Panagiota Daskalopoulos:
Nonlinear Diffusion Equations and Related Elliptic Problems
Summary:
This project involves the study of certain Nonlinear Diffusion Equations and related Elliptic Problems. The equations we propose to study have both physical and theoretical significance. For example, from the physical point of view, they arise as models for the dynamics of thin liquid films and also as models for the limiting density distribution in the kinetics of two gases obeying Boltzmann equation. From the mathematical point of view, they exhibit a qualitative behavior that has remarkable features, different than in previously studied cases. Their investigation leads to the development of new techniques of mathematical analysis. Another attractive feature of the proposed problems is their connection to Differential Geometry, as they arise in the "so called" Ricci flow, which has recently gained remarkable attention.
Paul Eklof
His research has generally dealt with applications of mathematical logic, specifically model theory and set theory, to algebra, specifically abelian group theory, module theory, and homological algebra. This has included the model theory of modules as well as properties of abelian groups expressible in various extensions of first order logic. In recent years, the main focus has been on algebraic results which are provably unsolvablein ordinary set theory. A major monograph on this subject, co-authored with Alan Mekler, was published in 1990.
Matthew Foreman
Problems in descriptive set theory, ergodic theory and set theory
Summary:
In this project Foreman proposes to work on two main types of problems. The first type of problem deals with applications of descriptive set theory to ergodic theory.
In the early twentiieth century, ergodic theory arose as a technique for studying complicated differential equations by looking at the statistical behavior of the associated flows. Dynamical systems arising in physics with otherwise intractible behavior were amendable to thsi kind of analysis. A prominent problem in ergodic theory deals with the extent that general statistical behavior of measure preserving systems models the behavior of flows on compact C<> manifolds with smooth densities. A precise statement of this problem is to ask whether evry flow of finite entropy can be realized as a smooth flowon a compact C<> manifold.
In recent papers Foreman has constructed two new classes of zero entropy flows. Foreman proposes to study the problem of realizability of flows as differentiable flows using techniques developed in thesepapers together with techniques developed recently by Kechris et. al. for studying equivalence relations determined by Polish group actions.
The second family of problems Foreman proposes to study are those related to the theory of saturated ideals and their consequences. Foreman proposes to extend his work that yielded the consistency of a countably complete, uniform, <>-dense ideal on <> to similar constructions that should yield <>-dense idealson <> for each natural number n. He hopes that he will be able to find more applications of these ideals to combinatorial problems on the <>'s such as the problem of whether it is consistent for all <>.
Michael D. Fried
Goals as a mathematician: Mathematics has a language for breaking tough problems into easier pieces. Trying to solve complicated equations reveals practical aspects of a theoretical difficulty. Rarely can one solve equations in one important sense. Though solutions may exist, they have no presentation displaying their properties from known functions. Rather than solutions, however, most scientists want properties of solutions. Fried uses group representation theory to avoid solving equations. This is the monodromy method. It often reveals symmetries connecting problems from one area to research tools from another. This has solved some renown problems.
*Schur's conjecture *Davenport's problem *The Galois stratification procedure for the theory of finite fields *Carlitz's conjecture *An enhancement of Shafarevich's conjecture to present the absolute Galois group of the rationals as an extension of known groups *Diophantine reduction of the Inverse Galois problem
Results from the Monodromy Method
Summary:
We describe some unsolved problems related to successes we've had
with the monodromy method.
The Successes include the following.
(i) We proved Carlitz's conjecture on exceptional polynomials over
finite fields [FGS].
(ii) We interpreted embedding problems over arbitrary fields using
rational points on decorated versionsof Hurwitz spaces [FV1].
Item (ii) led to proofs of analogs of Shafarevich's conjecture and tro
the first sytematic realizations of Chevalley groups over non-prime
finite fields as Galois groups ([FV2], [FV3], [V1] and [V2]).
The monodromy method takes the following form. A precise
statement S about diophantine equations translates to this: Rational
points on certain varieties produce a desired Galois theory situation.
Byforgetting where the Galois theory arises, we get a group theory
situation.
The method applies group theory to solve <> It then eliminates
geometric situations in the list of <> that don't correspond to what
remains from <>. This leaves the harder problem of deciding if what
geometricallyremains produces the arthmetic situation under
investigation.
Often, uneliminated geometric situations appear classically in
mathematics. We outline a programto give deeper insight for this
direction-Group Theory to Number Theory-by applying it to one
example. This is the relation between involution realizations of
dihedral groups and torsion points on abelian varieties.
Svetlana Jitomirskaya
Singular continuous spectrum and localization type effects in disordered systems
Summary:
Singular continuous spectrum of Schrodinger operators.
Localization type effects for Schrodinger operators and for the quantum spin systems, percolation and contact processes in disordered environments.
The proposed research can be divided as:
1. Schrodinger operators.
Singular continuous spectrum and wave functions.
-Stability and critical growth.
-Hausdorff dimension.
-Anderson model on the Bethe lattice.
-Almost Mathieu operator.
Localization.
-Uniform, semiuniform and dynamical localization.
-Almost Mathieu operator: semiunifromity; absolutely continuous
spectrum for all rotation numbers.
-Schrodinger operators with other quasiperiodic potentials.
-Kicked rotator and related models.
-Schrodinger operators with hyperbolic disorder.
Symmetries for multidimensional quasiperiodic operators.
2. Quantum spin systems, percolation and contact processes in disordered environments.
-Systems with hyperbolic disorder.
-Intermediate types of disorder.
-Disordered array of quantum rotators.
Abel Klein
The main subject of Klein's research is the study of phenomena that occur in disordered systems. Impurities can change dramatically the properties of certain materials. The best known examples are transistors, which are made of certain materials whose properties are modified by impurities. Such materials are modeled by disordered systems. Klein's research include disordered systems in Quantum Mechanics and in Classical and Quantum Statistical Mechanics. Topics being investigated are: *Random Schrodinger operators *Classical spin systems in a random environment *Disordered quantum spin models *Percolation and contact processes in random environments.
Peter Li
Primary research interest is in geometric analysis. In particular, the application of partial differential equations to the study of geometry. Topics of current research include eigenvalue problems on compact Riemannian manifolds, harmonic functions and the heat equation on complete manifolds, harmonic mappings between complete manifolds, analysis on singular algebraic varieties, and the prescribed curvature equation.
Bernard Russo
Early research interests were in algebras of operators on Hilbert space (von Neumann algebras) and their applications to abstract harmonic analysis (group representation theory) and operator theory. In the last 15 years Jordan operator algebras and triples and their connections with several complex variables and quantum mechanics have been the focus of study. Current main interest is in classical harmonic analysis and its application to operator theory in function spaces of several complex variables.
Ronald Stern
Professor Stern's research is in the general area of geometry and topology with special interest in problems in dimensions 3 and 4. His current research focuses on the structure of 4-dimensional manifolds, i.e. objects which are locally modelled on space-time.
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