Mathematics _|_Grafiku i Kurseve
The Master of Science degree in Mathematics has two options:- the thesis option
- non thesis option
The thesis option requires the completion of the thesis under the guidance of the Thesis advisor.
The non-thesis option requires passing successfully the qualifying exams in Algebra and Analysis.
Formimi i pergjithshem: 15 kredite
Lendet Baze:
MAT 521,
MAT 522,
MAT 551,
MAT 552,
Formimi baze: 12 kredite
Lendet Baze:
MAT 553,
MAT 631,
Lendet me Zgjedhje: , MAT 632, MAT 641, MAT 642,
Formimi special: 9 kredite
Lendet me Zgjedhje: MAT 632, MAT 641, MAT 642, MAT 651, MAT 652, MAT 655, MAT 657, MAT 658, MAT 661, MAT 662, MAT 771, MAT 772, MAT 851, MAT 852,
Pershkrimet e Lendeve
MAT 521 - Analysis I (3)
Lebesgue measure, measurable functions and the Lebesgue integral; convergence theorems; monotone functions, bounded variation and absolute continuity. The Lp spaces; product measures and Fubini's theorem; the Radon-Nikodym theorem.
MAT 522 - Analysis II (3)
Lebesgue measure, measurable functions and the Lebesgue integral; convergence theorems; monotone functions, bounded variation and absolute continuity. The Lp spaces; product measures and Fubini's theorem; the Radon-Nikodym theorem.
MAT 551 - Algebra I (3)
Groups, Sylow theorems, solvable and simple groups, computation in permutation groups. GAP will be used to perform computations with groups. Free groups, finitely generated abelian groups, semi-direct products, extension of groups. Introduction to rings, Euclidean domains, PID's, UFD's, polynomial rings, irreducibility criteria for polynomials.
MAT 552 - Algebra II (3)
A detailed study of module theory, decomposition theorems, linear algebra. Theory of fields, field extensions, finite fields, geometric constructions, Galois theory, solvability by radicals, computing Galois groups of polynomials.
MAT 553 - Probability and Statistics (3)
The distribution of random variables, conditional probability and stochastic independence, special distributions, functions of random variables, interval estimation, sufficient statistics and completeness, point estimation, tests of hypothesis and analysis of variance.
MAT 631 - Complex Analysis (3)
Rapid survey of properties of complex numbers, linear transformations, geometric forms and necessary concepts from topology. Complex integration. Cauchy's theorem and its corollaries. Taylor series and the implicit function theorem in complex form. Conformality and the Riemann-Caratheodory mapping theorem. Theorems of Bloch, Schottky, and the big and little theorems of Picard. Harmonicity and Dirichlet's problems.
MAT 632 - Riemann Surfaces (3)
An introduction to Riemann Surfaces from both the algebraic and function-theoretic points of view. Topics include projective algebraic curves, differential forms, integration, divisors of poles and zeroes, linear systems, the Riemann-Roch theorem, Serre duality, and applications.
MAT 641 - Computational Algebra I (3)
A study of the mathematics and algorithms which are used in symbolic algebraic manipulation packages.
MAT 642 - Computational Algebra II (3)
Topics include computer representation of symbolic mathematics, polynomial ring theory, field theory and algebraic extensions, modular and p-adic methods, subresultant algorithm for polynomial GCD's, Groebner bases for polynomial ideals and Buchberger's algorithm, factorization and zeros of polynomials.
MAT 651 - Commutative Algebra I (3)
Rings and ideals, modules, exact sequences, tensor products, integral dependence and valuations, the going-up and going -down theorem, chain conditions, Notherian rings, dicrete valuation rings, Dedekind domains. Basic knowledge of commutative ring theory, field theory, Galois theory, and group theory will be assumed.
MAT 652 - Commutative Algebra II (3)
Rings and ideals, modules, exact sequences, tensor products, integral dependence and valuations, the going-up and going -down theorem, chain conditions, Notherian rings, dicrete valuation rings, Dedekind domains. Basic knowledge of commutative ring theory, field theory, Galois theory, and group theory will be assumed.
MAT 655 - Computational Group Theory (3)
An introduction to computational group theory using computer algebra packages such as GAP.
MAT 657 - Coding Theory I (3)
We will be focusing on channel coding theory. In the first part of the course, a brief introduction will be given to information and coding theory in order to see what is the best one should expect from a good code. Then we will continue with the introduction to the basic algebra concepts needed in codding theory. Next we will follow will be more on the computational aspects of groups, finite fields, polynomials, etc other than the rigorous mathematical approach.
MAT 658 - Coding Theory II (3)
We will use software to do many computational problems (see below). These concepts will be utilized for the construction of polynomial and cyclic codes. BCH
codes and Reed-Solomon (RS) codes will be covered in detail.
MAT 661 - Mathematics of Communications I (3)
An introduction to mathematical concepts of digital communications. Random processes, Shanon's theorem, communication channels, antena theory, source coding, etc.
MAT 662 - Mathematics of Communications II (3)
A continuation of MAT 661, algebraic coding, turbo codes, LDPC codes, new developments in digital communications.
MAT 771 - Analytic Number Theory (3)
Basic concepts of analytic number theory are covered.
MAT 772 - Algebraic Number theory (3)
Algebraic number fields, integrality and Notherian properties, Dedekeind Domains, Extensions, ramified and non-ramified extensions, ramification in Galois extensions, class groups and units, cyclotomic fields, L-functions, Dedekind zeta-function, Brauer relations.
MAT 851 - Algebraic Geometry I (3)
Introduction to affine and projective spaces, algebraic varieties, maps between varieties, Hilbert's Nullstellensatz, Zariski topology, abelian varieties, the Riemann-Roch theorem, Jacobians of curves, sheaves and cohomology.
MAT 852 - Algebraic Geometry II (3)
Introduction to affine and projective spaces, algebraic varieties, maps between varieties, Hilbert's Nullstellensatz, Zariski topology, abelian varieties, the Riemann-Roch theorem, Jacobians of curves, sheaves and cohomology.