Wednesdays at 4 PM in Eckhart 202.
We shall review some recent mathematical contributions related to two fields of micro-macro simulations. First, models for the solid crystal phase that couple the continuum scale together with the atomic scale will be dealt with. Second, models for polymeric fluid flows, involving deterministic and probabilistic type equations will be examined. These are respectively joint works with X. Blanc and PL.Lions, and B. Jourdain and T. Lelievre.
The speaker will show that a wide class of Fredholm determinants arising in the representation theory of "big" groups such as the infinite -dimensional unitary group, solve Painlev'e equations. The method is based on the theory of integrable operators and the theory of Riemann-Hilbert problems. This is joint work with Alexei Borodin.
The hypothesis of microscopic chaos, i.e., the hypothesis of the chaotic nature of the collisions at molecular level, allows to study the relations between the macroscopic irreversibility of transport and reaction processes, and the reversibility of microscopic evolution laws. We consider here several reaction-diffusion processes, simulated with microscopic deterministic chaotic models. We derive expressions relating quantities characterizing the microscopic chaotic dynamics of the system, and coefficients describing the macroscopic transport and reaction processes.
Over the past two decades, domain decomposition has grown from an elegant mathematical technique to the dominant paradigm of large-scale scientific simulation for systems governed by partial differential equations. Two major families of domain decomposition methods -- Newton-Krylov-Schwarz and FETI-DP -- have been scalably employed on the ASCI platforms of the U.S. Department of Energy for mechanics problems, up to several thousand processors. One of the many scientific software projects intent on "packaging" the fruits of research in domain decomposition methods for mainstream computational scientists is a five-year, nine-institution "Terascale Optimal PDE Simulations Integrated Software Infrastructure Center", a component of DOE's new Scientific Discovery through Advanced Computing (SciDAC) initiative. The speaker, who serves as the TOPS project lead, will review the current algorithmic state of the art, outline the philosophy and goals of the center, and highlight some of the research challenges ahead, in algorithms and in software.
Recent experiments in Swinney's lab in Texas show an interesting departure from conventional wisdom concerning spectra of rotating turbulence: there is evidence for a k^{-2} spectrum. I'll give a brief description of the experimental results and discuss some of the related mathematical issues.
The talk will describe ongoing work in collaboration with a group of design engineers at MIT to develop and apply mathematical and computational techniques as a tool for device design. In particular, I will focus on the optimization of a MEMS relay switch invented by my collaborators. By changing the shapes of the components of both the switch and the electrostatic actuator in a fashion consistent with fabrication constraints, the size of the device can be shrunk substantially. Mathematical challenges and future directions will be described.
For a model equation of Allen-Cahn type, we will show how elementary minimization arguments can be used to find a large variety of mixed equilibrium states.
Friday, May 10 Surfing with wavelets
Monday, May 13 From nonlinear approximation theorems to rate-distortion bounds.
Tuesday, May 14 Smoothness for subdivision schemes.
A common theme in statistical physics studies of dynamical systems out of equilibrium is the appearance of ``aging'', a weak form of ergodicity breakup in a ``thermo-dynamical'' limit, often associated with violation of the fluctuation-dissipation theorem. I shall provide a mathematical perspective via the analysis of three systems: A spherical version of the Sherrington-Kirkpatrick model, Sinai's nearest neighbor random walk in random environment, and a Ginzburg-Landau massless free field. The talk is based on joint works with Ben-Arous, Deuschel, Guionnet and Zeitouni.
I will present joint works with Tristan Riviere / Francois Alouges + Tristan Riviere, on a vector-valued phase-transition problem arising in the theory of micromagnetics. Looking at a singular perturbation-type family of energy functionals, we identify limiting problems which involve planar divergence-free unit-norm vector-fields, with jump singularities (like for real-valued phase-transition models). We introduce "entropy" quantities to prove optimal energy lower bounds and identify explicit optimal transition profiles which are purely two-dimensional (and not one-dimensional) microstructures, called in physics "cross-tie walls".
The purpose of the talk is to give an introduction to determinantal random point fields. Determinantal random point fields appear naturally in random matrix theory, probability theory, quantum mechanics, combinatorics, representation theory and some other areas of mathematics and physics. The first part of the talk will be devoted to the examples. In the second part we will concentrate on the CLT type results for the liner statistics. In particular we will talk about the Costin-Lebowitz theorem for the counting function of particles.
Dopaminergic neurons of the basal ganglia are believed to be important in signaling rewards, and their death is involved in Parkinson's disease. In slice preparations, each such cell is capable of oscillations, and there is reason to suspect that spatial interactions along the dendrites of the cell are responsible for some of the transient behavior of the cells that partially mimics the reward response. This talk describes work with Georgi Medvedev, Charles Wilson and Jay Callaway on the dynamics of the transient response. The model describes a dendrite as a chain of compartments, each capable of oscillating at a frequency that is related to the diameter of the compartment. The mathematics shows that the strong electrical coupling between compartments leads to a slow transient response, and in the limit of strong coupling there is an integral for the system. It also gives insight into the physical processes that produce the transient response.
Stochastic PDEs have become important models for many phenomenon. Nonetheless, many fundamental questions about their behavior remain poorly understood. Often such SPDE contain different processes active at different scales. Not only does such structure give rise to beautiful mathematics and phenomenon, but I submit that it also contains the key to answering many seemingly unrelated questions. Questions such as ergodicity and Mixing. Given a stochastically forced dissipative PDE such as the 2D Navier Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion equation; is the system Ergodic ? If so, at what rate does the system equilibrate ? Is the convergence qualitatively different at different physical scales ? Answers to these an similar questions are basic assumptions of many physical theories such as theories of turbulence. I will try both to convince you why these questions are interesting and explain how to address them. The analysis will suggest strategies to explore other properties of these SPDEs as well as numerical methods. In particular, I will show that the stochastically forced 2D Navier Stokes equations converges exponentially to a unique invariant measure. I will discuss under what minimal conditions one should expect ergodic behavior. The central ideas will be illustrated with a simple model systems. Along the way I will explain how to exploit the different scales in the problem and how to overcome the fact that the problem is an extremely degenerate diffusion on an infinite dimensional function space. The analysis points to a class of operators in between STRICTLY ELLIPTIC and HYPOELLIPTIC operators which I call EFFECTIVELY ELLIPTIC. The techniques use a representation of the process on a finite dimensional space with memory. I will also touch on a novel coupling construction used to prove exponential convergence to equilibrium.
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