Mathematics – (Ph.D.)

The department of Mathematics offers graduate courses leading to Ph.D. degree in Mathematics. The department emphasizes both pure and applied mathematics. Research in the department covers algebra, algebraic geometry, number theory, functional analysis, differential geometry, differential equations, combinatorics, topology, biomathematics, statistics, probability, stochastic analysis and mathematical physics. In addition to the following courses, students in this program can take any of the courses listed under the “M.S. in Mathematics” program or from other courses not listed here in accordance with their areas of specialization and subject to the approval of their advisors.

Research Areas
Number Theory

* Ring Theory and Module Theory, especially Krull dimension, torsion theories, and localization
* Algebraic Theory of Lattices, especially their dimensions (Krull, Goldie, Gabriel, etc.) with applications to Grothendieck categories and module categories equipped with torsion theories
* Field Theory, especially Galois Theory, Cogalois Theory, and Galois cohomology
* Algebraic Number Theory, especially rings of algebraic integers
* Iwasawa Theory of Galois representations and their deformations, Euler and Kolyvagin systems, Equivariant Tamagawa Number Conjecture

Combinatorics

* Combinatorial design theory, in particular metamorphosis of designs, perfect hexagon triple systems
* Graph theory, in particular number of cycles in 2-factorizations of complete graphs
* Coding theory, especially relation of designs to codes
* Random graphs, in particular, random proximity catch graphs and digraphs

Differential Equations

* Partial Differential Equations
* Nonlinear Problems of Mathematical Physics
* Dissipative Dynamical Systems
* Scattering of classical and quantum waves
* Wavelet analysis
* Molecular dynamics

Analysis

* Banach algebras, especially the structure of the second Arens duals of Banach algebras
* Abstract Harmonic Analysis, especially the Fourier and Fourier-Stieltjes algebras associated to a locally compact group
* Geometry of Banach spaces, especially vector measures, spaces of vector valued continuous functions, fixed point theory, isomorphic properties of Banach spaces

Mathematical Physics

* Differential geometric, topologic, and algebraic methods used in quantum mechanics
* Geometric phases and dynamical invariants
* Supersymmetry and its generalizations
* Pseudo-Hermitian quantum mechanics
* Quantum cosmology

Numeric Analysis

* Numerical Linear Algebra
* Numerical Optimization
* Perturbation Theory of Eigenvalues
* Eigenvalue Optimization

Probability and Stochastic Processes

* Mathematical finance
* Stochastic optimal control and dynamic programming
* Stochastic flows and random velocity fields
* Lyapunov exponents of flows
* Unicast and multicast data traffic in telecommunications
* Probabilistic Inference

Statistics

* Inference on Random Graphs (with emphasis on modeling email and internet traffic and clustering analysis)
* Graph Theory (probabilistic investigation of graphs emerging from computational geometry)
* Statistics (analysis of spatial data and spatial point patterns with applications in epidemiology and ecology and statistical methods for medical data and image analysis)
* Classification and Pattern Recognition (with applications in mine field and face detection)

Algebraic Geometry

* Arithmetical Algebraic Geometry, Arakelov geometry, Mixed Tate motives
* p-adic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties

Geometry and Topology

* Topology of low-dimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings
* Symplectic topology and geometry, Seiberg-Witten theory, Floer homology
* Foliation and Lamination Theory, Minimal Surfaces, and Hyperbolic Geometry