dr. Snigdhayan Mahanta
University of TorontoIntroduction to homotopical algebraic geometry (after Toen-Vezzosi)
Abstract: Typically, a space (or a scheme) in algebraic geometry is locally defined by a commutative ring, which should be viewed as its ring of local functions. However, the local model of functions can be generalizedto some mathematical structure which is "ring-like". A convenient choice of this ring-like structure is a differential graded algebra. Based on this observation recently Toen-Vezzosi developed a framework of geometry which they call homotopical algebraic geometry (HAG). This setting is general enough to subsume both classical algebraic geometry and certain aspects of noncommutative geometry a la Connes. In this talk I shall discuss some of the features of HAG more elaborately.
Ms. Tracy Stepien
University at Buffalo, State University of New YorkTubuloglomerular Feedback - Mediated Dynamics in Three Coupled Nephrons
Abstract: A model of three coupled nephrons branching from a common cortical radial artery is developed to further understand the effects of equal and unequal coupling on tubuloglomerular feedback. The integral model of Pitman et al. (2002), which describes the fluid flow up the thick ascending limb of a single, short-looped nephron of the mammalian kidney, is extended to a system of three nephrons through a model of coupling proposed by Pitman et al. (2004). Analysis of the system, verified by numerical results, indicates that stable limit-cycle oscillations emerge for sufficiently large feedback gain magnitude and time delay through a Hopf bifurcation, similar to the single nephron model, yet generally at lower values. Previous work has demonstrated that coupling induces oscillations at lower values of gain, relative to uncoupled nephrons. The current analysis extends this earlier finding by showing that asymmetric coupling among nephrons further increases the likelihood of the model nephron system being in an oscillatory state.
Dr. Eduardo Martinez-Pedroza
Oklahoma UniversityAmalgamation of Quasiconvex Subgroups in Relatively Hyperbolic Groups
Abstract: The relatively hyperbolic groups, introduced by Gromov, generalize the class of fundamental groups of complete finite volume hyperbolic manifolds. When considering a relatively hyperbolic group as a geometric object, the quasiconvex subgroups are the natural subgroups to consider. In this talk, we will discuss combination theorems for quasiconvex subgroups of relatively hyperbolic groups.
Dr. Tobias Schafer
CUNY, Staten IslandCoarse-graining of noise in nonlinear systems with scale-separation
Abstract: I will discuss three methods to coarse-grain small noise in weakly nonlinear systems with scale-separation. The first method is based of a method of multiple scales on the level of the stochastic equation, the second method employs an asymptotic expansion of the associated Fokker-Planck equation and the third method is based on a hierachy of path integrals. Examples from optics and fluid dynamics will illustratethe application of the discussed methods to concrete problems.
Dr. Avner Peleg
University at Buffalo, State University of New YorkEnergy exchange in fast soliton collisions as a random cascade model - Part II
Abstract: We study the effects of delayed Raman response on a probe soliton propagating in an optical fiber and undergoing fast collisions with a random sequence of pump solitons. In a recent paper we showed that the probe soliton exhibits intermittent dynamics in the sense that the normalized moments of its parameters grow exponentially with propagation distance [1]. This is a surprising result since optical fiber systems are typically weakly nonlinear, whereas intermittency is usually associated with strongly nonlinear phenomena such as turbulence and chaotic dynamics. Here we show that this similarity in behavior is not coincidental, but rather a consequence of the fact that the equation for the dynamics of the probe soliton’s amplitude has the same form as the equation for the local space average of energy dissipation in random cascade models employed in turbulence theory. Furthermore, the statistics of the probe soliton’s amplitude can be characterized by means of the τq exponents, which are used in the analysis of multifractal sets. We find that the error probability (BER) and the soliton’s frequency shift exhibit power-law behavior with propagation distance, where the exponents can be characterized in terms of τq . We show that similar behavior is expected in systems where the collision-induced energy exchange is due to nonlinear loss/gain (instead of delayed Raman response).
[1] A. Peleg, Phys. Lett. A 360, 533 (2007).
Dr. Terry Bisson
Canisius CollegeA homotopical algebra of graphs related to zeta series
Abstract: Quillen's axioms for homotopical algebra generalize some of the standard notions and constructions from algebraic topology to other categories. Methods of homotopy theory in the category of simplicial sets provided a major motivating example.It should be possible to illustrate Quillen's ideas, at a simple combinatorial level, in various categories of graphs related to the category of simplicial sets.
I will sketch a Quillen model structure on a category of directed graphs, and examine the resulting homotopical algebra. It seems to fit well with traditional concepts of trees, cycles, and coverings in graph theory, and zeta functions and spectra of graphs from algebraic graph theory. (Joint work with Aristide Tsemo (preprint on the ArXiv))
Cynthia Cornelius
University at Buffalo Center for Computational Research, State University of New York,Introduction to Running Applications at CCR
Abstract: This seminar serves as an introduction to running programs on CCR’s U2 computer cluster. The U2 cluster is the largest computational platform in UB’s Center for Computational Research, with over 2000 processors. Users run on U2 by submitting jobs through a batch scheduler. A brief overview of batch computing will be presented. Topics include the creation and submission of jobs, as well as analysis of queue and job status. Examples and demonstrations will be given as time permits.
Dr. Kelly Delp
Buffalo State College, State University of New YorkConvex projective structures
Dr. E. Bruce Pitman
University at Buffalo, State University of New YorkA New Approach to Volcanic Hazard Mapping using HPC and Statistics
Dr. Yi-Jun Yao
Vanderbilt UniversityOn Rankin-Cohen Deformations
Abstract: Arise from the study of deformation question related to modular forms, Rankin-Cohen brackets are bidifferential operators which produce new modular forms from two “old” ones. They entered the domain of noncommutative geometry via some work of Connes-Moscovici in which some Hopf algebra structure hidden behind was revealed. We will discuss some results both on the modular form side and the Hopf algebra side.
The University at Buffalo will host a three day conference featuring talks on recent results in knot theory, hyperbolic geometry and geometric group theory.
The conference is sponsored by the Department of Mathematics and the College of Arts and Sciences, State University of New York at Buffalo.
Speakers:
Ian Agol, University of California, Berkeley
Joan Birman, Barnard College, Columbia University
Steven Boyer, University of Quebec at Montreal, Canada
Daryl Cooper, University of California, Santa Barbara
Daniel Groves, University of Illinois at Chicago
Thang Le, Georgia Institute of Technology, Atlanta
Tao Li, Boston College
Dr. Benson Farb
University of Chicago
“Hidden symmetry”