Seminars and Colloquia by Series

Irregular activity and propagation of synchrony in complex, spiking neural networks

Series
Mathematical Biology Seminar
Time
Wednesday, February 17, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Raoul-Martin MemmesheimerCenter for Brain Science, Faculty of Arts and Sciences Harvard University
Mean field theory for infinite sparse networks of spiking neurons shows that a balanced state of highly irregular activity arises under a variety of conditions. The state is considered to be a model for the ground state of cortical activity. In the first part, we analytically investigate its irregular dynamics in finite networks keeping track of all individual spike times and the identities of individual neurons. For delayed, purely inhibitory interactions, we show that the dynamics is not chaotic but in fact stable. Moreover, we demonstrate that after long transients the dynamics converges towards periodic orbits and that every generic periodic orbit of these dynamical systems is stable. These results indicate that chaotic and stable dynamics are equally capable of generating the irregular neuronal activity. More generally, chaos apparently is not essential for generating high irregularity of balanced activity, and we suggest that a mechanism different from chaos and stochasticity significantly contributes to irregular activity in cortical circuits. In the second part, we study the propagation of synchrony in front of a background of irregular spiking activity. We show numerically and analytically that supra-additive dendritic interactions, as recently discovered in single neuron experiments, enable the propagation of synchronous activity even in random networks. This can lead to intermittent events, characterized by strong increases of activity with high-frequency oscillations; our model predicts the shape of these events and the oscillation frequency. As an example, for the hippocampal region CA1, events with 200Hz oscillations are predicted. We argue that these dynamics provide a plausible explanation for experimentally observed sharp-wave/ripple events.

A variational method for a class of parabolic PDEs

Series
PDE Seminar
Time
Tuesday, February 16, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Wilfrid GangboGeorgia Tech
Let $\mathbb{H}$ be a Hilbert space and $h: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}$ be such that $h(x, \cdot)$ is uniformly convex and grows superlinearly at infinity, uniformy in $x$. Suppose $U: \mathbb{H} \rightarrow \mathbb{R}$ is strictly convex and grows superlinearly at infinity. We assume that both $H$ and $U$ are smooth. If $\mathbb{H}$ is of infinite dimension, the initial value problem $\dot x= -\nabla_p h(x, -\nabla U(x)), \; x(0)=\bar x$ is not known to admit a solution. We study a class of parabolic equations on $\mathbb{R}^d$ (and so of infinite dimensional nature), analogous to the previous initial value problem and establish existence of solutions. First, we extend De Giorgi's interpolation method to parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. These interpolation reveal to be powerful tool for proving convergence of a time discrete algorithm. (This talk is based on a joint work with A. Figalli and T. Yolcu).

Orthogonal Polynomials and their Ph.D. Theses

Series
Research Horizons Seminar
Time
Tuesday, February 16, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Doron LubinskySchool of Mathematics, Georgia Tech

Please Note: Hosted by: Huy Huynh and Yao Li

Orthogonal Polynomials and their generalizations have a great many applications in areas ranging from signal processing to random matrices to combinatorics. We outline a few of the connections, and present some possible Ph. D Problems

Multidimensional chaotic maps with hyperbolic attractors

Series
CDSNS Colloquium
Time
Monday, February 15, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Vladimir BelykhNizhny Novgorod University
In this lecture, I will discuss a class of multidimensional maps with one nonlinearity, often called discrete-time Lurie systems. In the 2-D case, this class includes Lozi map and Belykh map. I will derive rigorous conditions for the multidimensional maps to have a generalized hyperbolic attractor in the sense of Bunimovich-Pesin. Then, I will show how these chaotic maps can be embedded into the flow, and I will give specific examples of three-dimensional piece-wise linear ODEs, generating this class of hyperbolic attractors.

State polytopes and GIT

Series
Algebra Seminar
Time
Monday, February 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
David SwinarskiUniversity of Georgia
State polytopes in commutative algebra can be used to detect the geometric invariant theory (GIT) stability of points in the Hilbert scheme. I will review the construction of state polytopes and their role in GIT, and present recent work with Ian Morrison in which we use state polytopes to estabilish stability for curves of small genus and low degree, confirming predictions of the minimal model program for the moduli space of curves.

Applied and Computational Multilinear Algebra

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 15, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Lek-Heng LimUC Berkeley
Numerical linear algebra is often regarded as a workhorse of scientific and engineering computing. Computational problems arising from optimization, partial differential equation, statistical estimation, etc, are usually reduced to one or more standard problems involving matrices: linear systems, least squares, eigenvectors/singular vectors, low-rank approximation, matrix nearness, etc. The idea of developing numerical algorithms for multilinear algebra is naturally appealing -- if similar problems for tensors of higher order (represented as hypermatrices) may be solved effectively, then one would have substantially enlarged the arsenal of fundamental tools in numerical computations. We will see that higher order tensors are indeed ubiquitous in applications; for multivariate or non-Gaussian phenomena, they are usually inevitable. However the path from linear to multilinear is not straightforward. We will discuss the theoretical and computational difficulties as well as ways to avoid these, drawing insights from a variety of subjects ranging from algebraic geometry to compressed sensing. We will illustrate the utility of such techniques with our work in cancer metabolomics, EEG and fMRI neuroimaging, financial modeling, and multiarray signal processing.

Introduction to the Latex

Series
SIAM Student Seminar
Time
Friday, February 12, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 156 (undergraduate computer lab)
Speaker
Mitch KellerSchool of Mathematics, Georgia Tech
This is an introductory talk to everyone who wants to learn skills in Latex. We will discuss including and positioning graphics and the beamer document class for presentations. A list of other interesting topics will be covered if time permits.

Algebraic structures and legendrian contact homology

Series
Geometry Topology Working Seminar
Time
Friday, February 12, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
John EtnyreGeorgia Tech
After, briefly, recalling the definition of contact homology, a powerful but somewhat intractable and still largely unexplored invariant of Legendrian knots in contact structures, I will discuss various ways of constructing more tractable and computable invariants from it. In particular I will discuss linearizations, products, massy products, A_\infty structures and terms in a spectral sequence. I will also show examples that demonstrate some of these invariants are quite powerful. I will also discuss what is known and not known about the relations between all of these invariants.

Gasper's identity and the Markov sequence problem

Series
Analysis Seminar
Time
Wednesday, February 10, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jeff GeronimoGeorgia Tech
Gasper in his 1971 Annals of Math paper proved that the Jacobi polynomials satisfy a product formula which generalized the product formula of Gegenbauer for ultraspherical polynomials. Gasper proved this by showing that certains sums of triple products of Jacobi polynomials are positive generalizing results of Bochner who earlier proved a similar results for ultraspherical polynomials. These results allow a convolution structure for Jacobi polynomials. We will give a simple proof of Gasper's and Bochner's results using a Markov operator found by Carlen, Carvahlo, and Loss in their study of the Kac model in kinetic theory. This is joint work with Eric Carlen and Michael Loss.

On the Chvatal Closure of a Strictly Convex Body

Series
ACO Student Seminar
Time
Wednesday, February 10, 2010 - 13:30 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom
Speaker
Daniel DadushISyE ACO, Georgia Tech
The analysis of Chvatal Gomory (CG) cuts and their associated closure for polyhedra was initiated long ago in the study of integer programming. The classical results of Chvatal (73) and Schrijver (80) show that the Chvatal closure of a rational polyhedron is again itself a rational polyhedron. In this work, we show that for the class of strictly convex bodies the above result still holds, i.e. that the Chvatal closure of a strictly convex body is a rational polytope.This is joint work with Santanu Dey (ISyE) and Juan Pablo Vielma (IBM).

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