Program Description The
Department of Mathematics offers graduate courses leading to M.Sc., and
eventually to Ph.D., degree in Mathematics. The Master of Science
program aims to provide a sound foundation for the students who wish to
pursue a research career in mathematics as well as other related areas.
The department emphasizes both pure and applied mathematics. Research
in the department covers algebra, number theory, combinatorics,
differential equations, functional analysis, abstract harmonic
analysis, mathematical physics, stochastic analysis, biomathematics and
topology.
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
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
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 The requirements for the degree of Master of Science in Mathematics are as follows.
The
master program in Mathematics consists of a) at least 21 credit hours
of course work, b) a master thesis and c) graduate seminar course. All
students have to take
- MATH 521 Algebra I (4 credits) (*)
- MATH 531 Real Analysis I (4 credits) (*)
- MATH 590 Graduate Seminar (0 credit)
- MATH 595 Thesis
The
total credit of these required courses is 8. In addition to these
courses students have to take two sequence courses among the courses in
areas listed below as a) to g), such as Math 531-532, Math 521-522,
etc... These courses are not allowed to be 400 level courses. The areas
are;
- Algebra and Number Theory
- Analysis
- Topology and Geometry
- Probability and Statistics
- Discrete Mathematics
- Applied Mathematics
- Logic and Foundations of Science
To
complete the credit requirements the courses may be chosen among
courses offered by the department of Mathematics and the other
departments of the Graduate School of Sciences and Engineering. At most
2 of the elective courses can be 400 level undergraduate courses. The
choice of courses must be approved by the student's graduate advisor.
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 503 Applied Mathematics
- Math 504 Numerical Methods I
- Math 506 Numerical Methods II
- MATH 521 Algebra I
- MATH 522 Algebra II
- MATH 525 Algebraic Number Theory
- MATH 527 Number Theory
- MATH 528 Analytic Number Theory
- MATH 531 Real Analysis I
- MATH 532 Real Analysis II
- MATH 533 Complex Analysis I
- MATH 534 Complex Analysis II
- MATH 535 Functional Analysis
- MATH 536 Applied Functional Analysis I
- MATH 537 Applied Functional Analysis II
- MATH 538 Differential Geometry
- MATH 541 Probability Theory
- MATH 544 Stochastic Processes and Martingales
- MATH 545 Mathematics of Finance
- MATH 550 Advanced Ordinary Differential
- MATH 551 Partial Differential Equations I
- MATH 552 Partial Differential Equations II
- MATH 563 Algebraic Coding Theory
- MATH 564 Combinatorial Design Theory
- MATH 565 Graph Theory
- MATH 566 Topics in Module and Ring Theory
- MATH 571 Topology
- MATH 572 Algebraic Topology
- MATH 579 Readings in Mathematics
- MATH 580 Selected Topics in Topology I
- MATH 581Selected 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 503 Applied Mathematics Review
of Linear Algebra and Vector Fields: Vector Spaces, Eigenvalue
Problems, Quadratic Forms, Divergence Theorem and Stokes' Theorem.
Sturm-Liouville Theory and Orthogonal Polynomials, Methods of Solution
of Boundary Value Problems for the Laplace Equation, Diffusion Equation
and the Wave Equation. Elements of Variational Calculus.
Math 504 Numerical Methods I Review
of Linear Algebra: linear spaces, orthogonal matrices, norms of vectors
and matrices, singular value decomposition. Projectors, QR
Factorization Algorithms, Least Squares, Conditioning and Condition
Numbers, Floating Point Representation, Stability, Conditioning and
Stability of Least Squares, Conditioning and Stability Analysis of
Linear Systems of Equations.
Math 506 Numerical Methods II Numerical
Solution of Functional Equations, the Cauchy Problem and Boundary Value
Problems for Ordinary Differential Equations. Introduction to the
Approximation Theory of One Variable Functions. Finite - difference
Methods for Elementary Partial Differential Equations. Monte Carlo
Method and Applications.
MATH 521 Algebra I Free
groups, group actions, group with operators, Sylow theorems,
Jordan-Hölder theorem, nilpotent and solvable groups. Polynomial and
power series rings, Gauss's lemma, PID and UFD, localization and local
rings, chain conditions, Jacobson radical.
MATH 522 Algebra II Galois
theory, solvability of equations by radicals, separable extensions,
normal basis theorem, norm and trace, cyclic and cyclotomic extensions,
Kummer extensions. Modules, direct sums, free modules, sums and
products, exact sequences, morphisms, Hom and tensor functors, duality,
projective, injective and flat modules, simplicity and semisimplicity,
density theorem, Wedderburn-Artin theorem, finitely generated modules
over a principal ideal domain, basis theorem for finite abelian groups.
MATH 525 Algebraic Number Theory Valuations
of a field, local fields, ramification index and degree, places of
global fields, theory of divisors, ideal theory, adeles and ideles,
Minkowski's theory, extensions of global fields, the Artin symbol.
MATH 527 Number Theory Method
of descent, unique factorization, basic algebraic number theory,
diophantine equations, elliptic equations, p-adic numbers, Riemann zeta
function, elliptic curves, modular forms, zeta and L-functions, ABC-
conjecture, heights, class numbers for quadratic fields, a sketch of
Wiles' proof.
MATH 528 Analytic Number Theory Primes
in arithmetic progressions, Gauss' sum, primitive characters, class
number formula, distribution of primes, properties of the Riemann zeta
function and Dirichlet L-functions, the prime number theorem, Polya-
Vinogradov inequality, the large sieve, average results on the
distribution of primes.
Prerequisite: MATH 533 or consent of the instructor.
MATH 531 Real Analysis I Lebesgue
measure and Lebesgue integration on Rn, general measure and
integration, decomposition of measures, Radon-Nikodym theorem,
extension of measures, Fubini's theorem.
MATH 532 Real Analysis II Normed
and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, Baire
category and uniform boundedness theorems, strong, weak and
weak*-convergence, open mapping theorem, closed graph theorem.
Prerequisite: MATH 531 or consent of the instructor.
MATH 533 Complex Analysis I Review
of the complex number system and the topology of C, elementary
properties and examples of analytic functions, complex integration,
singularities, maximum modulus theorem, compactness and convergence in
the space of analytic functions.
MATH 534 Complex Analysis II Runge's
theorem, analytic continuation, Riemann surfaces, harmonic functions,
entire functions, the range of an analytic function.
Prerequisite: MATH 533 or consent of the instructor.
MATH 535 Functional Analysis Topological
vector spaces, locally convex spaces, weak and weak* topologies,
duality, Alaoglu's theorem, Krein-Milman theorem and applications,
Schauder fixed point theorem, Krein-Shmulian theorem, Eberlein-Shmulian
theorem, linear operators on Banach spaces.
Prerequisite: MATH 532 or consent of the instructor.
MATH 536 Applied Functional Analysis I Review
of linear operators in Banach spaces and Hilbert spaces; Riesz
-Schauder theory; fixed point theprems of Banach and Schauder;
semigroups of linear operators; Sobolev spaces and basic embedding
theorems; boundary - value problems for elliptic equations; eigenvalues
and eigenvectors of second order elliptic operators; initial
boundary-value problems for parabolic and hyperbolic equations.
MATH 537 Applied Functional Analysis II Existence and uniqueness of solutions of
abstract evolutionary equations. Global non-existence and blow up
theorems. Applications to the study of the solvability and asymptotic
behavior of solutions of initial boundary-value problems for reaction
diffusion equations, Navier-Stokes equations, nonlinear Klein-Gordon
equations and nonlinear Schrödinger equations.
MATH 538
Differential Geometry Differentiable manifolds; differentiable forms; integration on manifolds; de Rhamm cohomology; connections and curvature
MATH 541 Probability Theory An
introduction to measure theory, Kolmogorov axioms, independence, random
variables, product measures and joint probability, distribution laws,
expectation, modes of convergence for sequences of random variables,
moments of a random variable, generating functions, characteristic
functions, distribution laws, conditional expectations, strong and weak
law of large numbers, convergence theorems for probability measures,
central limit theorems.
MATH 544 Stochastic Processes and Martingales
Stochastic
processes, stopping times, Doob-Meyer decomposition, Doob's martingale
convergence theorem, characterization of square integrable martingales,
Radon-Nikodym theorem, Brownian motion, reflection principle, law of
iterated logarithms.
Prerequisite: MATH 541 or consent of the instructor.
MATH 545
Mathematics of Finance
From
random walk to Brownian motion, quadratic variation and volatility,
stochastic integrals, martingale property, Ito formula, geometric
Brownian motion, solution of Black-Scholes equation, stochastic
differential equations, Feynman-Kac theorem, Cox-Ingersoll-Ross and
Vasicek term structure models, Girsanov's theorem and risk neutral
measures, Heath-Jarrow-Morton term structure model, exchange-rate
instruments.
MATH 550
Advanced Ordinary Differential
Existence
and uniqueness theorems; continuation of solutions; continuous
dependence and stability, Lyapunovs direct method; differential
inequalities and their applications; boundary-value problems and
Sturm-Liouville theory.
MATH 551
Partial Differential Equations I
First
order equations, method of characteristics; the Cauchy-Kovalevskaya
theorem; Laplace's equation: potential theory and Greens's function,
properties of harmonic functions, the Dirichlet problem on a ball; heat
equation: the Cauchy problem, initial boundary-value problem, the
maximum principle; wave equation: the Cauchy problem, the domain of
dependence, initial boundary-value problem.
MATH 552
Partial Differential Equations II
Review
of functional spaces and embedding theorems; existence and regularity
of solutions of boundary-value problems for second-order elliptic
equations; maximum principles for elliptic and parabolic equations;
comparison theorems; existence, uniqueness and regularity theorems for
solutions of initial boundary-value problems for second-order parabolic
and hyperbolic equations.
Prerequisite: MATH 551 or consent of the instructor.
MATH 563
Algebraic Coding Theory
Error
correcting coding theory. Hamming, Golay, cyclic, 2-error correcting
BCH codes, Reed-Solomon, Convolutional, Reed-Muller and Preparata
codes. Interaction of codes and combinatorial designs.
MATH 564
Combinatorial Design Theory
Balanced
incomplete block designs, group divisible designs and pairwise balanced
designs. Resolvable designs, symmetric designs and designs having
cyclic automorphisms. Pairwise orthogonal latin squares. Affine and
projective geometries. Embeddings and nestings of designs.
MATH 565
Graph Theory
Matchings,
edge colorings and vertex colorings of graphs. Connectivity, spanning
trees, and disjoint paths in graphs. Cycles in graphs, embeddings.
Planar graphs, directed graphs. Ramsey Theory, matroids, random graphs.
MATH 566
Topics in Module and Ring Theory
Generalities
on modules, categories, and functors. The socle and the Jacobson
radical of a module. Semisimple modules. Chain conditions on modules.
The Hopkins-Levitzki Theorem. The Wedderburn-Artin Theorem and its
applications to linear representations of finite groups. The "Hom"
functors and exactness. Injective modules. Essential monomorphisms,
injective hulls. Projective modules. Superfluous epimorphisms,
projective covers. Indecomposable direct sum decompositions of modules.
The Krull-Remak-Schmidt-Azumaya Theorem. Krull dimension and Goldie
dimension of modules and lattices.
MATH 571
Topology
Fundamental
concepts, basis, subbasis of a topology, neighborhoods, continuous
functions, subspaces, product spaces and quotient spaces, weak
topologies and embedding theorem, convergence by nets and filters,
separation and countability, compactness, local compactness and
compactifications, paracompactness, metrization, complete metric spaces
and Baire category theorem, connectedness.
MATH 572
Algebraic Topology
Basic
notions on categories and functors, the fundamental group, homotopy,
covering spaces, the universal covering space, covering
transformations, simplicial complexes and their homology.
MATH 579
Readings in Mathematics
Literature survey and presentation on a subject determined by the instructor.
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
MATH 590
Graduate Seminar
TEAC 500Teaching 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
Non-credit
presentation of topics of interest in mathematics through seminars
offered by faculty, guest speakers and graduate students.