Un of Tenn | Math Department

Seminars and Colloquiums
for the week of February 15, 2010

Speaker:

Professor Jim Conant, Monday
Professor Vladimir Vysotsky, University of Delaware, Tuesday
Mr. Chad Kilpatrick, Wednesday
Professor Jerzy Dydak, Friday
Ms. Lei Wang, University of Michigan, Friday

Monday, February 15

TOPOLOGY SEMINAR
TIME: 11:15 – 12:05
ROOM: Temple 303
SPEAKER: Professor Jim Conant
TITLE: “Khovanov Homology III”
ABSTRACT: In this talk, we will discuss Khovanov's fascinating generalization of the Jones polynomial. This theory assigns a combinatorially defined chain complex to every link diagram, in such a way that the homology of this chain complex is a link invariant. Moreover, the graded Euler characteristic is nothing other than the Jones polynomial. I will try to make this talk accessible to newcomers.

Tuesday, February 16

COLLOQUIUM
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKER: Professor Vladimir Vysotsky, University of Delaware
TITLE: “Sticky particles, limit theorems, and random walks”
ABSTRACT: Consider the model of a one-dimensional gas, whose particles have random initial positions and random initial velocities. Particles attract each other because of gravity, and stick together at collisions. As time goes, the number of particles decreases while their sizes increase until there forms a giant single particle of the total mass.

The problem is to get a suitable probabilistic description of this process of mass aggregation. Our results are given in the form of limit theorems as the number of initial particles tends to infinity. For example, we prove that the stochastic processes of the total number of particles satisfy a functional version of the central limit theorem. Then we show how this problem of the number of particles brought us to the study of some important general properties of random walks. We present the problem of finding small deviation probabilities of partial sums of a random walk, and show our progress towards solution of this open question.

Wednesday, February 17

COURSE GEOMETRY SEMINAR
TIME: 11:15 – 12:05
ROOM: Temple 303
SPEAKER: Mr. Chad Kilpatrick
TITLE: “Property A is Equivalent to Nuclearity and Exactness for Countable
Discrete Groups G” II”
ABSTRACT: In this talk, we will see that for a countable discrete group G, property A is equivalent to:
1.    Cr*(G) is exact
2.    Cu*(G) is nuclear
3.    Cu*(G) is exact.
Here, a C*-algebra A is defined to be nuclear if for any finite subset F of A, the identity map may be approximated on F by unital completely positive maps which factorize through Mn(C), for some n. In this talk, I closely follow Ch. 4: "Connections with C*-Algebras” from Rufus Willet’s Some Notes on Property A.

COLLOQUIUM
TIME: 3:35 – 4:25 P.M.
ROOM: HBB 102
SPEAKER: Professor Tuoc Van Phan, Mathematics Department, The University of British Columbia
TITLE: “Small Solutions of Nonlinear Schrödinger Equations with Many Bound States”
ABSTRACT: Consider the nonlinear Schrödinger equation i@t = H0 + _j j2 , where : R3 _ [0;1) ! C, H0 = ??_ + V and _ = _1. Here _ denotes the three dimensional Laplacian and V = V (x) with x 2 R3 is a real smooth potential which decays sufficiently fast as jxj ! 1. The linear Hamiltonian H0 is assumed to have three or more eigenvalues satisfying some resonance conditions. In this talk, I present some recent results on the asymptotic behavior at time infinity of solutions with small initial data in H1(R3)\L1(R3). The results include the case that all of the eigenvalues of H0 are simple as well as the case where there is a degeneracy. These results are joint works with Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai.

Friday, February 19

TOPOLOGY SEMINAR
TIME: 11:15 – 12:05
ROOM: Temple 303
SPEAKER: Professor Jerzy Dydak
TITLE: “Introduction to CAT(0) spaces II”

COLLOQUIUM
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKER: Ms. Lei Wang, University of Michigan
TITLE: “A Cartesian Treecode for Radial Basis Functions and Applications of Particle Methods to Fluid Flow on a Sphere”
ABSTRACT: Particle methods are Lagrangian techniques that have been proposed as an alternative to more conventional methods, such as finite difference, finite element and spectral methods. In this talk, we’ll present a treecode algorithm for evaluating the multiquadric radial basis function (RBF) approximation, which is one type of particle method. For a given level of accuracy, the treecode reduces the operation count from $O(N2)$ to $O(N\log N)$ and has the memory usage $O(N)$ , where $N$ is the number of nodes in the system. We implement the treecode in Cartesian coordinates and use a far field Taylor expansion of the multiquadric. The Taylor coefficients are computed by a recurrence relation. I'll also mention work in progress on solving the Barotropic Vorticity Equation (BVE) on a rotating sphere by vortex particle methods, including different meshes on the sphere and an adaptive mesh refinement strategy.

If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.

If you have questions, or a date you would like to confirm, please contact Dr. Steve Wise.

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