# Neuronal Dynamics

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Oscillations and other patterns of neuronal activity arise throughout

the central nervous system. This activity has been observed in sensory

processing, motor activities, and learning, and has been implicated in

the generation of sleep rhythms, epilepsy, and parkinsonian tremor.

Mathematical models for neuronal activity often display an incredibly

rich structure of dynamic behavior. In this lecture, I describe how the

neuronal systems can be modeled, various types of activity patterns that

arise in these models, and mechanisms for how the activity patterns are

generated. In particular, I demonstrate how methods from geometric

singular perturbation theory have been used to analyze a recent model

for activity patterns in an insect's antennal lobe.

# The Ubiquity of Fluid Instability

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The unstable nature of fluid motion is a classical problem whose

mathematical roots go back to the 19th Century. It has important applications

to many aspects of our life from such disparate issues as predicting the

weather to regulating blood flow. Instabilities might lead to turbulence or

to new nonlinear flows which themselves might become unstable. We will

discuss some of the mathematical techniques which can be used to gain

insight into fluid instabilities. These tools include nonlinear PDE,

spectral theory and dynamical systems.

# Digital biology: data-mining with a physical chemistry lens

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The digital nature of biology is crucial to its functioning

as an information system. The hierarchical development of

biological components (translating DNA to proteins which form

complexes in cells that aggregate to make tissue which form

organs in different species) is discrete (or quantized) at

each step. It is important to understand what makes proteins

bind to other proteins predictably and not in a continuous

distribution of places, the way grease forms into blobs.

Data mining is a major technique in bioinformatics. It has been

used on both genomic and proteomic data bases with significant

success. One key issue in data mining is the type of lens that

is used to examine the data. At the simplest level, one can just

view the data as sequences of letters in some alphabet. However,

it is also possible to view the data in a more sophisticated

way using concepts and tools from physical chemistry. We will

give illustrations of the latter and also show how data mining

(in the PDB) has been used to derive new results in physical

chemistry. Thus there is a useful two-way interaction between

data mining and physical chemistry.

We will give a detailed description of how data mining in the

PDB can give clues to how proteins interact. This work makes

precise the notion of hydrophobic interaction in certain cases.

It provides an understanding of how molecular recognition and

signaling can evolve. This work also introduces a new model of

electrostatics for protein-solvent systems that presents

significant computational challenges.

# Time Reversal in a finite cavity effect of ergodicity and randomness

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This talk is devoted to a mathematical analysis of the time reversal

method which was promoted by Mathias Fink, his group and others.

It involves source and transductors. The challenge is to

understand how to use as few transductors as possible.

Emphasis is put on examples of problems in a closed bounded cavity In

this situation I will describe the effect of ergodicity both when the

transductors are in the media or when they are at the boundary.

Results and methods are compared with what has already been done for

random media. Some of the proof are very similar sharper results are

obtained but only in domains with no boundary.

# Estimates for the tangential Cauchy-Riemann equations with minimal smoothness

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We study the regularity for the tangential Cauchy-Riemann equations and the associated Laplacian on CR manifolds with minimal smoothness assumption. One application is to extend the embedding theorem of Boutet De Monvel

to strongly pseudoconvex CR manifolds of class C^2.

(Joint work with Lihe Wang).

# One direction and one component regularity for the Navier-Stokes equations

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We consider sufficient conditions for regularity of weak solutions of the Navier-Stokes equation. By a result of Neustupa and Panel, the weak solutions are regular provided a single component of the velocity is bounded. In this talk we will survey existing and present new results on one component and one direction regularity.

# Multiscale analysis in micromagnetics: an example

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From the point of view of mathematics,

micromagnetics is an ideal playground for a pattern forming system in materials

science: There are abundant experiments on a wealth of visually

attractive phenomena and there is a well--accepted continuum model.

In this talk, I will focus on a specific

experimental observation for thin film ferromagnetic elements: Elements

with elongated rectangular cross--section are saturated along the longer

axis by a strong external field. Then the external field is slowly

reduced. At a certain field strength, the uniform magnetization buckles

into a quasiperiodic domain pattern which resembles a concertina.

Our hypothesis is that the period of this pattern is the

frozen--in period of the unstable mode at critical field.

Starting point for the analysis is the micromagnetic model which has three

length scales. We rigorously identify four scaling regimes for the

critical field. One of the regimes has been

overseen by the physics literature. It displays an

oscillatory unstable mode, which we identify asymptotically.

In this parameter regime, we identify

a scaling limit for the bifurcation.

The analysis amounts to the combination of an asymptotic

limit with a bifurcation argument. This is carried out by a suitable

``blow--up'' of the energy landscape in form of Gamma--convergence.

Numerical simulation of the normal form visualizes a homotopy

from a turning point to

the strongly nonlinear concertina pattern.

This is joint work with Ruben Cantero--Alvarez and Jutta Steiner.

# Volatility surface estimation

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A central problem in modern mathematical

finance is that of estimating the volatility

of financial time series, whether they are

equity prices, exchange rates, interest rates

or something else, such as options. A recent trend is to try to

estimate the implied volatility of an asset from

the fluctuations in the price of derivatives

whose underlying it is. This is the volatility

surface estimation problem. I will review briefly the

background and status of this problem, including

computational issues, and I will present a variational

theory for volatility surface estimation within

stochastic volatilty models. I will show the form

this theory takes under a fast mean reverting hypothesis

and I will conclude with a calibration of the theoretical

framework using SP500 options data.