Seminars and Colloquia by Series

L^p Estimates for a Singular Integral Operator motivated by Calderón's Second Commutator

Series
Analysis Seminar
Time
Wednesday, December 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Eyvindur Ari PalssonCornell University
When Calderón studied his commutators, in connection with the Cauchy integral on Lipschitz curves, he ran into the bilinear Hilbert transform by dropping an average in his first commutator. He posed the question whether this new operator satisfied any L^p estimates. Lacey and Thiele showed a wide range of L^p estimates in two papers from 1997 and 1999. By dropping two averages in the second Calderón commutator one bumps into the trilinear Hilbert transform. Finding L^p estimates for this operator is still an open question. In my talk I will discuss L^p estimates for a singular integral operator motivated by Calderón's second commutator by dropping one average instead of two. I will motivate this operator from a historical perspective and give some comments on potential applications to partial differential equations motivated by recent results on the water wave problem.

On Ulam's Problem

Series
Research Horizons Seminar
Time
Wednesday, December 8, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Christian HoudreSchool of Mathematics - Georgia Institute of Technology

Please Note: Hosts: Yao Li and Ricardo Restrepo

Ulam's problem has to do with finding asymptotics, as $n \to +\infy$, for the length of the longest increasing subsequence of a random permutation of $\{1, .., n\}. I'll survey its history, its solutions and various extensions emphasizing progresses made at GaTech.

Hyperbolicity of hyperplane complements

Series
Geometry Topology Seminar
Time
Monday, December 6, 2010 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Igor BelegradekGeorgia Tech
We will discuss properties of manifolds obtained by deleting a totally geodesic ``divisor'' from hyperbolic manifold. Fundamental groups of these manifolds do not generally fit into any class of groups studied by the geometric group theory, yet the groups turn out to be relatively hyperbolic when the divisor is ``sparse'' and has ``normal crossings''.

Shape Optimization of Chiral Propellers in 3-D Stokes Flow

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 6, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shawn WalkerLSU Mathematics Dept.
Locomotion at the micro-scale is important in biology and in industrialapplications such as targeted drug delivery and micro-fluidics. Wepresent results on the optimal shape of a rigid body locomoting in 3-DStokes flow. The actuation consists of applying a fixed moment andconstraining the body to only move along the moment axis; this models theeffect of an external magnetic torque on an object made of magneticallysusceptible material. The shape of the object is parametrized by a 3-Dcenterline with a given cross-sectional shape. No a priori assumption ismade on the centerline. We show there exists a minimizer to the infinitedimensional optimization problem in a suitable infinite class ofadmissible shapes. We develop a variational (constrained) descent methodwhich is well-posed for the continuous and discrete versions of theproblem. Sensitivities of the cost and constraints are computedvariationally via shape differential calculus. Computations areaccomplished by a boundary integral method to solve the Stokes equations,and a finite element method to obtain descent directions for theoptimization algorithm. We show examples of locomotor shapes with andwithout different fixed payload/cargo shapes.

Recent Progress in Delay-Differential Equations

Series
CDSNS Colloquium
Time
Monday, December 6, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
John Mallet-ParetBrown University
We examine a variety of problems in delay-differential equations. Among the new results we discuss are existence and asymptotics for multiple-delay problems, global bifurcation of periodic solutions, and analyticity (or lack thereof) in variable-delay problems. We also plan to discuss some interesting open questions in the field.

Nonlinear Science Webminar - Multiple Time Scale Dynamics in Chemical Oscillators

Series
Other Talks
Time
Monday, December 6, 2010 - 10:00 for 1 hour (actually 50 minutes)
Location
Physics Howey 501
Speaker
Chris ScheperCenter for Applied Mathematics, Cornell University
Dynamical systems with multiple time scales have invariant geometric objects that organize the dynamics in phase space. The slow-fast structure of the dynamical system leads to phenomena such as canards, mixed-mode oscillations, and bifurcation delay. We'll discuss two projects involving chemical oscillators. The first is the analysis of a simple chemical model that exhibits complex oscillations. Its bifurcations are studied using a geometric reduction of the system to a one-dimensional induced map. The second investigates the slow-fast mechanisms generating mixed-mode oscillations in a model of the Belousov-Zhabotinsky (BZ) reaction. A mechanism called dynamic Hopf bifurcation is responsible for shaping the dynamics of the system. This webminar will be broadcast on evo.caltech.edu (register, start EVO, webminar link is evo.caltech.edu/evoNext/koala.jnlp?meeting=MMMeMn2e2sDDDD9v9nD29M )

Non-commutative Geometry V - Riemannian Geometry of Ultrametric Cantor Sets

Series
Geometry Topology Working Seminar
Time
Friday, December 3, 2010 - 14:00 for 2 hours
Location
Skiles 171
Speaker
Jean BellissardGa Tech

Please Note: This will be a 2 hour talk.

In this lecture the analog of Riemannian manifold will be introduced through the notion of spectral triple. The recent work on the case of a metric Cantor set, endowed with an ultrametric, will be described in detail during this lecture. An analog of the Laplace Beltrami operator for a metric Cantor set will be defined and studied

Invariant Manifolds in Dynamical Systems

Series
SIAM Student Seminar
Time
Friday, December 3, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Nan LuSchool of Mathematics, Georgia Tech
In this talk, I am going to give a elementary introduction of invariant manifold theory in dynamical systems. I will start with the motivation and definition of invariant manifolds. Then I will discuss how to construct various invariant manifolds of maps and flows. Finally, I will discuss some applications. If time is permitted, I will also discuss a little about invariant foliation.

Planted Cliques and Random Tensors

Series
Stochastics Seminar
Time
Thursday, December 2, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
Santosh VempalaCollege of Computing, Georgia Tech
For general graphs, approximating the maximum clique is a notoriously hard problem even to approximate to a factor of nearly n, the number of vertices. Does the situation get better with random graphs? A random graph on n vertices where each edge is chosen with probability 1/2 has a clique of size nearly 2\log n with high probability. However, it is not know how to find one of size 1.01\log n in polynomial time. Does the problem become easier if a larger clique were planted in a random graph? The current best algorithm can find a planted clique of size roughly n^{1/2}. Given that any planted clique of size greater than 2\log n is unique with high probability, there is a large gap here. In an intriguing paper, Frieze and Kannan introduced a tensor-based method that could reduce the size of the planted clique to as small as roughly n^{1/3}. Their method relies on finding the spectral norm of a 3-dimensional tensor, a problem whose complexity is open. Moreover, their combinatorial proof does not seem to extend beyond this threshold. We show how to recover the Frieze-Kannan result using a purely probabilistic argument that generalizes naturally to r-dimensional tensors and allows us recover cliques of size as small as poly(r).n^{1/r} provided we can find the spectral norm of r-dimensional tensors. We highlight the algorithmic question that remains open. This is joint work with Charlie Brubaker.

Traveling Salesman Problems

Series
Graph Theory Seminar
Time
Thursday, December 2, 2010 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Bill CookISyE, GT
We discuss open research questions surrounding the traveling salesman problem. A focus will be on topics having potential impact on the computational solution of large-scale problem instances.

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