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Series: Research Horizons Seminar

An assembly of atoms in a solid phase will be described through the notion
of Delone sets and related to tilings. The Hull and the tiling space wiill
be defined. It will be shown that the tiling space and the Hull can be
constructed through an inverse limit of CW-complexes built out of the
tiles and of the local patches. From then various cohomologies can be
defined and allow to distinguish between these atomic distributions. The
question of whether these topological invariant can be seen in experiments
will be addressed.

Series: Mathematical Biology Seminar

Bursting, tonic spiking, sub-threshold oscillations and silence are basic
robust regimes of activity of a single neuron. The talk will be focused on
the co-existence of regimes of activity of neurons. Such multistability
enhances potential flexibility to the nervous system and has many
implications for motor control and decision making. I will identify
different scenarios leading to multistability in the neuronal dynamics and
discuss its potential roles in the operation of the central nervous system
under normal and pathological conditions.

Series: PDE Seminar

In this talk, we will discuss the global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. This is joint work with Bin Cheng.

Series: Geometry Topology Seminar

In 3-dimensional contact topology one of the main problem is classifying Legendrian (transverse) knots in certain knot type up to Legendrian ( transverse) isotopy. In particular we want to decide if two (one in case of transverse knots) classical invariants of this knots 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.

Series: Analysis Working Seminar

We are going to continue explaining the proof of Seip's Interpolation Theorem for the Bergman Space. We are going to demonstrate the sufficiency of these conditions for a certain example. We then will show how to deduce the full theorem with appropriate modifications of the example.

Monday, November 9, 2009 - 13:00 ,
Location: Skiles 255 ,
Nicola Guglielmi ,
Università di L'Aquila ,
guglielm@univaq.it ,
Organizer: Sung Ha Kang

This is a joint work with Michael Overton (Courant Institute, NYU). The epsilon-pseudospectral abscissa and radius of an n x n matrix are respectively the maximum real part and the maximal modulus of points in its epsilon-pseudospectrum. Existing techniques compute these quantities accurately but the cost is multiple SVDs of order n, which makesthe method suitable to middle size problems. We present a novel approach based on computing only the spectral abscissa or radius or a sequence of matrices, generating a monotonic sequence of lower bounds which, in many but not all cases, converges to the pseudospectral abscissa or radius.

Series: CDSNS Colloquium

Many dynamical systems may be subject to stochastic excitations, so to find an efficient method to analyze the stochastic system is very important. As for the complexity of the stochastic systems, there are not any omnipotent methods. What I would like to present here is a brief introduction to quasi-non-integrable Hamiltonian systems and stochastic averaging method for analyzing certain stochastic dynamical systems. At the end, I will give some examples of the method.

Series: Combinatorics Seminar

This is joint work with Dr. Yi Zhao.

Graph tiling problems can be summarized as follows: given a graph H, what conditions do we need to find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. The most familiar example of a graph tiling problem is finding a matching in a graph. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove a degree condition that guarantees an arbitrary bipartite graph G will be tiled by an arbitrary bipartite graph H. We also prove this degree condition is best possible up to a constant. This answers a question of Zhao and proves an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

Friday, November 6, 2009 - 15:00 ,
Location: Skiles 269 ,
Meredith Casey ,
Georgia Tech ,
Organizer:

The goal of this talk is to describe simple constructions by which we can construct any compact, orientable 3-manifold. It is well-known that every orientable 3-manifold has a Heegaard splitting. We will first define Heegaard splittings, see some examples, and go through a very geometric proof of this therem. We will then focus on the Dehn-Lickorish Theorem, which states that any orientation-preserving homeomorphism of an oriented 2-manifold without boundary can by presented as the composition of Dehn twists and homeomorphisms isotopic to the identity. We will prove this theorm, and then see some applications and examples. With both of these resutls together, we will have shown that using only handlebodies and Dehn twists one can construct any compact, oriented 3-manifold.

Series: Other Talks

We consider the Stochastic Differential Equation
$dX_\epsilon=b(X_\epsilon)dt + \epsilon dW$ . Given a domain D, we
study how the exit time and the distribution of the process at the time
it exits D behave as \epsilon goes to 0. In particular, we cover the
case in which the unperturbed system $\frac{d}{dt}x=b(x)$ has a unique
fixed point of the hyperbolic type. We will illustrate how the behavior
of the system is in the linear case. We will remark how our results
give improvements to the study of systems admitting heteroclinic or
homoclinic connections. We will outline the general proof in two
dimensions that requires normal form theory from differential
equations. For higher dimensions, we introduce a new kind of non-smooth
stochastic calculus.