Predoctoral exam

2022/2023
Programme:
Doctoral Programme Mathematics and Physics
Orientation:
Mathematics
Year:
1 year
Semester:
first or second
Kind:
mandatory
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
0
Seminar
0
Tutorial
0
Lab
0
Content (Syllabus outline)

Revision of basic mathematical notions from the fields of analysis, algebra, discrete mathematics, theoretical computer science and probability.

Readings

S. Lang, "Undergraduate Algebra", Springer, New York, 2005.

S. Lang, "Undergraduate Analysis, 2nd edition", Springer, New York, 1997.

W. Rudin, "Principles of Mathematical Analysis, 3rd edition", McGraw-Hill, New York-Auckland-Düsseldorf, 1976.

J. H. van Lindt, R.M. Wilson, "A course in combinatorics", Cambridge University Press, Cambridge, 1992.

D. Stirzaker, "Probability and Random Variables, A beginner's guide", Cambridge University Press, 1999.

Michael Sipser, Introduction to the Theory of Computation, Cengage Learning (2013).

Objectives and competences

The goal of the course is to provide the students with the background material needed for following the advanced classes and for mathematical research.

Intended learning outcomes

The students revise some basic background material needed for their future work.

Learning and teaching methods

Individual learning and concultations.

Assessment

Written exam
Grades: pass, fail.

Lecturer's references

Matej Brešar:
BREŠAR, Matej. Functional identities on tensor products of algebras. Journal of algebra, ISSN 0021-8693, 2016, vol. 455, str. 108-136. [COBISS-SI-ID 17625945]
BREŠAR, Matej. Algebras in which non-scalar elements have small centralizers. Linear and Multilinear Algebra, ISSN 0308-1087, 2015, vol. 63, no. 9, str. 1864-1871. [COBISS-SI-ID 17160537]
BREŠAR, Matej, ŠPENKO, Špela. Functional identities of one variable. Journal of algebra, ISSN 0021-8693, 2014, vol. 401, str. 234-244.[COBISS-SI-ID 16842329]

Miran Černe:
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. On Beurling's boundary differential relation. Israel journal of mathematics, ISSN 0021-2172, 2014, vol. 199, iss. 2, str. 831-840. [COBISS-SI-ID 17144153]
ČERNE, Miran. Beurling's boundary differential relations on multiply connected domains. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2015, vol. 428, iss. 1, str. 544-562. [COBISS-SI-ID 17270873]

Primož Potočnik:
BERČIČ, Katja, POTOČNIK, Primož. Two-arc-transitive two-valent digraphs of certain orders. Ars mathematica contemporanea, ISSN 1855-3966. [Tiskana izd.], 2016, vol. 11, no. 1, str. 127-146. [COBISS-SI-ID 1538308036]
POTOČNIK, Primož, WILSON, Stephen. Linking rings structures and semisymmetric graphs: Cayley constructions. European journal of combinatorics, ISSN 0195-6698, 2016, vol. 51, str. 84-98. [COBISS-SI-ID 17462361]
POTOČNIK, Primož, SPIGA, Pablo, VERRET, Gabriel. Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs. Journal of combinatorial theory. Series B, ISSN 0095-8956, 2015, vol. 111, str. 148-180. [COBISS-SI-ID 1537132228]