Erasmus Mundus Master Course on Discrete Mathematics

Transcription

1 Erasmus Mundus Master Course on Discrete Mathematics Universidade de Aveiro (UA) Aveiro, Portugal Uniwersytet Adama Mickiewicza (UAM) Poznań, Poland Universitat Politècnica de Catalunya (UPC) Barcelona, Spain

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3 1 GENERAL INFORMATIONS 3 1 General informations 1.1. Discipline: mathematics; 1.2. The full-degree programme covers: two years (120 ECTS credits); 1.3. Student population and number of staff involved in the Master Course: 15 (minimum) - 30 (maximum) students and 15 professors for core courses; 1.4. Final degree delivered: Multiple Master degree in Mathematics by Universidade de Aveiro, Uniwersytet Adama Mickiewicza and Universitat Politècnica de Catalunya. 2 Detailed Description of the Master Course Study programme and recognition 2.1. Objectives of the Master Course: Many models of real-world problems are discrete or can be discretized. On the other hand the development of computer science allows the resolution of discrete problems of increasing complexity. Discrete Mathematics play an essencial role for solving this type of problems and has strong connections with theoretical computer science, operations research and several other branches of mathematics, as it is the case of algebra and number theory. This Master Course provides theoretical and practical background in Discrete Mathematics Structure and programme: Course Status h/ects 1 st Semester (UA) Seminar obligatory 30/2 Graph Theory I obligatory 45/7 Combinatorics obligatory 45/7 Algorithms and Complexity I elective 45/7 Combinatorial Algorithms I elective 45/7 Orders and Finite Structures elective 45/7 Network Optimization elective 45/7 Courses from other local M.Sc.C. elective 45/7 Expected number of h/ects 210/30

4 2 DETAILED DESCRIPTION OF THE MASTER COURSE 4 Course Status h/ects 2 nd Semester (UAM) Seminar obligatory 30/2 Probabilistic Methods in Combinatorics obligatory 45/7 Graph Algorithms obligatory 45/7 Random Structures elective 45/7 Combinatorial Algorithms II elective 45/7 Graph Theory II elective 45/7 Randomize Algorithms elective 45/7 Courses from other local M.Sc.C. elective 45/7 Expected number of h/ects 210/30 Course Status h/ects 3 rd Semester (UPC) Seminar obligatory 30/2 Comb. and Alg. Methods in Geometry obligatory 45/7 Algebraic Combinatorics obligatory 45/7 Mathematical Foundations of Cryptology elective 45/7 Algebraic Graph Theory elective 45/7 Finite Geometries elective 45/7 Algorithms and Complexity II elective 45/7 Courses from other local M.Sc.C. elective 45/7 Expected number of h/ects 210/30 4 o Semester (University of supervisor) Seminar obligatory 30/2 M.Sc. Thesis preparation obligatory /28 Expected number of h/ects 30/30 Short Description of the Courses 1 st Semester (UA) Graph Theory I. The basic concepts and results are introduced and a wide variety of applications both to other branches of mathematics and to real-world problems are analyzed. This course includes: connectivity; Euler and Hamilton cycles; planar graphs; independent sets and cliques; matchings; edge and vertex colorings. Furthermore, the study of several algebraic tools, usually used in graph theory, as it is the case of subspaces associated with graphs, adjacency and Laplacian eigenvalues and eigenvectors and Farkas lemma for graphs are also included. Combinatorics. This course is intended as an introduction to the study of how to deal with discrete sets to conclude the existence of

5 2 DETAILED DESCRIPTION OF THE MASTER COURSE 5 some pattern structures, and to enumerate or to construct them. The course includes the basic combinatorial principles of counting; arrangements and combinatorial identities; recurrence relations; generating functions; combinatorial numbers; divisibility and modular arithmetic; latin squares and magic squares; combinatorial designs. Orders and Finite Structures. The main focus of this course is the study of partial ordered sets from different points of view with applications. The course covers the following topics: preliminares on relations and functions; posets; sequences and subsequences; orderpreserving functions and fixed points; chains and antichains; subrelations and extensions; weak, interval, and semitransitive order relations; dimension of posets and other parameters; comparability graphs and comparability invariants; lattices and Boolean algebras; partial order relations on oriented matroids of sign vectors; abstract simplicial complexes. Algorithms and Complexity I. This course aims to provide the general principles and strategies in the design of fundamental algorithms and abstract data types, as well as the tools for the analysis of algorithm complexity. Particular attention is given to the study of powerful data structures for the efficient representation of graphs. Network Optimization. Network Optimization lies in the middle of the two big classes of optimization problems: continuous and discrete. The links between Linear Programming and Combinatorial Optimization can be achieved by the representation of the constrained polyhedron as the convex hull of its extreme points. Because of their structure and also because of their intuitive character, network models provide ideal techniques for explaining many of the fundamental ideas in both continuous and discrete optimization. The purpose of the course is to provide a fairly comprehensive and up-to-date study of algorithms and methods of discrete and continuous models of network optimization. Combinatorial Algorithms I. Three classes of combinatorial algorithms are considered. The algorithms of generating combinatorial structures such as binary sequences, permutations, number partitions, set partitions, graphs, etc. The heuristic search algorithms such as hill-climbing, simulated annealing, tabu ou genetic search. The last discussed group of algorithms are there based on group theory and symmetry. 2 nd Semester (UAM) Probabilistic Methods in Combinatorics. Renowned Hungarian mathematician Paul Erdös observed over fifty years ago that it was sometimes easier to show that the majority of objects of a family have

6 2 DETAILED DESCRIPTION OF THE MASTER COURSE 6 a certain property than to specify one object with this particular property. This type of argumentation is the most typical example of the so-called probabilistic method. The aim of the course is to present a variety of different probabilistic techniques of this type, both elementary and advanced. Random Structures. A random structure is defined as a probability space whose elements are combinatorial structures. A random structure can also be deemed to be the result of a random experiment. Such probabilistic objects were introduced to discrete mathematics in order to show the existence of combinatorial structures displaying certain unusual properties but soon the theory of random structures become a separate branch of combinatorics. The lectures present the basic models of random subsets and permutations and their elementary properties. Selected topics on the evolution of random graphs are also presented. Graph Algorithms. The aim of the course is to present the basic algorithms of the graph theory and their applications to real-world problems. The lectures focus on the algorithms related to connectivity, optimal trees, the shortest path problem, optimal colorings, Euler tours (the Chinese Postman Problem) and Hamiltonian cycles (the Travelling Postman Problem), the optimal assignment problem, planarity and flows in networks. Apart from the pure graph theoretical tools, the Boolean functions methods and linear programming approach are also employed. Combinatorial Algorithms II. Combinatorial algorithms deal with the problems of generating combinatorial objects such as binary sequences, permutations, graphs etc. Some of the problems discussed during the lectures concern representation and generation of combinatorial objects, sorting and searching. Graph Theory II. The course is a continuation of the Graph Theory I course and will supplement the knowledge provided during Graph Theory I by presenting the latest developments in the graph theory. Particular attention will be paid to structural theorems such as the Erdös-Stone theorem or the Szemeredy regularity lemma. The course is based on the 1997 Diestel s textbook and its character is purely theoretical. Randomized Algorithms. The course presents a new trend in the theory of algorithms which introduces random elements to the computations. Particular attention will be paid to the Las Vegas and Monte Carlo algorithms, their classes of computational complexity and basic analytic tools used for these algorithms. The core of the course is the presentation of the applications of such random algorithms in geometric problems (computer graphics), cryptographic problems (checking whether a given number is a prime), pattern matching (DNA structure), and the theory of graphs and computer

7 2 DETAILED DESCRIPTION OF THE MASTER COURSE 7 networks. 3 rd Semester (UPC) Combinatorial and Algorithmic Methods in Geometry. The course presents the basic topics and structures of discreta and algorithmic geometry: convex hulls; triangulations and other subdivisions; Voronoi diagrams and Delaunay triangulations; hyperplane arrangements. Particular attention is paid to the combinatorics of convex polytopes, covering topics such us: lattice of faces, cyclic polytopes, and the upper bound theorem. Algebraic Combinatorics. The course consists of two main topics. First, classical problems in combinatorial enumeration are introduced through the use of power series and symbolic methods. Besides classical combinatorial structures, such as permutations and partitions, more advanced topics are covered, such us lattices paths and polyominoes. The second part of the course focusses on combinatorial number theory, with emphasis on sum-set problems and connections to graph coloring. Mathematical Foundations of Cryptology. The course presents the basic concepts of number theory, finite fields, and elliptic curves, that are essential for the applications in cryptology. Then it is shown how these topics ar are an essential tool for the study and development of cryptology. Concrete examples are given of cryptographic systems that make use of the tools introduced. Algebraic Graph Theory. This course is devoted to the interplay between graph theory and algebra, which has produced beautiful and important results in the field. After introducing the basic eigenvalue techniques, it turns to the topics of interlacing, strongly regular graphs, two-graphs, line-graphs, and the Laplacian. Applications to combinatorial chemistry are also discussed. Finite Geometries. The first topics covered are projective and affine planes, as prototypes of finite combinatorial geometries. General projective and affine spaces are introduced later. As more advanced topics, polar spaces and generalized quadrangles are introduced. Connections to Latin squares and codes are also discussed. Algorithms and Complexity II. This is an advanced course, covering the following topics: dynamic programming, branch and bound, random algorithms, derandomization, parametrized complexity (FPT classes), approximation algorithms. Both theoretical and practical aspects of algorithmics are discussed.

8 2 DETAILED DESCRIPTION OF THE MASTER COURSE Competencies and learning outcomes: The Master Course graduates are able to analyze and model complex problems from different areas and implement efficient algorithms for their resolution. They are also prepared for doing research in discrete mathematics and related topics and are particularly qualified to work in modern organizations, dealing with digital technology, cryptography, networking, finance, assurance companies, etc. Admission, application, selection, examination criteria 2.4. The ranking of the applications is done taking into account the following criteria: (a) The certificate of graduation degree or equivalent document (Mathematics or Computer Science diploma is preferred); (b) List of approved courses with marks and scale; (c) Curriculum vitae, including information about English language skills; All documents have to be translated to English and sent by regular mail to the following address: European Master Course on Discrete Mathematics Department of Mathematics, University of Aveiro Aveiro, Portugal. For the 2004/2006 edition, only the students whose applications reach the University of Aveiro before May 27, 2004, are candidates to a Erasmus Mundus scholarship of euros per year. In future editions, beyond 2004/2006, if necessary additional admission examinations can be introduce. Mobility arragements 2.5. The course is organized in four semesters with core courses given in the three first semesters. The fourth semester has a seminar and is devoted to the finalization of the Master Thesis. The mobility of the students must be as follows: (a) The first semester will be realized at UA, which is the co-ordinate institution; (b) The second semester will be realized at UAM; (c) The third semester will be realized at UPC; (d) The fourth semester will be realized for each student at university of her/his supervisor. Each course lasts one semester and ends with an examination (except seminars). In each of the three universities the students have to complete all the obligatory courses, the seminar and at least one elective

9 2 DETAILED DESCRIPTION OF THE MASTER COURSE 9 course, which totalize 23 ECTS. In each university the students are allowed to complete one additional elective course. Language policy 2.6. Language policy applied within the Master consortium: The core courses (all obligatory and some elective) will be given in English as well as its final examinations. The students are allowed to choose some courses of local Master Courses in local languages. The Master Course thesis will be written in English and also the Master degree examination will be in English. In each university a course of national survival language will be available for students and scholars. Quality assurance and evaluation 2.7. Quality mechanism and evaluation: The study of Erasmus Mundus Master Course on Discrete Mathematics is a credit points (ECTS points) study. It means that in order to complete the programme, the student has to score a specified number of credit points granted for both obligatory and elective courses. The system is governed by the following body of rules: (a) Each course lasts one semester and ends with an examination (except seminars). Each examination has at most to trials. The course credit points are credited to the student s account upon completion of a given course. The number of points credited for each course is given in the course list table. (b) The student has to complete all obligatory and some elective courses, totalizing (jointly with the seminars) 120 ECTS points. (c) The student is free to select not more than one elective course. (d) The total number of credit points scored by the student in each semester cannot be less than 23 (with the exception of the last semester of studies where the minimum is 30) and no more than 37. (e) During the first two weeks of each semester the student can change the courses he enrolled in this semester. The student cannot, however, overtake the minimal and maximal number of points scored during a semester. Withdrawing from a given course later than in the first two weeks of the semester will be equal to failing this course. All the changes should be registered. The grading scale used for evaluation in the three universities of the Master Course consortium is Failed (0-4.9/10), Approved (5-6.9/10), Good (7-8.9/10) and Very Good (9-10/10) Qualitative assessment mechanism: The final mark is rounded to units weighted average of the following components:

10 2 DETAILED DESCRIPTION OF THE MASTER COURSE 10 ECTS weighted average of a all courses, with weight 2; mark obtained for the thesis, with weight 1; mark obtained from the Master degree examination, with weight 1;