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Talk Abstracts
The IMA at 20: Mathematics and its Impact
June 6-7, 2003

Probability and Statistics in Complex Systems: Genomics, Networks, and Financial Engineering, September 1, 2003 - June 30, 2004

Bjorn Engquist (Program in Computational and Applied Mathematics, Princeton University)  engquist@princeton.edu

New Classes of Multi-Scale Methods

Multi-scale modeling and computation is an increasingly important area of research with profound impact on computational science and applied mathematics. Recently a number of techniques have been developed that link the micro-scale and the macro-scale together in the same simulation. We will discuss a computational framework that allows for the accuracy of a micro-scale technique but with a computational cost that is closer to that of a macro-scale method.

Nancy Kopell (Center for BioDynamics (CBD), Department of Mathematics, Boston University)  nk@math.bu.edu  http://cbd.bu.edu/members/nkopell.html

Rhythms of the Nervous System: From Cells to Behavior via Dynamics

The nervous system produces rhythmic electrical activity in many frequency ranges, and the rhythms displayed during waking are tightly tied to cognitive state. This talk describes ongoing work whose ultimate aim is to understand the uses of these rhythms in sensory processing, cognition and motor control. The method is to address the biophysical underpinnings of the different rhythms and transitions among them, to get clues to how specific important subsets of the cortex and hippocampus process and transform spatio-temporal input. We focus in this talk on the gamma rhythm (30-80 hz), which is associated with attention and awareness, and theta (4-12), associated with active exploration and learning of sequences. Via case studies, we show that different biophysics corresponds to different dynamical structure in the rhythms, with implications for function. The mathematical tools come from dynamical systems, and include the use of low-dimensional maps, probability and geometric singular perturbations.

Graeme Walter Milton (Department of Mathematics, University of Utah)  milton@math.utah.edu  http://www.math.utah.edu/%7Emilton/

Composite Materials: An Old Field of Study Full of New Surprises    Slides:   html

Composite materials have been studied for centuries, and have attracted the interest of reknown scientists such as Poisson, Faraday, Maxwell, Rayleigh, and Einstein. Their properties are usually not just a linear average of the properties of the constituent materials and can sometimes be strikingly different. The beautiful red glass one sees in old church windows is a suspension of small gold particles in glass. Sound waves travel slower in bubbly water than in either water or air. In the last few decades composites have been found to have some surprising properties. Most materials, such as rubber, get thinner when they are stretched, but it is possible to design composites which get fatter as they are stretched. Electromagnetic signals can travel faster in a composite than in the constituent phases. It is possible to combine materials which expand when heated, with voids, to obtain a material which contracts when heated. It is still an open question as to what properties can be achieved when one mixes two or more materials with known properties. This lecture will survey some of the progress which has been made and the role the IMA played in the development of the field.

Andrew Odlyzko (Digital Technology Center, University of Minnesota)  odlyzko@dtc.umn.edu  http://www.dtc.umn.edu/~odlyzko

Zeros of the Riemann Zeta Function: Computations and Implications

The Riemann Hypothesis is now left as the most famous unsolved problem in mathematics. Extensive computations of zeros have been used not only to provide evidence for its truth, but also for the truth of deeper conjectures that predict fine scale statistics on the distribution of zeros of various zeta functions. These conjectures connect number theory with physics, and are regarded by many as the most promising avenue towards a proof of the Riemann Hypothesis. However, as is often true in mathematics, numerical data is subject to a variety of interpretations, and it is possible to argue that the numerical evidence we have gathered so far is misleading. Whatever the truth may be, the computational exploration of zeros of zeta functions is flourishing, and through projects such as the ZetaGrid is drawing many amateurs into contact with higher mathematics.

George C. Papanicolaou (Mathematics Department, Stanford University)  papanico@math.stanford.edu  http://georgep.stanford.edu

Imaging in Clutter

Array imaging, like synthetic aperture radar and Kirchhoff migration in seismic imaging, does not produce good reflectivity images when there is clutter, or random scattering inhomogeneities, between the reflectors and the array. Can the blurring effects of clutter be controlled? I will discuss this issue in some detail and explain why it is a central one for the recent developments in the mathematics of imaging. I will also review briefly the current status of array imaging and I will show in particular that if array data is collected carefully, and there is lots of it, then a good deal can be done to minimize blurring by clutter.

Panagiotis Souganidis (Department of Mathematics, University of Texas at Austin)  souganid@math.utexas.edu

Viscosity Solutions: Theory and Applications. Brief history and Future Directions    Slides:   html

Since their introduction in 1981, viscosity solutions have become one of the fundamental tools in the theory of nonlinear partial differential equations. This lecture presents a brief history of the theory and the main applications. Moreover it identifies two important future directions (stochastic homogenization and stochastic partial differential equations) which are related to the understanding of the role of randomness in the theory of nonlinear equations and related fields.

Wim Sweldens (Vice President, Computing Sciences Research Bell Labs, Lucent Technologies)  wim@lucent.com  http://cm.bell-labs.com/who/wim/

Wavelets and Digital Geometry Processing    Slides:   pdf

Over the last 50 years we have seen a tremendous evolution in digital signal processing. As computers become more and more powerful they are able to deal with ever increasing amounts of digitized media. So far we have witnessed three waves: audio (1D), images (2D), and video (3D). Each wave of digitization comes with its own need for algorithms and sets off a new branch of digital signal processing. Today a forth wave in digital signal processing is emerging: digital geometry processing. New technology exist for quickly and accurately acquiring 3D geometry of objects: A sub-millimeter digitization of Michelangelo's David for example consists of over one billion samples. While audio, images, and video are defined on Euclidean geometry and therefore often used Fourier based algorithms, this no longer works for digital geometry. We will describe new multiresolution and wavelet based geometry representations and show how they are used to build a digital geometry processing toolbox, including denoising, filtering, editing, morphing, and compression.

Margaret H. Wright (Computer Science Department, Courant Institute of Mathematical Sciences, New York University)  mhw@cs.nyu.edu

The Latest Score in Optimization

During the past twenty years, continuous optimization has become (in the admittedly biased view of the speaker) ever more important across applied mathematics, computer science, and real-world applications of all kinds. This talk will survey selected highlights, emphasizing two areas of active research (direct search and interior-point methods) as well as the growing association between optimization and other fields of applied mathematics, computer science, science, engineering, and medicine (such as partial differential equations, combinatorial optimization, design, and data analysis).

Lai-Sang Young (Courant Institute of Mathematical Sciences, New York University)  lsy@courant.nyu.edu  http://www.cims.nyu.edu/~lsy/

Deterministic Chaos    Slides:   html

Recent results illustrating state-of-the-art analytic, geometric and probabilistic tecnhniques in the theory of chaos will be presented, along with a discussion of the limitations and potential applications of this theory as it stands today.

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