Program Description
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.
Degree Requirements
Students can apply to the Ph.D. programs with a B.S. or M.S. degree. The Ph.D. degree requires successful completion of 14 courses beyond the B.S. degree or 7 courses beyond the M.S. degree. All students must pass the Ph.D. Qualifying Examination in the first year after they have been admitted to the Ph.D. program. Students are encouraged to begin research early. Students who have passed the Ph.D. qualifying examination are assisted in matters pertaining to their thesis research by a faculty thesis advisory committee. The research advisor serves as the chair of this committee. The committee meets with the student at least once each semester. Ph.D. students must submit a satisfactory written Ph.D. thesis proposal in their second year of study. At the completion of the Ph.D. research, the students must submit a written Thesis and pass an oral defense to complete the degree requirements.
Research Areas
Algebra and 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
Combinatorics
 Combinatorial design theory, in particular metamorphosis of designs, perfect hexagon triple systems
 Graph theory, in particular number of cycles in 2factorizations of complete graphs
 Coding theory, especially relation of designs to codes
 Random graphs, in particular, random proximity catch graphs and digraphs
Differential Equations
 Nonlinear ordinary differential equations of molecular dynamics
 PDE’s of quantum mechanics: time dependent Schrodinger equation
 Weak, in particular viscosity solutions, of second order equations
 Asymptotic analysis of reaction diffusion equations
 Gamma limits of nonconvex functionals
 Geometric flows and level set equations
 Global behavior of solutions to nonlinear PDE’s
 Dissipative dynamical systems generated by evolutionary PDE’s
 PDE’s modeling nonlinear problems of continuum mechanics
Analysis
 Banach algebras, especially the structure of the second Arens duals of Banach algebras
 Abstract Harmonic Analysis, especially the Fourier and FourierStieltjes 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
 PseudoHermitian quantum mechanics
 Quantum cosmology
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
 Spatial Statistics, mostly on nearest neighbor methods and multispecies spatial patterns of segregation and association
 Statistical Pattern Recognition, Classification
 Statistical Depth
 Statistics of Medicine concerning morphometric changes in organs and tissues, say, due to a disease
 Scale, size, and shape comparisons of organs or tissues based on MRI data
 Linear Models
 Computationally Intensive Methods: Bootstrap and Randomization
Algebraic Geometry
 Arithmetical Algebraic Geometry, Arakelov geometry, Mixed Tate motives
 padic methods in arithmetical algebraic geometry, Ramification theory of arithmetic varieties
Geometry and Topology
 Topology of lowdimensional manifolds, in particular Lefschetz fibrations, symplectic and contact structures, Stein fillings
 Symplectic topology and geometry, SeibergWitten theory, Floer homology
 Foliation and Lamination Theory, Minimal Surfaces, and Hyperbolic Geometry
Faculty
 Attila Askar, Differential Equations
 Mine Caglar, Probability and Stochastic Processes
 Emre Alkan, Number Theory
 Elvan Ceyhan, Probability and Statistics
 Tolga Etgu, Topology
 Varga Kalantarov, Differential Equations
 Sinan Unver, Algebraic Geometry
 Selda Kucukcifci, Combinatorics
 Ali Mostafazadeh, Mathematical Physics
 Burak Ozbagci, Topology
 Baris Coskunuzer, Geometric Topology
 Ali Ulger, Functional Analysis
 Emine Sule Yazici, Combinatorics
Curriculum
Students who are admitted with an M.S. degree must complete at least 21 credits of coursework. Students with a B.S. degree must complete an additional 21 credits of coursework by taking courses in the M.S. program. They must also complete the core courses in the “M.S. in Mathematics” program.

In addition, each student has to take a seminar course, MATH 590 Seminar.

Students working towards the thesis register for MATH 695 Ph.D. Thesis.

Students who have TA assignments must take TEAC 500: Teaching Experience during the semesters of their assignments.

Students must also take ENGL 500: Graduate Writing course.
 MATH 580 Selected Topics in Topology I
 MATH 581 Selected Topics in Analysis I
 MATH 582 Selected Topics in Analysis II
 MATH 583 Selected Topics in Foundations of Mathematics
 MATH 584 Selected Topics in Algebra and Topology
 MATH 585 Selected Topics in Probability and Statistics
 MATH 586 Selected Topics in Differential Geometry
 MATH 587 Selected Topics in Differential Equations
 MATH 588 Selected Topics in Applied Mathematics
 MATH 589 Selected Topics in Combinatorics
Course Descriptions
MATH 590 Graduate Seminar
Noncredit presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.
MATH 695 Ph.D.
Thesis Independent research towards Ph.D. degree.
TEAC 500 Teaching Experience
Provides handson teaching experience to graduate students in undergraduate courses. Reinforces students' understanding of basic concepts and allows them to communicate and apply their knowledge of the subject matter.
ENGL 500 Graduate Writing
This is a writing course specifically designed to improve academic writing skills as well as critical reading and thinking. The course objectives will be met through extensive reading, writing and discussion both in and out of class. Student performance will be assessed and graded by Satisfactory/Unsatisfactory.