Bernard Deconinck
Department of Applied Mathematics,
University of Colorado at Boulder, Boulder, CO, 80309-0526
The underlying topic of most of my research is the study of nonlinear wavephenomena. In the past I have been interested in nonlinear waves in fluids, plasmas and lattice systems. I have used mathematical techniques from such different areas as partial differential equations, ordinary differential equations and dynamical systems, asymptotics, soliton theory, algebraic geometry and lie algebras.
In my Ph.D. thesis, which is my most recent research, I solve the initial-value problem for quasiperiodic solutions of the Kadomtsev-Petviashvili (KP) equation. This equation is interesting because it is
(a) of physical importance,
(b) integrable in (2n+1)-dimensions,
(c) intimatelly connected with the theory of Riemann surfaces.

I present two methods of solution; both are exact and constructive. The first method uses algebraic geometry and the theory of Riemann surfaces. The second method constructs finite-dimensional Hamiltonian systems which are completely integrable. A complete set of conserved quantities is constructed explicitly.

For more information, please see my webpage http://amath-www.colorado.edu/appm/student/deconinc/Home.html

Birk Huber
My main research interests include computational aspects of algebraic geometry and computational convexity. In particular, a large component of my research centers on continuation methods for solving polynomial systems. Here my emphasis is on designing homotopies which take advantage of geometric knowledge inherent in a given class of problems. For example, in my thesis work I focussed on the use of combinatorial techniques motivated by toric geometry to set up optimal homotopies for solving sparse polynomial systems. More recent work on continuation includes an investigation of homotopy methods for effective Schubert calculus. Other applications of algebraic geometry that I have worked on include an application of geometric invariant theory to single view recognition.

Charles Yanguang Li
My main interest is in two areas.
1. Chaos in partial differential equations focusing on perturbed soliton systems. I prove the existence of chaos on three canonical equations representing three classes of soliton equations. One is the focusing cubic nonlinear Schrodinger equation representing (1+1)-dimensional soliton equations with hyperbolic structures. Second is the discrete focusing cubic nonlinear Schrodinger equation representing soliton lattices with hyperbolic structures. Third is the Davey -Stewartson II equation representing (1+2)-dimensional (and higher dimensional) soliton equations with hyperbolic structures.
2. Two-dimensional turbulence. Mainly I utilize the machinery developed in my work in area 1 to study and develop new models and new theories on two-dimensional turbulence, and try to understand two-dimensional turbulence from dynamical system point of view. The main equation is the two-dimensional Euler equation in kinetic form (with or without forcing and damping).

Gregorio Malajovich Munoz
Main research interests:
(1) Solution of polynomial equations. This includes algorithms for solving univariate polynomials and systems of polynomial equations. The main classes of algorithms I have been working with are homotopy methods and (more recently) a modern version of Graeffe iteration.
(2) Complexity theory over a ring. NP-completeness theory over the complex or over number fields, and connections with the classical NP-completeness theory.
For more information please see my webpage: www.labma.ufrj.br/~gregorio

Marta Mazzocco
RESEARCH INTERESTS
PAINLEVÉ VI EQUATION
My interest in this field is to study the global analytic properties of the solutions of Painlevé VI equation, a second order nonlinear ordinary differential equation in the complex variable x, depending on four parameters alpha, beta, gamma, delta in C. Its solutions define some new special functions called Painlevé VI transcendents. I try to find all the values of the parameters (alpha, beta, gamma, delta) such that there exist particular solutions y(x;c₁,c₂), determined by the integration constants (c₁,c₂), that can be expressed via classical functions. Let me recall that the six Painlevé equations are new nonlinear second order ordinary differential equations of the type:

The main idea of the work is that if y(x;c₁,c₂) is a branch of a solution of PVI, then its analytic continuation along any closed path gamma avoiding the critical points of the eqution (in PVI case they are 0,1,infinity), is a new branch , with new integration constants . Since, due to the Painlevé property, all the singularities of the solution on are poles, the result of the analytic continuation depends only on the homotopy class of the loop gamma on . As a consequence the structure of the analytic continuation is described by an action of the fundamental group on (c₁,c₂). I study this action. As an application I want to classify all the algebraic solutions of Painlevé VI, by classifying all the finite orbits of this action. The above scheme is completed for a particular case of Painlevé VI equation specified by the following choice of the parameters:
in the complex plane, mu is an arbitrary complex parameter. The obtained results are written in the pepers
Boris Dubrovin and Marta Mazzocco: "Monodromy of certain Painlevé VI transcendents and Reflection Groups", SISSA-preprint n. 149/97/FM (1997).
Marta Mazzocco: ``A brief survey on the algebraic solutions of a particular case of the PVI equation'', quaderni CNR 54, G. Gaeta and D. Bambusi editors (1998). Marta Mazzocco: ``Picard and Chazy solutions for the PVI equation'', SISSA-preprint.
DYNAMICAL SYSTEMS
My interest in this field is within the classical Hamiltonian systems, the Liouville integrability and the classical perturbation theory. In particular, the subject of my research, under the supervision of Prof. Giancarlo Benettin, is KAM theory and its application to the problem of fast rotations of a rigid body, or, equivalently, to a small perturbation of the Euler system. The difficulty is that the Euler system is properly degenerate, and one has to remove the degeneration, in order to apply the KAM theorem. This study is now concluded and the results are published in Z. angew. Math. Phys. 48, (1997).

John McCuan
I am interested in various aspects of geometry and partial differential equations. I also have some interest in certain aspects of materials science and mathematical physics.

Most of my work centers around classical problems of the calculus of variations. In particular, I am interested in problems of least area (and least length). Equilibria of fluids (and fluid membranes) in various circumstances provide the framework for much of my research. As a consequence I am also interested in the adhesion properties of fluids.

Here is a short list of keywords indicative of my interests: Prescribed mean curvature, constant mean curvature, minimal surfaces, adhesion, wetting, elliptic pde, calculus of variations, drops and bubbles, maximum principles, nonlinear boundary conditions, contact angle, capillary surfaces.

Eugene Mukhin.
I am currently interested in applications of Representation Theory to Integrable Systems in statistical mechanics and 1+1 dimensional field theory. Key words: affine Lie algebras, quantum groups, Yangians, form factors, random matrix models, KdV.

Recently, my research was related to the integral formulas of solutions to the quantized Knizhnik-Zamolodchikov equation. Key words: hypergeometric integrals, hypergeometric pairing, quantized conformal blocks, elliptic (dynamical) quantum groups.

Dimitri Shlyakhtenko
Research interests. My research interests center on the "Free probability theory" of Voiculescu, especially to its applications to von Neumann algebras and random matrices. In algebraic approach to probability, a measure space is replaced by a commutative algebra of functions on it, and the measure by a linear functional on this algebra. Then a random variable is just an element of this algebra. The notion of independence of random variables can be formulated in terms of tensor product. In non-commutative probability, one drops the requirement that the algebra be commutative. An example of a "non-commutative probability space" is the algebra generated by a Gaussian random matrix and a fixed deterministic matrix. Voiculescu found that this non-commutative framework allows for a notion of "freeness" of non-commutative random variables, which is different from independence. Amazingly, he showed that in the example above, as the size of a (Gaussian) random matrix increases, it becomes free from the deterministic matrix. He found other applications of these results, such as in the theory of free group factors in von Neumann algebras.

My particular area of interest lies around the theory of Gaussian random band matrices. Unlike the usual GUE, one assumes that the independent Gaussian entries of such a random matrix do not have equal covariances, but that the covariances vary with the position of the element in the matrix. It turns out that there is also a connection between such matrices and free probability theory, which may be useful for deriving limit eigenvalue distrubutions of such matrices. This connection turns out to be extremely useful for free probability applications to von Neumann algebras, where one finds a remarkable class of von Neumann algebras generated by "limits" of such Gaussian random band matrices. The widely open classification problem for these algebras in spirit is analogous the classification problem of Krieger factors arising out of ergodic theory; however, the current methods are quite different, and are motivated by random matrices.

For more information, please see my webpage http://www.math.ucla.edu/~shlyakht

Frank Sottile
I am interested in mathematics related to the interface of algebraic geometry, algebraic combinatorics, and representation theory. This includes studying the geometry and combinatorics of flag manifolds, in particular the Schubert calculus. In combinatorics, I study symmetric functions, Schubert polynomials, Bruhat orders, Young tableaux, quasi-symmetric functions, and enumerative combinatorics on partially ordered sets. In geometry, my work includes the geometry of Schubert varieties and seeking effective, geometric, and elementary derivations of the Schubert calculus. Of particular interest to me are algorithms for solving enumerative problems, the existence of real solutions to enumerative problems, and applications of these ideas. I often do experimentation in computer algebra systems (Macaulay, Maple, Singular) or write routines in C for larger computations.

Irena Swanson
I do commutative algebra and have interests in computational algebra and algebraic geometry. I have worked in tight closure, a theory for equicharacteristic rings due to Melvin Hochster and Craig Huneke which has simplified and extended a lot of results in commutative algebra and algebraic geometry.

My contribution there was a multi-ideal algebraic version of the Briancon-Skoda theorem. An open question in the theory of tight closure is whether it commutes with localization. One approach to solving this is to find some "linear" properties of primary decompositions of Frobenius powers of ideals. In this pursuit I have established several "linear" properties of powers, symbolic powers, and integral closures of powers of ideals, having to do with the Artin-Rees lemma, primary decompositions, Castelnuovo-Mumford regularity, equivalence of topologies, zero divisors, derivations, etc. My collaborators in this area have been Donatella Delfino, William Heinzer, Reinhold Hubl, and Karen Smith.

In addition to linearizing ideal properties, I am also interested in monomial and binomial ideals, adjoints, cores, primary decompositions, and computability of these.

Peter Topping
My research is centred on the theory of nonlinear partial differential equations, with an emphasis on those arising in the Calculus of Variations. I am studying such equations arising in geometric analysis, applied analysis and differential geometry.

Particular areas of specialisation currently include
1) Harmonic maps and their heat flow; bubbling phenomena.
2) Compensation properties of Jacobian determinants
3) Isoperimetric inequalities.
4) Minimal surfaces and mean curvature flow.
5) Willmore surfaces.
6) Fluid dynamics.

Mark van Hoej
Research interests
*) Linear recurrence equations, in particular computation of hypergeometric solutions.
*) Linear differential equations, factorization of linear differential operators, computation of exponential and Liouvillian solutions.
*) Algorithms for algebraic curves, computing a parametrization of a curve with genus 0.
See http://math.fsu.edu/~hoeij/papers.html for my papers on these topics.

Please see my webpage for more information: http://math.fsu.edu/~hoeij/papers.html

Jan Verschelde
"Homotopy Continuation Methods for Solving Polynomial Systems" isthe title of my PhD thesis and still covers my ongoing project. My research interests stem from the study of methods to obtain numerical approximations to all isolated solutions of systems of polynomial equations.

Tracking solution paths defined by a parametric family of systems is done by standard predictor-corrector methods. These methods slow down when approaching singularities, which in this setting occur at the end of the paths, most often at infinity. Producing certificates for non-isolated singularities is challenging to the numerical path-following techniques.

Solving polynomial systems is one of the main problems in the field of Computational Algebraic Geometry. Here symbolic and numerical computations arise in a natural combination. Counting the roots can be regarded as a symbolic operation, whereas in general numerical techniques must be used for approximation. The root counts are exact for certain generic classes of polynomial systems and lead to an optimal number of solution paths. The exploitation of symmetry is one of my research topics.

Newton polytopes provide a fruitful combinatorial and geometric model to exploit sparsity in solving polynomial systems. My research is aimed at the construction of triangulations of Newton polytopes that ensure a numerically stable path-following. To understand the structure of regular subdivisions better I am studying secondary polytopes.

Polynomial systems are ubiquitous in various fields of science and engineering. The creation of scientific software in this area struggles with complexity issues that are intrinsic to the hardness of the problem. One of my interminable jobs is the development of the software package PHC.

Divakar Viswanath
My current interest is focused on infinite matrix products --- that is products P_n=M_n... M₁ where the M_i are d*d matrices generated according to some rule --- and their Lyapunov exponents. Roughly, the d Lyapunov exponents give the asymptotic exponential rates of increase of the d singular values of the product P_n with n. This is a very broad framework; sensitive dependence on initial conditions of chaotic phenomena, stability of stochastic and deterministic differential equations, asymptotic behaviour of random matrix products can all be expressed in the framework. My aim is to take an approach that mixes mathematics with computer calculations, algorithms and numerical analysis.

A memorable result of my research is about random Fibonacci sequences. Random Fibonacci sequences are defined by the random recurrence $t₁ = t₂ = 1 and . I have proven that

At the numerical end, I have worked on and am interested in the accuracy and reliability of Lyapunov exponents computed for chaotic dynamical systems. In a recent paper, I have shown with an example that numerical discretizations can introduce transients that are easily mistaken for asymptotic chaos, and offered some suggestions for avoiding this pitfall.