**Research at Department of Mathematics**

The Department of Mathematical Sciences at Nazarbayev University aims to develop a strong team in research in mathematical science and encourages interdisciplinary interests.

The Department of Mathematics boasts a variety of research topics in applied and pure mathematics. For more detail on research in Pure and Applied Mathematics, see the links below to individual staff pages.

**Current research areas**

Math Department at NU carries out research in the following areas:

**Finite Element Analysis of Partial Differential Equations, Modeling of Engineering Problems Arising from Nonlinear Mechanics in Advanced Materials, Biomedical Materials, and Engineering (Dongming Wei)**

Professor Wei’s main research areas are in the fields of analysis and numerical solutions of nonlinear differential equations and modeling of engineering problems arising from design of devices of nonlinear structural materials, from modeling of polymer extrusion of non-Newtonian thermal flows, from cancer cells research, and from study of human eyes. The mathematical equations include Navier-Stokes equations for non-Newtonian flows, nonlinear convective heat transfer equations, nonlinear structural dynamical systems in mechanical and civil and biomedical engineering design problems.

**Nonlinear Differential Equations, Dynamical Systems, Chaos, Fractals and Complexity (Anastasios Bountis)**Prof. Bountis’ research has focused in the past on Nonlinear Differential Equations and Chaotic Dynamics particularly in connection with low-dimensional Hamiltonian systems. In parallel he also worked on integrable Hamiltonian systems, as well as more applied questions such as stability of particle beams in high energy accelerators and the analysis of chaotic time series obtained from temperature measurements, electrocardiograms, water spring flow data and other sources. Since the early 2000’s, Prof. Bountis has worked on the dynamics of high dimensional Hamiltonian systems and more recently on their thermostatistics. He is interested in the entropic analysis of these systems where by he has been able to distinguish weak from strong chaos, highlighting the role of long vs. short range interactions. More recently, Prof. Bountis has been studying topics of complex systems, particularly with respect to the solution of nonlinear differential equations describing neuron dynamics in problems of neuroscience. Currently, his research involves problems of higher dimensional maps, dynamics and statistics of Hamiltonian lattices and, more generally, mathematical models of complex systems of physical, biological and engineering interest.

**Generalized families of continuous and discrete distributions (Ayman Alzaatreh)**

Ayman Alzaatreh’s research area focuses on generalized families of continuous and discrete distributions, a branch of statistics that seeks to develop more flexible statistical distributions which can be applied to describe various phenomena. In particular, his research focuses on generalized families of distributions arising from the hazard function. Other interests include univariate and multivariate weighted distributions, statistical inference for probability models and characterization of statistical distributions.

**Harmonic Analysis and its applications to Partial Differential Equations (Alejandro J. Castro)**

Alejandro J. Castro’s research concerns fundamental operators in harmonic analysis (maximal operators, singular integrals, Littlewood-Paley functions, Riesz transforms, multipliers, . . . ) associated with classical orthogonal systems (mainly Bessel, Hermite, Laguerre and Jacobi settings). In particular, he is interested in the vector valued situation, that is, when the previous operators act on functions taking values in a Banach space B. The motivation is to determine relations between the mapping properties of such an operator in certain function spaces (Lp, Hardy, BMO) and geometric properties (UMD, martingale type or cotype, . . . ) of the underlying space B. Currently he is working on the well-posedness (i.e. existence and uniqueness of solutions) of low regularity parabolic boundary value problems and on the initial data problem for the Schrödinger equation.

**Numerical solution of PDE (Yogi Erlangga)**Dr. Erlangga’s research interests are in the numerical solution of partial differential equations, iterative solvers for large systems of equations, and PDE-constrained optimizations, with applications that involve modelling of fluid flows (Navier-Stokes and Euler equations), the indefinite Helmholtz equation, and the biharmonic equations. In the area of iterative methods, he focuses on the deflation-type method, its convergence analysis and applications, which include large scale computational inverse problems.

**Commutative algebra, singularities in positive prime characteristic (Zhibek Kadyrsizova)**

Zhibek Kadyrsizova works in the area of commutative algebra that studies singularities of rings (and hence of the associated geometric objects) using a natural endomorphism that can be defined in positive prime characteristic. It is called a Frobenius endomorphism. Currently, the work is focused on the varieties of nearly commutative matrices, their singularities and invariants.

**Probability, Stochastic Processes, Mathematical Physics, Applied Harmonic Analysis (Eunghyun Lee)**Eunghyun Lee’s current research focuses on the integrable probability. He is studying various stochastic particle systems in the KPZ universality class and their connections to random matrix theory, symmetric function theory, and non-equilibrium statistical mechanics.

**Mathematical logics (Manat Mustafa)**Manat Mustafa is working on Mathematical Logic, Computability theory and Set theory.

**Numerical Linear Algebra, Structured Matrices, and Polynomials (Thomas Mach)**Prof. Mach’s research focus lies on structured preserving and structure exploiting algorithms. Exploiting available structure in matrices often leads to giantic improvements in numerical linear algebra algorithms. Structure preservation on the other hand can be expensive, like in the Hamiltonian eigenvalue problem, but necessary to preserve physical properties. Some years ago, Thomas Mach investigated the solution of boundary element and eigenvalue problems with hierarchical matrices. Recently, Prof. Mach has worked on backward stable algorithm for the unitary and Hamiltonian eigenvalue problem, and for polynomial root finding. In parallel he has studied the inverse eigenvalue problem, adaptive cross approximation, and inverse problems. Currently, his research involves inverse integral equations, and polynomial and structured eigenvalue problems.

**Numerical Analysis (Piotr Skrzypacz)**Piotr is an expert in Numerical Analysis. His research is focused on Finite Elements, Superconvergence, Computational Fluid Dynamics and Control Problems.

**Cryptography (Francesco Sica)**Dr. Sica is interested in the use of computational methods in number theory to approach basic problems of public-key cryptography. In particular, how to use algebraic and geometric properties of elliptic curves to speed up elliptic scalar multiplication. His research also includes understanding some attacks to the discrete logarithm problem on special elliptic curves as well as using analytic methods for diophantine problems such as the factorization of large integers.

**Constraint satisfaction problems and machine learning (Rustem Takhanov)**Rustem Takhanov is interested in computational complexity of constraint satisfaction problems (CSPs) and their soft versions. Historically the first example of an NP-hard problem discovered by Stephen Cook was a famous Boolean satisfiability problem which is a special case of CSP. Along with pointing to its NP-hardness, S. Cook showed that its special case, 2-satisfiability, can be solved via polynomial resolution technique. Rustem Takhanov’s research interests lie in the classification of various fragments of CSPs that can be solved by polynomial algorithms. Specifically, he is interested in a dichotomy hypothesis that claims that for a certain class of CSPs intermediate complexity between P and NP-hard never happens. Another topic of his interest is the supervised learning, especially the structured learning. In this field he works on such topics as the sequence labeling problems (including applications in computational linguistics and bioinformatics), pattern-based conditional random fields, the monotonization of the training set.