Mathematical Physics Seminar
Rutgers University
Hill Center, Room 705

April Schedule

Organized by: Joel L. Lebowitz

Speaker:M. Douglas, Rutgers University
Title:"Extra Dimensions - Beyond Kaluza and Klein"
Time/Place:4/1/04, 11:30am, Hill 705
Abstract:The old idea that there are extra dimensions of space, not apparent to everyday observations, has gradually evolved from outlandish speculation, to become one of the foundations of present-day thinking in fundamental physics. We trace the developments which led to this, from theoretical motivations in supergravity and superstring theory, through the geometry of extra dimensions, to possible experimental tests of the idea.

There will be a brown bag lunch in between the seminars
Please bring your lunch
Coffee & cookies will be available

Speaker:H. Tasaki, Gakushuin University
Title:"Steady State Thermodynamics - Towards a --Universal Phenomenological Framework for Nonequilibrium Steady States"
Time/Place:4/1/04, 1:30pm, Hill 705
Abstract:I will describe a recent attempt with Shin-ichi Sasa (University of Tokyo), where we try to construct a thermodynamics which applies to nonequilibrium steady states. Our basic strategy is to examine the general structures (e.g., additivity, scaling) of thermodynamics carefully, and to define all the quantities using (experimentally realizable) operational procedures. We then make some predictions (including the existence of a new osmotic pressure caused by nonequilibrium flow) which enable one to check for the quantitative validity of our theory.

Speaker:Y. Li, Rutgers University
Title:"The Method of Moving Planes and Some Applications"
Time/Place:4/8/04, 11:30am, Hill 705
Abstract:This will be an expository talk. I will first introduce the method of moving planes and then describe a number of applications. The applications include a joint work with Aobing Li which extends classical Liouville type theorems of Gidas-Ni-Nirenberg and Caffarelli-Gidas-Spruck to all conformally invariant fully nonlinear elliptic equations of second order.

There will be a brown bag lunch in between the seminars
Please bring your lunch
Coffee & cookies will be available

Speaker:M.N. Hounkonnou, University of Abomey-Calavi
Title:"New Families of Orthogonal Polynomials"
Time/Place:4/8/04, 1:30pm, Hill 705
Abstract: Please see attachment.

Speaker:A. Sengupta, Rutgers University
Title: "Epigenetic Switching in Genetic Networks"
Time/Place:4/15/04, 11:30am, Hill 705
Abstract:Different developmental states of a single cell organism could be thought of as multiple fixed points of non-linear genetic networks. Stochasticity of reactions however allows transitions between such states. We employ tools for calculating rates of rare events (instantons for physicists/ optimal path methods for queueing theorists) to analyze stability of genetic states in lambda phage and in a synthetic biochemical switch. These methods are obviously superior to direct simulation. We also point out when we need to go beyond the Langevin approximation to noise in this context.

There will be a brown bag lunch in between the seminars
Please bring your lunch
Coffee & cookies will be available

Speaker:D. Vanderbilt, Rutgers University
Title:"Electronic Structure of an Insulator in a Finite Electric Field: What to do When There is No Ground State"
Time/Place:4/15/04, 1:30pm, Hill 705
Abstract:I will discuss two related problems: (i) how to compute the electric polarization of a (non-centrosymmetric) crystalline insulator, even in zero electric field; and (ii) how to compute the properties of a crystalline insulator in a finite, homogeneous electric field. (The discussion will be in the context of Kohn-Sham density-functional theory, which provides a now-standard approach for the quantum-mechanical calculation of the electronic and structural properties of not-too-strongly correlated solids.) It might be thought that both of these problems should have standard textbook solutions, but in fact it is only in the last few years that these two problems have been adequately resolved. The problem of the electric field is quite subtle, for example, because the electric field acts as a singular perturbation: even for a small field, the Hamiltonian eigenstates lose their Bloch symmetry, the potential energy for the electrons is not bounded from below, and there is no ground state. I will review recent developments in this field, showing how a Berry phase figures prominently in the solution to (i), and explaining how the solution to (i) also provides the solution to (ii). Examples of applications to semiconductors and ferroelectrics will briefly be presented.

Speaker:J. Beck, Rutgers University
Title: "Limitations to Regularity"
Time/Place:4/22/04, 11:30am, Hill 705
Abstract: Does there exist a "perfectly balanced" 2-coloring of the lattice points on the plane? Well, the chessboard type alternating 2-coloring is certainly perfectly balanced with respect to Axis-Parallel Rectangles: the number of red points inside differs from the number of blue points inside by 0 or 1 depending on the parity. That was easy, but what happens if we switch from Axis-Parallel Rectangles to Circles? What is the "most balanced" 2-coloring with respect to Circles? How about if we fix the radius? How about "one-sided irregularity", that is, when can we guarantee (say) "at least 100 more red points than blue points" inside a Circle of fixed radius? The objective of my talk is to answer these hard questions.

There will be a brown bag lunch in between the seminars
Please bring your lunch
Coffee & cookies will be available

Speaker:V. Retakh, Rutgers University
Title:"A Possible Approach to Noncommutative and Multi-Dimensional Determinantal Point Processes"
Time/Place:4/22/04, 1:30pm, Hill 705
Abstract:We will discuss a possible approach to noncommutative and multi-dimensional determinantal point processes based on a theory of noncommutative determinants by Gelfand and Retakh and multi-dimensional determinants by Gelfand-Kapranov-Zelevinsky

Speaker:E. Allender, Rutgers University
Title:"Algorithmic Randomness and Derandomization"
Time/Place:4/29/04, 11:30am, Hill 705
Abstract: Kolmogorov complexity is a tool to measure the information content of strings. Strings with high Kolmogorov complexity are said to be "K-random".

The study of this notion of randomness has a long history and it has close connections with the theory of computability. The set of K-random strings has long been known to be undecidable. Derandomization is a fairly recent topic in complexity theory, providing techniques whereby probabilistic algorithms can be simulated efficiently by deterministic algorithms. In this talk, I will present some new and surprising (or bizarre?) connections between these fields. In particular, we will show that everything PSPACE is poly-time reducible to the set of K-random strings, and we investigate the question of whether or not PSPACE is PRECISELY the set of decidable sets poly-time reducible to the K-random strings.

Using related techniques, we will shed new light on the "Minimal Circuit Size Problem" - a problem in NP that seems to be intractible but does not seem to be NP-complete. This problem is very closely related to a type of time-bounded K-complexity. We will show that many cryptographic problems (such as factoring, discrete log, and various lattice-related cryptographic problems) are reducible to any approximate solution of the Minimal Circuit Size Problem.

Some of this material is from the FOCS 2002 paper "Power from Random Strings" and some is from a more recent paper "What is Efficiently Reducible to the K-Random Strings".

There will be a brown bag lunch in between the seminars
Please bring your lunch
Coffee & cookies will be available

Speaker:R. Marra, Universita degli Studi di L'Aquila, Italy
Title:"Interface Dynamics in Kinetic Systems"
Time/Place: 4/29/04, 1:30pm, Hill 705
Abstract: We consider kinetic models describing two species of particles interacting via a long range repulsive potential and a) with a reservoir at fixed temperature, b) by collisions. The dynamics for the first model conserves the total masses of the two species and its sharp interface limit is described by a kind of Mullins-Sekerka motion. The second dynamics models the behaviour of a binary fluid and conserves masses, momentum and energy. In the sharp interface limit in this case the velocity field satisfies the incompressible Navier-Stokes equations together with a jump boundary condition for the pressure across the interface which, in turn, moves with a velocity given by the normal component of the velocity field.

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