DEPARTMENT OF MATHEMATICS
University at Buffalo
State University of New York

coffee at 3:30 p.m in the Common Room, Rm. 240 Math Bldg.

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2009 - 2010 Seminars and Colloquia

Friday, November 20

Seminar - GEOMETRY/TOPOLOGY - 4:00 p.m., 122 Mathematics Building

Dr. Daniel Groves
University of Illinois at Chicago

Parametrizing surface bundles

Abstract: Let S be an orientable surface of finite type (not a torus or a sphere) and B a reasonable space (CW complex or manifold). The the set of S-bundles over B is naturally parametrised by the set Hom(π₁(B), Mod(S))/~ of conjugacy classes of homomorphisms from the fundamental group of B to the mapping class group of S.

I will discuss some recent work (still being written) which provides a description of this set of homomorphisms whenever π₁(B) is finitely generated.

Thursday, November 19

Colloquium - 4:00 p.m., 250 Mathematics Building

Dr. Bernard Deconinck
University of Washington

The stability of periodic finite-gap solutions of the KdV equation

Abstract: The periodic finite-gap solutions of the KdV equation are the largest class of periodic solutions of the KdV equation whose functional form is known explicitly. The one-gap solutions are stationary, and their stability has been a topic of much discussion in their literature. Recently, it was settled that they are (nonlinearly) orbitally stable with respect to perturbations that are periodic with periodic equal to an integer multiple of the one-gap solutions period. In this talk, I will show how one proceeds to show that this orbital stability results holds for all periodic finite-gap solutions.

Wednesday, November 18

Seminar - ANALYSIS - 4:00 p.m., 250 Mathematics Building

Dr. Byung-Jay Kahng
Canisius College

Some remarks on duality in the locally compact quantum group setting

Abstract: In abstract harmonic analysis, among the most important result is the Pontryagin duality, wheich holds at the level of locally compact abelian (LCA) groups. Also, at the LCA group level, the notion of Fourier transform is defined. For further generalization, we consider the category of quantum groups, where Pontryagin-type, self-duality holds. Our quantum groups are locally compact quantum groups, in the C*-algebra or von Neumann algebra framework.

By using the notion of the multiplicative unitary operators and the generalized Fourier transform, we can enhance our understanding of the duality picture at the quantum group level. In particular, we will consider a case of a certain coalgebra deformation of the quantum double, and its dual counterpart.

Friday, November 13

Seminar - GEOMETRY/TOPOLOGY - 4:00 p.m., 122 Mathematics Building

Dr. Bulent Tosun
Georgia Institute of Technology

On the Legendrian and transverse classification of cabled knot types

Abstract: In 3-dimensional contact topology one of the main problems is classifying Legendrian (transverse) knots in certain knot type up to Legendrian (transverse) isotopy. In particular we want to decide if two (one in the case of transverse knots) classical invarients of this knot are complete set of invariants. If it is, then we call this knot type Legendrian (transversely) simple knot type otherwise it is called Legendrian (transversely) non-simple. In this talk, by tracing the techniques developed by Etnyre and Honda, we will present some results concerning the complete Legendrian and transverse classification of certain cabled knots in the standard tight contact 3-sphere. Moreover we will provide an infinite family of Legendrian and transversely non-simple prime knots.

Monday, November 9

Seminar - ALGEBRA - 4:00 p.m., 150 Mathematics Building

Dr. Adam Glesser
Suffolk University

Sparse fusion systems

Abstract: Classical results on fusion in finite groups from Frobenius to Alperin to Glauberman and Thompson played a major role in the classification of finite simple groups. A modern treatment of fusion takes place in the category of fusion systems, a relatively new object that allows results to extend from groups to blocks as well as making a connection to algebraic topology via the classifying space of a finite group. In this talk, we discuss how certain minimal nontrivial elements in the lattice of subfusion systems of a system (called sparse fusion systems) are used to simplify several recent proofs of fusion results and how, in the case of a theorem of Navarro, fusion systems naturally lead to a stronger result about groups than was not previously known.

Wednesday, November 4

Seminar - ANALYSIS - 4:00 p.m., 250 Mathematics Building

Dr. Jingbo Xia
University at Buffalo, State University of New York

Defect Operators Associated with Submodules of the Hardy Module

Abstract: Let H²(S) be the Hardy space on the unit sphere S in Cⁿ, n ≥ 2. Then H²(S) is a natural Hilbert module over the ball algebra A(B) . Let M_z1, ..., M_zn be the module operators corresponding to the multiplication by the coordinated functions. Each submodule M ⊂ H² (S) gives rise to the module operators Z_M,j = M_zj|M, j = 1, ..., n, on M. In this paper we establish the following commonly believed, but never previously proven result: whenever M ≠ {0}, the sum of the commutators

[Z*_M,1, Z_M,1] + ... + [Z*_M,n, Z_M,n]

does not belong to the Schatten class C_n. (This is a joint work with Quanlei Fang.)

Monday, November 2

Seminar - APPLIED MATHEMATICS - CANCELLED - 4:00 p.m., 250 Mathematics Building

Dr. Gaurav Khanna
University of Massachusetts, Dartmouth

Title To Be Announced

Wednesday, October 28

Seminar - ANALYSIS - 4:00 p.m., 250 Mathematics Building

Dr. Quanlei Fang
University at Buffalo, State University of New York

Commutators and Localization on the Drury-Arveson Space

Abstract: Let f be a multiplier for the Drury-Arveson space H²_n of the unit ball, and let ζ₁, ..., ζ_n denote the coordinate functions. We show that for each 1 ≤ i ≤ n, the commutator [M*_f, M_ζi] belongs to the Schatten class C_p, p > 2n. This leads to a localization result for multipliers.

Tuesday, October 20

Seminar - APPLIED MATHEMATICS - presented jointly with the Electrical Engineering Department, University at Buffalo - 4:00 p.m., 414 Bonner Hall

Dr. Rene-Jean Essiambre
Bell Laboratories

Capacity Limits of Fiber-Optic Communication Systems

Abstract: The capacity of fiber-optic communication systems, or “fiber capacity”, that a single strand of fiber can carry has steadily increased for the last two decades. Such capacity growth has been driven by technological innovations, both in the electrical and optical domains. The question then arises: are there fundamental limits to fiber capacity?

In this talk, I will describe a procedure that has been developed to calculate a fiber-capacity estimate starting from Shannon’s information theory. I will present the main challenges associated to calculating a capacity for optical fibers, all revolving around the presence of the instantaneous Kerr nonlinearity of fibers. We will show that a series of advanced technologies is necessary to maximize capacity. Such technologies include distributed Raman amplification, arbitrary waveform generation for generating Nyquist signals and advanced modulation formats, coherent detection and optimum digital signal processing based on reverse nonlinear fiber propagation. The fiber-capacity estimate obtained will be compared to the capacity of the highest capacity ‘hero experiments’.

Friday, October 16

Seminar - APPLIED MATHEMATICS - 4:00 p.m., 250 Mathematics Building

Dr. Allen Tesdall
CUNY, Staten Island

High-resolution solutions for shock formation in transonic flow

Abstract: Shock waves that form as the result of an interaction of a rarefaction wave with a sonic line are a generic feature of solutions of transonic flow problems. Examples include (i) the sequence of shocks that occur in Guderley Mach reflection, (ii) the shock that forms at the rear of a supersonic bubble on an airfoil in a slightly subsonic free stream flow, and (iii) the shock wave that forms when a supersonic flow hits the corner of an expanding duct. Whether the shock forms on the sonic line or inside the supersonic region appears to be an open question. We present high-resolution numerical solutions of problems for the steady and unsteady transonic small disturbance equations that describe examples (ii) and (iii) above. Our solutions show that the shock forms strictly inside the supersonic region. These results appear to be the first that clearly show the supersonic nature of the shock formation point.

Thursday, October 15

Colloquium - 4:00 - 5:00 p.m., 250 Mathematics Building

Dr. Daniel Calegari
California Institute of Technology

Faces of the scl norm ball

Abstract: It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem - when does an immersed curve on a surface bound an immersed subsurface? - is unexpectedly related to linear programming in nonseparable Banach spaces, and gives rise to geometric and dynamical rigidity and discreteness of symplectic representations.