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All courses and course definitions of the Mathematics
department may be seen under the undergraduate catalogue and graduate catalogue
pages of Boğaziçi University. The links indicated below provide the most recent
information about new courses or changes in course titles, definitions or
credits as accepted by the university senate.
Please check Graduate Catalogue
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Graduate Courses Offered by Mathematics Department
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| MATH 521 |
Algebra I |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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. |
| MATH 531 |
Real
Analysis I |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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. |
| MATH 533 |
Complex Analysis I |
(4+0+0) 4 |
| 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 I |
(4+0+0) 4 |
| Runge's theorem, analytic continuation, Riemann
surfaces, harmonic functions, entire functions, the range of an analytic
function. |
| MATH 551 |
Partial Differential Equations I |
(4+0+0) 4 |
| 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 |
| 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. |
| MATH 541 |
Probability Theory |
(4+0+0) 4 |
| 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 |
| 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. |
| MATH 545 |
Mathematics of
Finance |
(4+0+0) 4 |
| 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 571 |
Topology |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| Basic notions on categories and functors, the
fundamental group, homotopy, covering spaces, the universal covering space,
covering transformations, simplicial complexes and their homology. |
| MATH 535 |
Functional Analysis |
(4+0+0) 4 |
| 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. |
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Prerequisite: |
MATH 531 and MATH
532 |
| MATH 579 |
Graduate Seminar |
(0+1+0) Non-credit |
| Presentation of topics of interest in mathematics
through seminars offered by faculty, guest speakers and graduate students. |
| MATH 590 |
Readings in
Mathematics |
(0+0+2) 1 |
| Literature survey and presentation on a subject to be
determined by the instructor. |
| MATH 581 |
Selected Topics in
Analysis I |
(3+0+0) 3 |
| MATH 582 |
Selected Topics in
Analysis II |
(3+0+0) 3 |
| MATH 583 |
Selected Topics in
Foundations of Mathematics |
(3+0+0) 3 |
| MATH 584 |
Selected Topics in
Algebra and Topology |
(3+0+0) 3 |
| MATH 585 |
Selected Topics in
Probability and Statistics |
(3+0+0) 3 |
| MATH 586 |
Selected Topics in
Differential Geometry |
(3+0+0) 3 |
| MATH 587 |
Selected Topics in Differential Equations |
(3+0+0) 3 |
| MATH 588 |
Selected Topics in
Applied Mathematics I |
(3+0+0) 3 |
| MATH 589 |
Selected Topics in
Combinatorics |
(3+0+0) 3 |
| MATH 601 |
Measure Theory |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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. |
| MATH 623 |
Integral Transforms |
(4+0+0) 4 |
| Fourier transforms, 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
|
(4+0+0) 4 |
| Parabolic differential equations, explicit and
implicit formulas, elliptic equations, hyperbolic systems, finite elements
characteristics, Volterra and Fredholm integral equations. |
| MATH 627 |
Optimization Theory
I |
(4+0+0) 4 |
| Fundamentals of linear and nonlinear optimization
theory. Unconstrained optimization, constrained optimization, saddlepoint
conditions, Kuhn-Tucker conditions, post-optimality, duality, convexity,
quadratic programming, multistage optimization. |
| MATH 628 |
Optimization Theory
II |
(4+0+0) 4 |
| Design and analysis of algorithms for linear and
non-linear optimization. The revised simplex method, algorithms for network
problems, dynamic programming, quadratic programming techniques, methods for
constrained nonlinear problems. |
| MATH 631 |
Algebraic Topology I |
(4+0+0) 4 |
| Basic notions on categories and functions, the
fundamental groups, homotopy, covering spaces, the universal covering space,
covering transformations, simplicial complexes and homology of simplicial
complexes. |
| MATH 632 |
Algebraic Topology
II |
(4+0+0) 4 |
| 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 |
(4+0+0) 4 |
| 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 processes, 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. |
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Prerequisite: |
MATH 552 or consent
of instructor. |
| MATH 644 |
Stochastic Processes
II |
(4+0+0) 4 |
| 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,
applications to mathematics of finance. |
| MATH 645 |
Mathematical
Statistics |
(4+0+0) 4 |
| 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 Bayes and minimax decision rules, distribution and sufficient
statistics, invariant statistical decision problem, testing hypotheses, the
Neyman-Pearson lemma, sequential decision problem. |
|
Prerequisite: |
MATH 552 or consent
of instructor. |
| MATH 660 |
Number Theory |
(4+0+0) 4 |
| 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 |
| Recent developments in pure mathematics. |
| MATH 681 |
Seminar in Pure
Mathematics II |
(4+0+0) 4 |
| Recent developments in pure mathematics. |
| MATH 682 |
Seminar in Applied
Mathematics I |
(4+0+0) 4 |
| Recent developments in applied mathematics. |
| MATH 683 |
Seminar in Applied
Mathematics II |
(4+0+0) 4 |
| Recent developments in applied mathematics. |
| MATH 699 |
Guided Research |
(2+4+0) 4 |
| 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. |
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