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Series: ACO Student Seminar

On-line graph coloring has a rich history, with a very large number of elegant results together with a near equal number of unsolved problems. In this talk, we will briefly survey some of the classic results including: performance on k-colorable graphs and \chi-bounded classes. We will conclude with a sketch of some recent and on-going work, focusing on the analysis of First Fit on particular classes of graphs.

Series: Mathematical Biology Seminar

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.

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.

Series: PDE Seminar

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).

$\mathbb{H}$ 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).

Series: Research Horizons Seminar

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

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

Series: CDSNS Colloquium

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.

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.

Series: Algebra Seminar

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.

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.

Monday, February 15, 2010 - 13:00 ,
Location: Skiles 255 ,
Lek-Heng Lim ,
UC Berkeley ,
Organizer: Haomin Zhou

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.

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.

Series: SIAM Student Seminar

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.

Friday, February 12, 2010 - 14:00 ,
Location: Skiles 269 ,
John Etnyre ,
Georgia Tech ,
Organizer:

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.

Series: Analysis Seminar

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.

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.