UNIVERSITY OF LATVIA
Academic Studies Programme
51461 Mathematics
(Name of Programme)
Doctor’s degree in Mathematics
(Name of Degree)
Dr.Math., Dr.mat.
Director of
Programme: Teodors Cīrulis(Name, surname)
Dr.Hab.Math., Professor(scientific degree, academic title)
AFFIRMED
at Board meeting of MathematicsStudies Programme
on February 11, 1999
…………………………………………………..
(signature, surname) /
AFFIRMED
at Council meeting of Faculty of Physics and Mathematics,on June 15, 1999
……………………………………………
(signature, surname)
AFFIRMED
at Science council Meeting of ULJan. 26, 2000
Resolution No…………………….
Scientific pro-rektor
…………………………………………………..
(signature, surname) /
AFFIRMED
at the LU Senate Meeting ULFebr.28, 2000
Resolution No…175
Chairmen of Senate
…………………………………
(signature, surname)
Table of Contents
Aim of Studies 3
Characterisation of Programme 3
The Contents of the Doctor’s Studies Programme 3
Preconditions to doctor’s studies in mathematics 4
Enrolment Order in doctor’s studies in Mathematics 4
Co-operation in Study Programme Realisation 4
The Characterisation of the promotion paper in Mathematics 5
Form of the promotion paper 5
Series of scientific papers 5
Thesis 5
The summary of the promotion paper. 5
Program in the leading courses of the sub-branches of mathematics 7
Aim of Studies
To train highly qualified scientific specialists and lecturers in mathematics and applied mathematics.
Characterisation of Programme
The co-ordinator of studies – professor T.Cīrulis.
The place of studies – Faculty of Physics and Mathematics and institutes associated with the faculty: Institute of Mathematics of UL and LAS and Institute of Mathematics and Informatics of UL.
The studies and research is carried out in the following sub-branches:
Subbranch / Leading Professor/ Algebra and mathematical Logic / Prof.A.Šostaks (Shostaks)
/ Differential Equations / Prof. A.Reinfelds
Prof. U.Raitums
/ Discrete Mathematics and Mathematical Informatics / Prof. R.Freivalds
/ Function Theory / Prof. T.Cīrulis
/ Geometry and Topology / Prof. A.Šostaks
/ Mathematical Analysis and Functional Analysis / Prof. T.Cīrulis
/ Mathematical Physics / Prof. A.Buiķis
/ Mathematical Modelling / Prof. A.Buiķis
/ Modern elementary Mathematics and Didactic of Mathematics / Prof. A.Andžāns
/ Methods of Optimisation / Prof. U.Raitums
/ Numerical Analysis / Prof. H.Kalis
/ Theory of Probabilities and Mathematical Statistics / Prof.A.Lorencs
The Contents of the Doctor’s Studies Programme
Full time studies in doctor’s study programme correspond to 144 credits which are distributed in the following way:
1 Acquirement of the theoretical courses
the main course of the sub-branch (the programme in supplement) 16 credits
the course of the specialisation (the contents is denoted individually and the specialist must approve his knowledge of English) 14 credits
2 Individual research work and elaboration of the promotion paper 90 credits
3 Practice (teaching practice at a higher educational establishment or practice in applied mathematics in one of the scientific institutions) 12 credits
4 Courses of option or individual additional courses (if necessary), preparation for and taking part in scientific conferences, seminars, workshops 12 credits
Preconditions to doctor’s studies in mathematics
Preconditions to doctor’s studies in mathematics are master’s degree in Mathematics (Mg. math.), Physics (Mg. phys.), Computer Sciences (Mg. comp.) and diplomas of the higher education adequate to these degrees.
As an exception those who have acquired master’s degree in other branches of sciences can also qualify for doctor’s studies in mathematics on condition that their doctor’s studies envisage common research work both in mathematics and in the branch they have acquired their master’s degree.
Enrolment Order in doctor’s studies in Mathematics
A commission consisting of experts on candidates for a doctor’s degree makes a decision about the applicant’s fitness to the studies. The commission consists of Prof. T.Cīrulis (chairman), Prof. A.Lorencs, Prof. A.Šostaks, Prof. H.Kalis, Prof. U.Raitums, Prof. A.Reinfelds.
The commission evaluates the applicant’s scientific research scheme, the general knowledge level in mathematics and, if necessary, points to the subject advisable to acquire during the studies.
According to the commission’s motion, the applicant is immatriculated for studies.
Working together with the scientific supervisor and taking into account the suggestions by the expert commission, the candidate develops his individual research and study programme. It is confirmed at the Administrative Unit meeting under the supervision of the professor of the corresponding sub-branch and is submitted to the Doctor’s study Department.
Co-operation in Study Programme Realisation
Studies and research work are carried out in co-operation with the following universities
Umea University (Sweden);
Bremen University (Germany);
Charle’s University (Czechia);
Helsinki University (Finland);
York University (Canada);
Kaiserslautern University (Germany);
Tartu University (Estonia);
Vilnius University (Lithuania);
Vilnius Technological University (Lithuania).
There are seven more universities with which the co-operation is little a bit less intensive.
The Characterisation of the promotion paper in Mathematics
Promotion paper consists of an independent scientific research of an actual and perspective problem in one of 12 sub-branches. It should involve new scientific conclusions and suggestions, it should give a proof that the applicant for the scientific degree has high theoretical and professional competence and abilities for independent scientific work.
Form of the promotion paper
The promotion paper may have two forms:
1 a series of scientific papers;
2 thesis.
Series of scientific papers
The series of scientific papers contains:
a) not less than 5 publications in which the most important results and usage’s have been well grounded. They have to be submitted in the original language of the publication. If the papers are published in Latvian, English, German, Russian or French, it is not necessary to translate them. Otherwise it is necessary to add translation in English or Latvian.
b) the description of the scientific papers. The author has to:
· reason the unity of the thematic of the submitted papers;
· show which papers have resulted in important effects and conclusions
· give a short information about the new methods, their applications, conceptions or new mathematical conclusions developed in the submitted papers.
The description must contain a full reference to the papers mentioned in point a).
The language of the description – English or Latvian; the advisable volume – from 10 to 30 pages.
The number of submitted sets of papers – at least 5.
Thesis
It shows more profoundly than scientific papers the object of the research, the applied methods, results and their significance, proofs and conclusions. The main results of thesis must be published in scientific journals at least 3 months before submitting thesis for defence.
The language of thesis - Latvian or English; the advisable volume – from 40 to 70 pages, but not more than 200 pages, together with the supplements and the list of references.
The number of submitted copies – at least 5.
The summary of the promotion paper.
The summary is a short report given by the applicant on the results of the research in order to inform the scientific society. The summary must be added to the promotion paper not withstanding its form.
It must include:
· clearly formulated thematic, the statement of the problem, the aims and tasks of the paper;
· a short historical survey of the problems under investigation and congenial problems;
· the structure and volume of the paper;
· the survey of the basic results. When formulating the most results, it must be precisely defined where they are taken from.
· it must have a clear definition what novelties compared with the previously known facts, does the paper contain;
· the information about conferences, seminars, etc. in which the applicant has delivered lectures on themes connected with his promotion paper;
· a list of publication to which the promotion papers refers, at first giving the list of own publications.
The summary must be identical in Latvian and English. In the case of necessity it can be submitted in some other language. The volume of the summary – 1 signature containing 20 – 25 pages. It should be paper – bound, printed or photocopied. The number of submitted copies – not less than 15 copies both in English and Latvian.
The applicant himself must sign the summary but the title page must be wised by the chairmen of the council.
Program in the leading courses of the sub-branches of mathematics
The program consists of 92 questions covering all sub-branches:
1. Algebra and Mathematical Logic
2. Differential Equations
3. Discrete Mathematics and Mathematical Informatics
4. Function Theory
5. Geometry and Topology
6. Mathematical Analysis and Functional Analysis
7. Mathematical Modelling
8. Mathematical Physics
9. Modern Elementary Mathematics and Didactics of Mathematics
10. Methods of Optimisation
11. Numerical Analysis
12. Theory of Probabilities and Mathematical Statistics.
Each doctoral student must study 20 of these 92 questions, chosen from the corresponding sub-branch or from related sub-branches.
For each doctoral student the proposal on the choice of these 20 questions is made by his supervisor and is certified by the director of the doctoral study program.
The amount of the program is 16 credits for each doctoral student.
1. Most important axiom systems of set theory: Zermelo – Fraenkel and Goedel – Bernays systems. Corollaries from the axioms. The axiom of choice and its equivalents: Zorn’s lemma, Kuratowski’s principle etc.
2. Cardinal numbers and their arithmetic. The continuum hypothesis CH and its generalisations and modifications (GCH, Martin’s axiom MA etc.)
3. Linearly ordered sets, completely ordered sets, minimally ordered sets. Ordinal numbers and their arithmetic. Transfinite induction. Transfinite recursion.
4. Elements of group theory: semigroup, monoid, group. Abelian groups. Normal divisor. Homomorphismus of groups and special classes of them (isomorphism, monomorphism, endomorphism, automorphism). Factorgroup and a theorem on the factorisation of a homomorphism. Group product. Free groups. Generators and identities. Representations of groups. The action of a group in the set. Transformation groups.
5. Rings. general properties of rings. Ideals. The ring of residue classes. Factorring. Field. A characteristic of a field. Field extensions. Elements of Galois theory. The basic theorem of Galois theory.
6. Topological structures in algebraic objects. Topological groups. Local properties of topological groups. Topological groups of transformations. Continuous functions in topological groups. Linear representations of compact topological groups. Topological algebra’s and topological rings.
7. The definition of category. Examples from topology, algebra, functional analysis etc. Concrete and abstract categories. Special morphisms (monomorphisms, epimorphisms, isomorphisms etc.) and special objects (initial, final etc.) Subcategories. Functors. Examples of functors. Natural transformation of functors. Isomorphism of functors. Equivalence and isomorphism of categories.
8. Limits and co-limits in categories. Initial and final structures in categories. Topological categories. Algebraic categories. Operations in categories. Operations over categories.
9. Partially ordered sets and lattices. Lattices as abstract algebra’s. Homomorphisms of lattices. Ideals in lattices. Lattices with involutions.
10. Distributive lattices. Modular lattices. Complete lattices. Galois relations in complete lattices. Infinitely distributive lattices. Completely distributive lattices. Chains.
11. Elements of algebraic and analytical number theory. Field of algebraic numbers and its basic properties. Basics of ideal theory. Classical transcendence proofs. Central results on prime distribution. Riemann z-function and Dirichlet L-function. Fast algorithms for factorisation and for primality.
12. Combinatorial structures. Elements of counting combinatorics. Transversals, latin squares, block-schemas, finite geometries. Generating functions and their algebra. The method of recurrence relations. Moebius inversion function. The orbit method.
13. Selected chapters of graph theory. Main classes of graphs. Euler and Hamilton cycles and paths. Planarity of graphs. Colouring of graphs. Numerations of graphs. The amount of some classes of graphs.
14. Ramsey theory. Classical Ramsey numbers and their generalisations. Ramsey-type structural theorems in number theory. Ramsey-type results in geometry, algebra, mathematical analysis, combinatorics. Classical minimax theorems.
15. Classial fast algorithms: Sorting algorithms. Algorithms for arithmetical operations. Algorithms for computing polynomials. Algorithms for operations with matrices.
16. Coding and cryptography. Main classes of codes: error-detecting and error-correcting codes, block codes, branching codes. Coding and decoding of messages in a discrete noisy channel. Classical (secret key) cryptosystems. Main public key cryptosystems. Cryptographic protocols. Relations between cryptography and complexity theory. Zero-information proofs. Systems for distribution of the secret information.
17. Main formal concepts of algorithm and their basic properties. Turing machines, normal algorithms, recursive functions and their equivalence. Reducibility and its formalisations. Algorithmically unsolvable problems. Kleene – Mostowski hierarchy. The characterisation of recursively enumerable sets through Diophantive predicates.
18. Basics of the classical automata theory. Finite automata, pushdown automata and their variations, Formal grammars. Classes of languages recognisable by various types of automata. Decomposition and structural synthesis of automata. Experiments with automata. Control and diagnostics of automata. Systems of automata.
19. Elements of complexity theory of algorithms. P-reducibility and equivalence of combinatorial problems. Nondeterministic Turing machines. Cook’s theorem. Examples of P-complete and NP-complete problems.
20. The concept of probabilistic algorithm. Probabilistic Turing machines, their principal possibilities in set recognition compared with those of deterministic Turing machines. Advantages of probabilistic Turing machines and various types of automata over their deterministic counterparts from the complexity point of view.
21. Probability measures. Probability spaces and probability measures. Random variables, random vectors, sequences of random variables, families of random variables. Stochastic processes, conformity and invariance of a finite dimension. Kolmogorov theorem on conformed distributions. Independence of random variables and random vectors. Conditional distributions and conditional mean values.
22. Limit problems of probability theory in the space Rn. Characteristic functions and their properties. The continuity of the correspondence between distribution functions and characteristic functions. Central limit theorem. Infinitely divisible distributions.
23. Theory of stochastic processes. Stationary processes, ergodicity theorem. Function of autocorrelation and spectral representation of stationary process. Markov chains and Markov processes. Poisson processes. Gaussian processes. Brownian motion.
24. Theory of statistical estimations. A concept of sufficient statistics. Neumann – Fischer factorisation theorem. Unbiasity, consistency and effectiveness of estimation. Moment method, sequential method and maximum likelihood method.