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 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
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 non-convex 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 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
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 multi-species 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
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
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 Non-credit 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 hands-on 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.