Graduate Programs in Mathematics

Head of Department: Sadik Deger

Professors: Talin Budak, Sadik Deger, Alp Eden, Ahmet Feyzioglu, Zerrin Gokturk, Nilgun Isik, Ercument Ortacgil, John Pym•, Ayse Soysal, Betul Tanbay, Yalcin Yildirim

Assistant Professors: Serdar Altok, Arzu Boysal, Olcay Coskun, Burak Gurel, Muge Kanuni, Ferit Ozturk, Muge Taskin Aydin

Instructors: Ozlem Beyarslan, Fatih Ecevit, Gulnihal Yucel

The M.S. program in Mathematics comprises 22 credits or more credits of course work (minimum 7 courses), graduate seminar and a Master's thesis. The total number of credits includes 9 credits of required courses. The required courses are:

The elective courses are chosen from the graduate courses offered by the Program or the Institute, and are subject to the approval of the student's advisor.

There is a breadth requirement and a depth requirement to fulfill as follows.

1. Breadth requirement :

Students are expected to take at least one course from four of the topics chosen from the list below. Two of these courses can be chosen from 400 level courses approved by the department.

2. Depth requirement :

Students must take a two course sequence chosen from one topic among the list above from a) to f) such as MATH 531-532, MATH 521-522 etc. None of these courses can be a reading course nor a 400 level course.

DOCTOR OF PHILOSOPHY PROGRAM

The Ph.D. program in Mathematics is composed of 21 or more credits of course work planned under the supervision of an advisor and Ph.D. thesis carried out according to the regulations of the Institute.

COURSE DESCRIPTIONS

MATH 521 Algebra I (Cebir I) (4+0+0) 4 ECTS 10
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 (Cebir II) (4+0+0) 4 ECTS 10
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 (Cebirsel Sayilar Kurami) (4+0+0) 4 ECTS 10
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 (Sayilar Kurami) (4+0+0) 4 ECTS 10
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 (Cözumsel Sayilar Kurami) (4+0+0) 4 ECTS 10
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.

MATH 531 Real Analysis I (Gercel Analiz I) (4+0+0) 4 ECTS 10
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 (Gercel Analiz II) (4+0+0) 4 ECTS 10
Normed and Banach spaces, Lp-spaces and duals, Hahn-Banach theorem, category and uniform boundedness theorem, strong, weak and weak-convergence, open mapping theorem, closed graph theorem.
Prerequisite: MATH 531.

MATH 533 Complex Analysis I (Karmasik Analiz I) (4+0+0) 4 ECTS 10
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.
Prerequisite: MATH 431 or consent of the instructor.

MATH 534 Complex Analysis II (Karmasik Analiz II) (4+0+0) 4 ECTS 10
Runge's theorem, analytic continuation, Riemann surfaces, harmonic functions, entire functions, the range of an analytic function.
Prerequisite: MATH 533.

MATH 535 Functional Analysis (Fonksiyonel Analiz) (4+0+0) 4 ECTS 10
Topological vector spaces, locally convex spaces, weak and weak* topologies, duality, Alaoglu's theorem, Krein-Milman theorem and applications, Schauder fixed point theorem, Krein-Smulian theorem, Eberlein-Smulian theorem, linear operators on Banach spaces.
Prerequisite: MATH 531 and MATH 532

MATH 541 Probability Theory (Olasilik Kurami) (4+0+0) 4 ECTS 10
An introduction to measure theory, Kolmogorov axioms, independence, random variables, expectation, modes of convergence for sequences of random variables, moments of a random variable, generating functions, characteristic functions, product measures and joint probability, 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 (4+0+0) 4 ECTS 10
(Rassal Surecler ve Martingaller)
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.

MATH 545 Mathematics of Finance (Finans Matematigi) (4+0+0) 4 ECTS 10
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 551 Partial Differential Equations I (4+0+0) 4 ECTS 10
(Kismi Diferansiyal Denklemler I)
Existence and uniqueness theorems for ordinary differential equations, continuous dependence on data. Basic linear partial differential equations : transport equation, Laplace's equation, diffusion equation, wave equation. Method of characteristics for non-linear first-order PDE's, conservation laws, special solutions of PDE's, Cauchy-Kowalevskaya theorem.

MATH 552 Partial Differential Equations II (4+0+0) 4 ECTS 10
(Kismi Diferansiyal Denklemler II)
Hölder spaces, Sobolev spaces, Sobolev embedding theorems, existence and regularity for second-order elliptic equations, maximum principles, second-order linear parabolic and hyperbolic equations, methods for non-linear PDE's, variational methods, fixed point theorems of Banach and Schauder.
Prerequisite: MATH 551.

MATH 571 Topology (Topoloji) (4+0+0) 4 ECTS 10
Fundamental concepts, subbasis, 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 (Cebirsel Topoloji) (4+0+0) 4 ECTS 10
Basic notions on categories and functors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and their homology.
Prerequisite: MATH 571.

MATH 579 Graduate Seminar (Lisansustu Seminer) (0+1+0) 0 Pass/Fail ECTS 1
Presentation of topics of interest in mathematics through seminars offered by faculty, guest speakers and graduate students.

MATH 581 Selected Topics in Analysis I (3+0+0) 3 ECTS 7
(Analizden Secme Konular I)

MATH 582 Selected Topics in Analysis II (3+0+0) 3 ECTS 7
(Analizden Secme Konular II)

MATH 583 Selected Topics in Foundations of Mathematics (3+0+0) 3 ECTS 7
(Matematigin Temellerinden Secme Konular)

MATH 584 Selected Topics in Algebra and Topology (3+0+0) 3 ECTS 7
(Cebir ve Topolojiden Secme Konular)

MATH 585 Selected Topics in Probability and Statistics (3+0+0) 3 ECTS 7
(Olasilik ve Istatistikten Secme Konular)

MATH 586 Selected Topics in Differential Geometry (3+0+0) 3 ECTS 7
(Turevsel Geometriden Secme Konular)

MATH 587 Selected Topics in Differential Equations (3+0+0) 3 ECTS 7
(Turevsel Denklemlerden Secme Konular)

MATH 588 Selected Topics in Applied Mathematics (3+0+0) 3 ECTS 7
(Uygulamali Matematikten Secme Konular)

MATH 589 Selected Topics in Combinatorics (3+0+0) 3 ECTS 7
(Kombinatorikten Secme Konular)

MATH 590 Readings in Mathematics (Matematikte Okumalar) (0+0+2) 1 ECTS 73
Literature survey and presentation on a subject determined by the instructor.

MATH 601 Measure Theory (Ölcu Teorisi) (4+0+0) 4 ECTS 10
Fundamentals of measure and integration theory, Radon-Nikodym Theorem, Lp spaces, modes of convergence, product measures and integration over locally compact topological spaces.

MATH 611 Differential Geometry I (Turevsel Geometri I) (4+0+0) 4 ECTS 10
Survey of differentiable manifolds, Lie groups and fibre bundles, theory of connections, holonomy groups, extension and reduction theorems, applications to linear and affine connections, curvature, torsion, geodesics, applications to Riemannian connections, metric normal coordinates, completeness, De Rham decomposition theorem, sectional curvature, spaces of constant curvature, equivalence problem for affine and Riemannian connection.

MATH 612 Differential Geometry II (Turevsel Geometri II) (4+0+0) 4 ECTS 10
Submanifolds, fundamental theorem for hypersurfaces, variations of the length integral, Jacobi fields, comparison theorem, Morse index theorem, almost complex and complex manifolds, Hermitian and Kaehlerian metrics, homogeneous spaces, symmetric spaces and symmetric Lie algebra, characteristic classes.
Prerequisite: MATH 611.

MATH 623 Integral Transforms (Integral Donusumleri) (4+0+0) 4
Exponential, cosine and sine, Fourier transform in many variables, application of Fourier transform to solve boundary value problems, Laplace Transform, use of residue theorem and contour integration for the inverse of Laplace transform, application of Laplace transform to solve differential and integral equations, Fourier-Bessel and Hankel transforms for circular regions, Abel transform for dual integral equations.

MATH 624 Numerical Solutions of Partial Differential and Integral Equations
(Kismi Turevsel Diferansiyel Denklemlerle Integral Denklemlerin Sayisal Cözulmesi) (4+0+0) 4 ECTS 10
Parabolic differential equations, explicit and implicit formulas, elliptic equations, hyperbolic systems, finite elements characteristics, Volterra and Fredholm integral equations.

MATH 627 Optimization Theory I (Eniyileme Kurami I) (4+0+0) 4 ECTS 10
Fundamentals of linear and non-linear optimization theory, unconstrained optimization, constrained optimization, saddlepoint conditions, Kuhn-Tucker conditions, post-optimality, duality, convexity, quadratic programming, multistage optimization.

MATH 628 Optimization Theory II (Eniyileme Kurami II) (4+0+0) 4 ECTS 10
Design and analysis of algorithms for linear and non-linear optimization. The revised simplex method, algorithms for network problems, dynamic programming techniques, methods for constrained nonlinear problems.
Prerequisite: MATH 627.

MATH 631 Algebraic Topology I (Cebirsel Topoloji I) (4+0+0) 4 ECTS 10
Basic notions on categories and factors, the fundamental group, homotopy, covering spaces, the universal covering space, covering transformations, simplicial complexes and homology of simplicial complexes.

MATH 632 Algebraic Topology II (Cebirsel Topoloji II) (4+0+0) 4 ECTS 10
Singular homology, exact sequences, the Mayer-Vietoris exact sequence, the Lefschetz fixed-point theorem, cohomology, cup and cap products, duality theorems, the Hurewicz theorem, higher homotopy groups.

MATH 643 Stochastic Processes I (Stokastik Surecler I) (4+0+0) 4 ECTS 10
Survey of measure and integration theory, measurable functions and random variables, expectation of random variables, convergence concepts, conditional expectation, stochastic processes with emphasis on Wiener process, Markov processes and martingales, spectral representation of second-order processes, linear prediction and filtering, Ito and Saratonovich integrals, Ito calculus, stochastic differential equations, diffusion processes, Gaussian measures, recursive estimation.
Prerequisite: MATH 552 or consent of the instructor.

MATH 644 Stochastic Processes II (Stokastik Surecler II) (4+0+0) 4 ECTS 10
Tightness, Prohorov's theorem, existence of Brownian motion, Martingale characterization of Brownian motion, Girsanov's theorem, Feynmann-Kac formulas, Martingale problem of Stroock and Varadhan, application to mathematics of finance.
Prerequisite: MATH 643.

MATH 645 Mathematical Statistics (Matematiksel Istatistik) (4+0+0) 4 ECTS 10
Review of essentials of probability theory, subjective probability and utility theory, statistical decision problems, a comparison game theory and decision theory, main theorems of decision theory with emphasis on Bayer and minimax decision rules, distribution and sufficient statistics, invariant statistical decision problem, testing hypotheses, the Newton-Pearson lemma, sequential decision problem.
Prerequisite: MATH 552 or consent of the instructor.

MATH 660 Number Theory (Sayilar Teorisi) (4+0+0) 4 ECTS 10
Basic algebraic number theory, number fields, ramification theory, class groups, Dirichlet unit theorem, zeta and L-functions, Riemann, Dedekind zeta functions, Dirichlet, Hecke L-functions, primes in arithmetic progressions, prime number theorem, cyclotomic fields, reciprocity laws, class field theory, ideles and adeles, modular functions and modular forms.

MATH 680 Seminar in Pure Mathematics I (4+0+0) 4 ECTS 10
(Sirfi Matematik Semineri I)
Recent developments in pure mathematics.

MATH 681 Seminar in Pure Mathematics II (4+0+0) 4 ECTS 10
(Sirfi Matematik Semineri II)
Recent developments in pure mathematics.

MATH 682 Seminar in Applied Mathematics I (4+0+0) 4 ECTS 10
(Uygulamali Matematik Semineri I)
Recent developments in applied mathematics.

MATH 683 Seminar in Applied Mathematics II (4+0+0) 4 ECTS 10
(Uygulamali Matematik Semineri II)
Recent developments in applied mathematics.

MATH 690 M.S. Thesis (Yuksek Lisans Tezi) ECTS 30

MATH 699 Guided Research (Yönlendirilmis Arastirmalar) (2+4+0) 4 ECTS 3
Research in the field of Mathematics, by arrangement with members of the faculty; guidance of doctoral students towards the preparation and presentation of a research proposal.