Research interests of the faculty

Here are brief statements from our faculty about their current research. You also can find out their research interests from Math Review

I am interested in the qualitative behavior of solutions to non-linear partial differential equations, more particularly in establishing the existence of singular solutions and providing a precise description of the singularities.

My main interest is in parabolic and elliptic equations with secondary interest in hyperbolic equations. Recently I have been focusing on mathematical problems arising from fluid mechanics.

Selected Publications:

- Stability result for the inverse transmissivity problem. Journal of Math Analysis and Applications, vol. 168, 1992, 13-27.
- Phenomenological Behavior of Multipolar Viscous Fluids. (with F. Bloom and J. Necas). Quarterly of Applied Mathematics, vol. L, number 3, 1992, 559-583.
- On some singular solutions of the equation $\DELTA u = -\lambda
e
^{u}$. Commun. in Partial Differential Equations, 15(4), 525-539(1990).

I am a mathematical biologist focusing on cryobiology, the study of life at low temperatures. Cryobiology presents a natural setting for mathematical problems involving heat and mass transfer in biological systems, often coupled to external environments. My contributions involve parameter identification, model formulation, and optimization of these models.

My current research is on the analytic and numeric optimal control of the heat and mass transfer problems intrinsic to cryobiology. I am also involved with the construction of thermodynamically accurate (and thus nonlinear) models of the interaction between cells and tissues and highly non-ideal envirionments. These problems couple classic free-boundary problems from phase field theory with solution theory along with considerations generated by cell-media interactions.

My area has proceeded from the representation theory of finite groups to a more general field of algebraic combinatorics focused on algebras with a distinguished basis. These algebras are variously called hypergroups, character algebras, and table algebras, among other names. Applications range from geometric symmetry via group algebras to combinatorial symmetry via adjacency algebras of association schemes to symmetries of quantum physics via the fusion rings that underlie fusion categories. Some specific projects of mine have examined (1) the generalization of Sylow theory from finite groups to table algebras; (2) the classification of integral table algebras and association schemes that are generated by a basis element of small degree; (3) classification of integral table algebras from data on representation-theoretic invariants such as multiplicities; (4) decomposition of products in nilpotent table algebras, generalizing results for conjugacy classes and irreducible characters of nilpotent groups.

References:

- Sylow theory for table algebras, fusion rule algebras, and hypergroups, with P.-H. Zieschang, J. Algebra 273 (2004), 551-570.
- Integral table algebras and Bose-Mesner algebras with a faithful nonreal element of degree three, J. Algebra 231 (2000), 484-545.
- Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal, J. Algebraic Combin. 31 (2010), 491-499.
- Decomposition of products in nilpotent table algebras, J. Algebra 323 (2010), 1581-1592.
- Table algebras, European Journal of Combinatorics 30 (2009), 1426-1455.

My specialty is computational number theory. Although I have stayed away (somewhat) from the well-worn path of primarily testing and factorization, I have used the computer to investigate several interesting combinatorial and number theoretic problems.

Over the last ten years I have worked with John Brillhart, from the University of Arizona and Irving Gerst, from S.U.N.Y. at Stony Brook (Prof. Gerst passed away in 1996) on infinite product identities, especially those involving the Jacobi triple product functions. These functions have very nice product and series representations. Although no one knows an easy way to tell the parity of the partition function, we have found two infinite classes of restricted partitions whose parity can be immediately determined. The three of us were funded by the N.S.F. for two years to study these functions via computer. Recently we have discovered combinatorial mappings we hope will lead to symbolic computer proofs and ultimately a classification of these interesting identities.

A few years ago Paul Erdos introduced the concept of d-completeness. A
sequence of positive integers u_{1}, u_{2}, ...
is said to be d-complete if
every sufficiently large integer n can be represented as a sum of the
u_{i} with the condition that no summand divides another. The sequence
{2^{a}3^{b}} is d-complete and it
is the only such sequence of that form.
Recently J. L. Selfridge and I settled one of Erdos' last conjectures
concerning the growth of the smallest term needed in these
representations. Selfridge and I were working on a joint project with
Erdos involving cluster primes. These are prime numbers p with the
property that every even number less than p-3 can be
written as a difference of
primes no larger than p itself. Computationally, the cluster primes
appear to have the same density as the twin primes.

My research centers around using combinatorics and analysis for attacking problems in special functions and, in particular, number theory.

The topics which interest me most are continued fractions,
multiple polylogarithms (and their special values), *q*-series,
and orthogonal polynomials. I am also interested in some aspects
of diophantine approximation.

My research is interdisciplinary in nature, blending theoretical and applied and computational linear algebra with application areas such as control and systems theory.

The control theory is a major source of beautiful linear algebra problems. The design and analysis of linear control systems give rise to well-known linear algebra problems such as Eigenvalue and Eigen-Structure Assignment Problems, Frequency Response Problems, Controllability and Observability Problems, Matrix Equations Problems, Stability and Inertia Problems, etc. The development of numerically effective algorithms for these problems, especially algorithms for Large Problems and those suitable for implementation on existing vector and parallel machines are of utmost importance. Numerical algorithms for control problems are still in their infancy. The control theory is lagging behind in this respect compared to other areas of applied sciences and engineering. Yet, there are control problems which are so large that they can be termed as ``Super Computer Problems''. An outstanding example is that of Large Space Structures (LSS).

My current research centers around understanding and analysis of the existing algorithms and developing new numerically viable algorithms, both sequential and parallel, for linear algebra problems in control. A particular attention is being given to the development of algorithms for large-scale solutions of problems arising especially from second order differential equations associated with large space structure problems. In the design of parallel algorithms for control problems, we make use of the existing sophisticated parallel algorithms for matrix computations and the associated software libraries presently being built both for distributed and shared-memory computers such as CRAY XMP and Hypercubes. My research addresses the urgent need clearly pointed out in the recent NSF panel report on ``FUTURE DIRECTIONS IN CONTROL THEORY.''

My research focuses on Mathematics teaching and learning at the undergraduate level. My previous studies investigated problem solving among calculus students and the development of defining and proving among advanced calculus (analysis) students.

I am particularly interested in the transition to advanced mathematical thinking, classroom communication, and the emergence of communal mathematical meaning.

My research interests are in optimization along with related areas of nonsmooth analysis. Currently I am working on

- sensitivity analysis in parametric optimization;
- error bounds for convex systems and their applications;
- multi-stage stochastic problems and multi-objective (vector) optimization problems.

My main area of research is in Reliability and Life Testing. My recent interest is to model the lifetime of a system through several stochastic processes and use the model to estimate its reliability.

My research can broadly be described as complex analysis with two main areas of study, both of which feature centrally the notion of quasiconformal mappings:

- Quasiregular dynamics is a natural generalization of the iteration theory of holomorphic mappings in the plane to higher dimensions. This is a relatively new area of research which shows plenty of promise in being fertile for years to come.

- Teichmueller theory is a beautiful subject bringing together complex analysis, topology, geometry and more to describe deformation spaces of surfaces. This is a deep, challenging subject.

Further details on my research and contributions to these areas can be found on my publications page here

I study representations of finite groups and am particularly interested in relationships between rationality questions and some classical conjectures in block theory such as Brauer's height zero conjecture, Brauer's k(B) conjecture, and McKay's conjecture.

These conjectures are usually studied in the context of particular families of simple or quasisimple groups (usually of Lie type). My approach differs substantially in that I attempt to draw as much as possible from the theory of vertices and sources. The techniques come down to integral representations of p-groups.

I work in Abstract Harmonic Analysis. This involved the analysis of functions, measures, and related structures in the setting of locally compact groups. My work often involves substantial amounts of functional analysis, as well as topology.

My specific interests concern generalizing certain relatively well-known results from classical Fourier analysis to the setting of locally compact abelian groups.

**BERNARD HARRIS** (Professor) – Ph.D., University College, Cardiff.
**Differential Equations. **

Ordinary differential equations and asymptotic analysis. In particular linear equations and questions relating to the spectral theory. I also work on questions involving the qualitative theory of ordinary differential equations.

My research interest is in complex matrices. Currently, I am working on the properties of nonsingular matrices under $\Phi$-maps, i.e., equivalences, canonical form, invariances, spectral structures, generalized singular or polar decomposition under a $\Phi$-map, and so on, Rank revealing QR factorization, and other problems in (numerical) matrix theory.

Recent Publications:

- A characterization of unitary congruence, Linear and Multilinear Algebra, 25 (1989), 105-119.
- The Jordan Canonical form of a product of a Hermitian and a positive semidefinite matrix (to appear, Linear Algebra and its Applications).
- A canonical form under $\phi$-equivalence (to appear, Linear Algebra and its Applications).

The focus of my mathematics education research has been on investigating students' mathematical learning with the intention of attempting to improve mathematical instruction. In specific, my research emphasis has been on understanding students' intellectual and constructive mathematical development. I have worked with students, in groups and individually, and I have observed and analyzed the reasoning strategies students apply in specific mathematical problem-solving situations, as well as the connections they form with their mathematical and othe domains of knowledge.

In my research, I have investigated the mathematical reasoning strategies which students use as they solve problems related to any of the following multiplicative-structured concepts: place-value, rational numbers and proportions, equation-solving, functions, probability, correlations, and conditional logic. I have worked with students of various ages: elementary school students, secondary school students, and college level students including preservice school teachers. In my research, I have applied both quantitative and qualitative research methodologies depending upon the questions that were investigated. Currently, along with the identification and analyses of students' developmental reasoning and problem-solving strategies, I am also investigating students' structures of conceptual units and their flexibility of reasoning during problem solving.

My research interests are mainly in the qualitative analysis of differential equations and its applications which include:

- Oscillation and nonoscillation properties of ordinary and functional differential equations, integral equations, and difference equations.
- Sturm-Liouville problems.
- Nonlinear boundary value problems of ordinary and functional differential equations.
- Dynamic equations on time scales.
- Differential and integral inequalities and their applications to differential equations.
- Stability and persistence problems.

My research interests cover a wide range that includes spectral theory of linear operators and linear relations as well as abstract and applied harmonic analysis. I am investigating problems in frame theory, wavelet and time-frequency analysis, sampling theory, causal pseudo-differential operators, etc. A usually up-to-date list of my publications and preprints is available on my home page.

My research is in the area of continuous and combinatorial optimization with a focus on:

- convex analysis; semidefinite optimization; semidefinite relaxations of hard combinatorial problems;

- sparse and low-rank optimization, matrix completion, and matrix approximation (least-squares) problems;

- Euclidean distance matrices and applications:

- wireless sensor network localization;

- protein structure determination;

- numerical optimization and linear algebra; large-scale computation.

My field of research is in Non-linear Partial Differential Equations. It can be mainly divided into 2 parts:

- Parabolic Equations – these are known as ``Diffusion Equations'' which take the form . The areas which I have been working on are: Asymptotic Behavior and Regularity of the weak solutions of this type of equation. Basically, the former one is concerning the behavior of the solutions when the time variable for some critical time. The latter one consists of investigating the smoothness (i.e. whether it is in or ) of the weak solutions as in general, there are no classical solutions for non-linear equations.
- Non-linear Elliptic Equations – I have been working on some 4th order Non-linear Elliptic boundary value problems with Professor Chaitan Gupta (and with Professor Nečas) in which we investigate the existence of weak solutions for 4th order elliptic problems of the form where is the bi-harmonic operator and and are different boundary operators of the domain under consideration.

- Interior and Boundary Regularity of the plasma type equation with homogeneous boundary condition and non-negative initial data. Proc. AMS. vol. 104, No. 2, 1988 (pp. 472-478).
- Asymptotic Behavior of the Plasma Equation. Applicable Analysis, Vol. 28, No. 2, 1988 (pp. 95-113).
- Asymptotic Behavior of Plasma Type Equations with finite extinction, Arch. Rat. Mechanics Analysis Vol. 104, no. 3, 1988 (pp. 277-294).

My current research interests are related to problems regarding periodic extremals. Many lasting non-chaotic physical phenomena can be viewed as extremals of this kind. By the use of essentially infinite dimensional methods it is possible to represent all periodic functions as a nonflat subset of the space of all functions. The extremals appear at points where the projected gradient vector field vanishes. In order to find these elusive extremals one attempts to follow the trajectories of the gradient vector field.

My methods have proven to be very useful when applied to so-called nonlinear splines in approximation theory. The techniques used involve differential geometry, global analysis, calculus of variations and optimal control. I also use Sobolev spaces, convexity, tensor analysis, numerical analysis, Mathematica, the C-language, computer graphics (real time and animated) and occasionally theories of physics. In the future some of this work will lead to computer implementations of new algorithms.

A good reference is my paper ``Curve straightening'' which appeared in ``Proceedings of Symposia in Pure Mathematics'' by the American Mathematical Society. This volume covers the AMS summer research institute in differential geometry at UCLA and it gives the state of the art as of 1990.

Applications of systems of differential equations (ordinary, reaction- diffusion or delay) to (bio)chemical reaction network models. Specifically we analyze the connection between the structure of a biochemical network and its biological properties, such as multistability and oscillations. Also, we study the influence of space diffusion or time delays on the capacity of a biochemical network for multistability or oscillations.

Representative publications:

M.Mincheva, G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proc. of the IEEE 96, 1281-1291, 2008.

M. Mincheva, M. Roussel, Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in mass-action kinetics models, J. Math. Biol. 55, 61-86, 2007.

M. Mincheva, M. Roussel, Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays, J. Math. Biol. 55, 87-104, 2007.

M. Mincheva, M. Roussel, A graph-theoretic method for detecting Turing bifurcations, J. Chem. Phys. 125, 204102, 2006.

My research interests include tensor categories and Hopf algebras. I also study the structure of algebras, specifically their Hochschild cohomology and their associated deformations.

My research interests are in the general area of nonparametric Statistics. Specifically, my research focuses on the following areas:

- Smoothing methods for distribution functions
- Nonparametric bootstrap confidence regions
- Smoothing methods for the bootstrap
- Semi-parametric transformation theory
- Randomization and permutation testing
- Nonparametric multivariate data depth theory
- Nonparametric applications in quality control

Professor Shafer's current research in mathematics education is studying the impact of the Master of Science in Teaching specialization in Middle School Mathematics Education on teachers' knowledge of mathematics, teaching practices, and professional growth and the impact on their students' mathematics achievement and mathematical dispositions. This research is supported by the Mathematics and Science Partnership grant Excellence in the Middle: Enhancing Mathematics Pedagogy with Connections in Science and Engineering, funded by U.S. Department of Education, NCLB, Title II, Part B, through the Illinois State Board of Education. Professors Shafer and Khoury are co-directors of this grant, which is currently supporting 32 teachers from several high-needs Illinois school districts to complete graduate coursework toward the Master of Science in Teaching specialization in Middle School Mathematics Education. Professor Shafer co-authored a book with Thomas Romberg entitled The Impact of Reform Mathematics Instruction on Student Achievement: An Example of Standards-Based Curriculum Research, which was published in 2008. Her other publications have focused on teachers' pedagogical decisions and classroom assessment practices.

I am interested in the following areas of Functional analysis and Operator theory.

* Geometry of Banach spaces (Hereditarily Indecomposable Banach spaces; Operators with and without invariant subspaces; Daugavet equation);

* Positive operators in Banach lattices (Invariant subspace problem for positive operators; Compact-friendly operators).

Present activity:

- Combinatorial problems in semigroup theory: similar to the study of groups given by generators and relations (see [Lal]).
- The Burnside problem for semigroups (see [Lal]).
- Rational subsets of groups (see [Ber]).
- Various categories of graphs.
- Lal - Lallemant, G., Semigroups and Combinatorial applications.
- Ber - Berstel, J., Transductions and Context - Free Languages.

My research interests are in Diophantine equations, Diophantine
approximation, arithmetic geometry, and the geometry of numbers. The
study of Diophantine equations deals with finding integral or rational
solutions to polynomial equations. This is closely related to
Diophantine approximation, where one studies approximations to real
numbers by rational numbers. Arithmetic geometry, broadly speaking,
deals with arithmetic properties (i.e., properties concerning the
integers) of geometric objects, usually affine or projective varieties.
This is an area where number theory and algebraic geometry come
together. The geometry of numbers deals with points with integer
coordinates in regions of real *n*-space (under what condition will a
region have such a point? how many? etc...), sphere packing, and
related subjects.

I work mainly in algebraic number theory at a relatively elementary level, dealing with problems of classnumbers and units in algebraic number fields. I also make an occasional sideline into diphantine equations, that is, equations in which integer solutions are required. Some recent references are the following:

- Bounding squares in second order recurrence sequences, Acta Arith. 54, 1989, 127-145.
- Bounding a unit index in terms of a ring index, Mathematika 442, 1995, 199-205.
- Algebraic integers of small discriminant, Acta Arith. 75, 1996, 375-382.

Motivated by P. Vojta's dictionary of number-theory-Nevanlinna-theory and related S. Lang's questions, my current research interests are finding analogies of Diophantine approximation in Nevanlinna theory of meromorphic mappings between two complex manifolds.