Vlora Conference
Series in Mathematics

ACA 2010

Vlora in pictures

NATO ASI Vlora 2008

NATO ASI, 2008
pictures

Courses

Undergraduate

MAT 121, Calculus for Business Majors, 10
MAT 154, Calculus, 8
MAT 155, Calculus, 8
MAT 221, Applied Mathematics, 6
STAT 226, Applied Probability and Statistics, 8
MAT 231, Geometry, 8
MAT 250, Analysis I, 7
MAT 251, Analysis II, 7
MAT 255, Introduction to Differential Equations, 7
MAT 263, Discrete Mathematics, 8
MAT 265, Teaching of Mathematics, 6
MAT 270, Algebra I, 14
MAT 275, Linear Algebra, 8
MAT 290, Topology I, 6
MAT 291, Topology II, 6
MAT 330, Complex Analysis, 7
MAT 355, Number theory and Cryptology, 7
MAT 361, Numerical Analysis, 6
MAT 370, Abstract Algebra II, 14
MAT 387, Analysis of algorithms, 12
MAT 390, Coding Theory, 6
MAT 398, Final Project, 6

Graduate

MAT 421, Real Analysis I, 8
MAT 422, Real Analysis II, 8
MAT 433, Numerical Methods, 6
MAT 462, Geometric Structures, 6
MAT 472, Number Theory with Cryptography, 8
MAT 475, Abstract Algebra, 6
MAT 451, Introduction to Algebra I, 8
MAT 452, Introduction to Algebra II, 8
MAT 472, Number Theory, 8
MAT 521, Analysis I
MAT 522, Analysis II
MAT 525, Topology
MAT 551, Algebra I, 8
MAT 552, Algebra II, 8
STAT 551, Statistics, 8

MAT 481, Cryptography, 8
MAT 631, Complex Analysis, 8
MAT 632, Riemann Surfaces, 8
MAT 641, Computational Algebra I, 8
MAT 642, Computational Algebra II, 8
MAT 651, Commutative Algebra I, 8
MAT 652, Commutative Algebra II, 8
MAT 655 Computational Group Theory, 8
MAT 657, Coding Theory I, 8
MAT 658, Coding Theory II, 8
MAT 661, Mathematics of Communications I, 8
MAT 662, Mathematics of Communications II, 8
MAT 771, Analitic Number Theory I, 8
MAT 773, Special Topics, 8
MAT 772, Algebraic Number theory II, 8
MAT 851, Algebraic Geometry I, 8
MAT 852, Algebraic Geometry II, 8
MAT 870, Independent Research, 8

Course description

MAT 121 Calculus for Business Majors: Principal Issues in the class "General Mathematics" for the Economy students. Mathematical language, sets.Numerical sequences and series. Real functions.Limit and continuity.   Derivation. Integration. Real matrices,determinants and the inverse matrix. The structure of Vector Spaces.Linear Systems of equations.Linear maps.Linear functions in many  variables. Recurrent equations.

MAT 154 Calculus: A comprehensive study of analytic geometry, limits, differentiation and integration of functions of one real variable, including transcendental functions, infinite series, indeterminate forms, polar coordinates,numerical methods and applications. Each is offered fall and winter semester. MAT 154 satisfies the university general education requirement in the formal reasoning knowledge
foundation area.

MAT 155 Calculus : MAT 155 satisfies the university general education requirement for the knowledge applications integration area.Prerequisite for knowledge applications:completion of the university general education requirement in the
formal reasoning knowledge foundation area.Prerequisite: MAT 154 or placement.

MAT 221 Applied Mathematics: Presentation of the Problem of a Linear Program: General definition of flux in a transportation net.Basic concepts of graphs.Flux in a transportation net. Maximal traffic.How to present a problem.Methods PERT. Relation Time-Cost.

STA 226 Applied Probability and Statistics : Introduction to probability and statistics as applied to the physical, biological and social sciences and to engineering. Applications of special distributions and nonparametric techniques.Regression
analysis and analysis of variance.Satisfies the university general education requirement in the formal reasoning knowledge foundation area.Prerequisite or corequisite:MAT 154.

MAT 231 Geometry: Transformational approach to Euclidean geometry including an in-depth study of isometries and their application to symmetry, geometric constructions, congruence, coordinate geometry, and non-Euclidean geometries

MAT 250 Analysis I: The topology of the real number line and of n-dimensional
Euclidean space,continuity and uniform continuity, derivatives, the Riemann integral, sequences and series, uniform convergence.Offered every fall.Prerequisite: MAT 155.

MAT 251 Analysis II: Improper integrals, derivatives and integrals in n- dimensional Euclidean space, implicit and inverse function theorems,differential geometry and vector calculus,and Fourier series. Offered every winter.Prerequisite: MAT 250.

MAT 255 Introduction to Differential Equations : An introduction to the basic methods of solving ordinary differential  equations, including the methods of undetermined coefficients,variation of parameters,series, Laplace transforms and numerical methods. Separable, exact and linear  equations. Applications. Prerequisite: MAT 155.

MAT 263 Discrete Mathematics: Concepts and methods of discrete mathematics with an emphasis on their application to computer science.Logic and proofs, sets and relations, algorithms, induction and recursion, combinatorics,graphs
and trees.Prerequisite:MAT 155.

MAT 265 Teaching of Mathematics: Basic concepts of teaching methods, documentation, didactic plans, programs, mathematic’s textbook in C.U. Basic methods of teaching, daily preparation of plans, main facts of adding and other mathematics applications. Mathematic’s roblems, types, methods of solutions. Methods of how the students will be evaluated.

MAT 270 Algebra I: An introduction to the theory of groups with an emphasis on the development of careful mathematical reasoning. Among the topics covered by the course are subgroups, quotient groups, cyclic groups, permutation groups, linear groups, homomorphism theorems and group actions. Each topic will be illustrated and explored through examples. The second semester will be an introduction to ring and field theory. A ring is an algebraic system with two operations (addition and multiplication) satisfying various axioms, eg., the integers. We will see that many properties of the integers are shared by other broad classes of rings (for example polynomial rings with coefficients over a field). In particular we will explore the general notion of unique factorization and formulate conditions under which a ring has this property. Later in the course we will apply some of the results of ring theory to construct and study field extentions. If time permits we will outline the main results of Galois theory which studies the structure of the extension generated by the roots of a polynomial in terms of an associated group.

MAT 275 Linear Algebra: Study of general vector spaces,linear systems of equations,
linear transformations and compositions, eigenvalues, eigenvectors, diagonalization,modeling and orthogonality. Provides a transition to formal mathematics.Prerequisite: MAT 154.

MAT 290 Topology I, MAT 291 Topology II: Basic concepts of topological spaces, continuous functions, connected spaces, compact spaces, and metric spaces.

MAT 330 Complex Analysis: Vector Analysis: Green's theorem, potential theory, divergence, and Stokes' theorem. Complex Analysis: Analyticity, complex integration, Taylor series, residues, conformal mapping, applications.

MAT 355 Number theory and cryptology: Divisibility theory, Congruences, Residue systems, Chinese remainder theorem, Arithmetic functions, primitive roots, Quadratic reciprocity, Diophanitive approximations, Continued fractions, rational approximations,
Mathematics of secure communications, Crypto-complexity, Public key encryption, Some algorithm for encryption and decryption, RSA crypto system, Some applications of number theory and algebraic coding theory.

MAT 361 Numerical Analysis: Computational methods for numerical solution of non-linear equations, differential equations, approximations, iterations, methods of least squares, and other topics.
Vector spaces and review of linear algebra, direct and iterative solutions of linear systems of equations, numerical solutions to the algebraic eigenvalue problem, solutions of general non-linear equations and systems of equations.  Interpolation and approximation, numerical integration and differentiation, numerical solutions of ordinary differential equations. Computer programming skills required.

MAT 370 Abstract Algebra II:A more advanced approach of abstract algebra. This is a year long course which  covers  four main parts of algebra: groups, rings, modules, and field theory.

MAT 387 Analysis of algorithms: Introduction to algorithms and its importance, mathematical foundations: growth functions, complexity analysis of algorithms, summations, recurrences, sorting algorithms design and analysis: Insertion sort, divide and conquer, merge sort, heap sort, radix sorting.
Hash table, B trees, Binomial Heaps, Fibonacci Heaps. Dynamic Programming: Introduction, Matrx chain multiplication, Greedy Algorithms.
Elementary Graph algorithms: Minimum spanning trees, Single source shortest path, all pair shortest path.
String matching: Robin – Karp algorithm, Knuth – Morris Pratt algorithm, Algorithm for parallel computers, parallelism, the PRAM models, simple PRAM algorithms. P and NP Class, some NP – complete problems.

MAT 390 Coding theory: Communication processes. Channel matrix. Probability relation in a channel. The measure of information. Entropy function – Properties of entropy function. Channel capacity. Special types of channels. Binary symmetric channel. Encoding. Block code. Binary code. Binary Huffman code. Shannon – Fano Encoding procedure. Noiseless coding theorem. Shannon’s first theorem.
Error – correcting codes. Examples of codes. Hadamard matrices and codes. Binary Colay code. Matrix description of linear codes. Equivalence of linear codes. The Hamming codes. The standard array. Syndrome decoding.

MAT 398 Final Project:

MAT 421-MAT 422: 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 433 Numerical Methods: Propagation of errors, approximation and interpolation, numerical integration, methods for the solution of equations,Runge- Kutta and predictor-corrector methods.Prerequisite:MAT 275,MAT 155 and MAT 250

MAT 462 Geometric Structures: A study of topics from Euclidean geometry, projective geometry, non-Euclidean geometry and transformation geometry. Offered every fall.

MAT 472 Number Theory with Cryptography: Structure of the integers,prime factorization, congruences, multiplicative functions,primitive roots and quadratic reciprocity, and selected applications including cryptography.Prerequisite:MAT 155.

MAT 475 Abstract Algebra: Groups,subgroups,cosets,and homomorphisms,rings
and ideals,integral domains,and fields and field extensions. Applications.Offered every winter.

MAT 451, Introduction to Algebra I:This is an introduction to the graduate algebra. Groups, normal and simple groups, permutation groups, Abelian groups, Sylow theorem, Jordan-Holder theorem.

MAT 452, Introduction to Algebra II:An introduction to rings and ideals; integral domains; and field and field extensions. Geometric constructions and an introduction to galois theory.

MAT 472, Number Theory, 8:Structure of the integers, prime factorization, congruences, multiplicative functions, primitive roots and quadratic reciprocity.

MAT 521, Analysis I: 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: 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 525, Topology: An introductory course with emphasis on the algebraic and differential topology of manifolds. Topics include simplicial and singular homology, de Rham cohomology, and Poincare duality.

MAT 551, Algebra I: 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: 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.

STAT 551, Introduction to Mathematical Statistics, 8: 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. Prerequisites: MAT 421

MAT 481, Cryptography: Elementary concepts in cryptography; classical cryptosystems; modern symmetric cryptography; public key cryptography; digital signatures, authentication schemes; modular arithmetic, primitive roots, primality testing. At least one mathematics course at or above the 3000 level and facility with either a programming language or a computer algebra system is required. 4176: Discrete logs; pseudoprime tests; Pollard rho factoring; groups; quadratic residues; elliptic curve cryptosystems and factoring; coding theory; quantum cryptography.

MAT 631 Complex Analysis: 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: 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. Prerequisites: MAT 421, MAT 451.

MAT 641, MAT 642 Computational Algebra: A study of the mathematics and algorithms which are used in symbolic algebraic manipulation packages. 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. Prerequisite: MTH 256 and knowledge of a scientific programming language or permission of instructor.

MAT 651, MAT 652Commutative Algebra I - II: 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, 8: An introduction to computational group theory using computer algebra packages such as GAP. 

MAT 657, MAT 658, Coding Theory I - II: 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. The approach that I will follow will be more on the computational aspects of   groups, finite fields, polynomials, etc other than the rigorous mathematical approach. 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: 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: A continuation of MAT 661, algebraic coding, turbo codes, LDPC codes, new developments in digital communications.

MAT 771, MAT 772, Algebraic Number Theory I - II: 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 773, Special topics: This course is offered every winter and is open only to students who are accepted in the Phd program. Special research topics are discussed. Permission of instructor is needed to enroll.

MAT 851, MAT 852, Algebraic Geometry I - II: 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. Prerequisites MAT 652.

MAT 870: Independent Research: Open only to students who have passed the PhD qualifying exams. Students are expected to complete a research projects at the end of this course.