- Department of Mathematical Sciences
- Academics
- Graduate Programs
- Faculty Research Interests

Here are brief statements from our faculty about their current research. You also can find out their research interests by reading reviews of their papers on Math Reviews.

**Assistant Professor**Ph.D., IMPA – Numerical Analysis and Optimization.

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

- Algorithms, convex analysis, smooth and nonsmooth optimization problems, projection methods.
- Generalized equations and inclusion problems, variational inequalities.
- Numerical optimization and linear algebra, large-scale problems.

Further details on my research and contributions can be found on my personal homepage.

**Professor**Ph.D., UC Los Angeles – Combinatorics.

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.
- Orthogonal polynomials.
- Some aspects of diophantine approximation.

**Professor**Ph.D., University of Ottawa – Numerical Analysis.

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 on 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."

**Professor**Ph.D., University of Washington – Optimization Theory, Variational Analysis and Applied Functional Analysis.

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

- Sensitivity analysis in parametric optimization.
- Error bounds for inequality systems and their applications.
- Multi-stage stochastic problems.
- Multi-objective (vector) optimization problems.

**Professor**Ph.D., University of Warwick – Complex Analysis.

My research is based in the classical complex analysis that NIU’s graduate students see in MATH 632, but then goes in various different directions from there. One of the best things about doing research in these areas is seeing how analysis, dynamics, geometry and topology can all be interweaved together in various ways.

Recently, my research has largely been in the area of quasiregular dynamics: this is a relatively new generalization of complex dynamics with many interesting unsolved problems still to be worked on. Additionally, I work in Teichmueller theory and on the Decomposition Problem for bi-Lipschitz mappings.

**Professor**Ph.D., The University of Chicago – Finite Group Theory.

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.

**Associate Professor**Ph.D., Kansas State University – Harmonic Analysis.

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.

**Professor**Ph.D., Voronezh State University, Russia – Harmonic Analysis & Operator Theory.

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.

**Associate Professor**Ph.D., University of Waterloo – Numerical Optimization.

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.

**Associate Professor**Ph.D., University of Wisconsin – Partial Differential Equations.

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

Parabolic Equations – these are known as "Diffusion Equations'' which take the form. The areas that I have been working on are Asymptotic Behavior and Regularity of the weak solutions of this type of equation. 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 are different boundary operators of the domain under consideration.

References:

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).

**Associate Professor**Ph.D., Case Western Reserve University – Differential Geometry and Global Analysis.

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.

**Professor**Ph.D., University of Wisconsin – Dynamical Systems and Algebraic Topology.

Much of my work has involved the use of topological methods, particularly those of algebraic topology, to address questions in dynamical systems. One strand of this work has been to expand and apply a generalization of Morse theory known as the Conley index. Another has been to study topological fixedpoint theory using the techniques of Nielsen fixed point theory. A third direction, which is my current focus, is the study of the global structures that arise in the N-body problem in celestial mechanics.

The "N-body problem," which dates back to Sir Isaac Newton's Principia, involves the study of the motion of a number of point-masses moving under their mutual gravitational attraction. While over 350 years old, the N-body problem continues to attract interest from multiple perspectives. Mine has been to study the global structure of the surfaces of constant center of mass, linear momentum, angular momentum and energy, known as the integral manifolds. The structure of these manifolds regulates (to some extent) the dynamics that can occur, and that structure can change as the energy level changes. My current interest is in identifying when those changes occur, and using the tools of algebraic topology, particularly homology theory, to describe the manifolds in between bifurcation values.

Representative publications:

- McCord. The Integral Manifolds of the N-Body Problem. Accepted for publication in Journal of Dynamics and Differential Equations, 65 manuscript pages, February, 2021.
- McCord, C. Pawlowski. A Markov model for assessing ecological stability properties. Ecological Modelling, 220 (2009) 86 – 95.
- McCord. Saari's Conjecture for the Three-Body Problem with Equal Masses. Celestial Mechanics. 89 (2004) 99 – 118.
- McCord, K. Meyer, D. Offin. Are Hamiltonian flows geodesic flows? Transactions of the American Mathematical Society. 355 (2003) 1237 – 1250.
- Cabral, C. McCord. Topology of the integral manifolds of the N-body problem with positive energy. Journal of Dynamics and Differential Equations 14 (2002) 259 - 293.
- McCord, K. Meyer. Integral manifolds in the restricted three-body problem, Ergodic Theory & Dynamical Systems 21 (2001) 885 - 914.
- McCord. Simplicial models for the global dynamics of attractors. Journal of Differential Equations 167 (2000) 316 - 356.

**Professor**Ph.D., University of Waterloo – Differential Equations applied to Biology.

My research interests involve applications of differential equations (ordinary, reaction-diffusion or delay) models to chemical reaction networks. I am interested in the connection between the structure of a chemical network and its properties, such as multistationarity (existence of several steady states) or oscillations. Methods from algebraic geometry are used to analyze the parameter space for the existence of multistationarity or oscillations. I also study the influence of space diffusion or time delays on the capacity of a chemical network for multistationarity 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.

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

**Associate Professor**Ph.D., University of New Hampshire – Algebra and Representation Theory.

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

**Assistant Professor**Ph.D., Virginia Tech – Undergraduate Mathematics Education.

My research focuses on undergraduate education in three main directions:

- Students’ and mathematicians’ conceptual understanding of isomorphism and homomorphism in abstract algebra and sameness more broadly across mathematics.
- Professors’ instructional practices and beliefs underpinning those practices.
- Connections between students’ and professors’ perceptions and understandings.

Further details on my research program and publications can be found on my webpage.

**Associate Professor**Ph.D., University of Wisconsin – Mathematics Education.

My 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 Khoury and I 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.

I 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. My other publications have focused on teachers' pedagogical decisions and classroom assessment practices.

**Associate Professor**Ph.D., Indiana University-Purdue University University Indianapolis – Functional Analysis.

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).

**Professor**Ph.D., University of Colorado – Number Theory.

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.

DeKalb, IL 60115

**Registration or class questions**815-753-6722

**Chair's office**815-753-6780

**Director of Graduate Studies**

Sien Deng, Professor

gradprog@niu.edu

815-753-6765