Topics in algebra

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

The content consists of a selection of standard topics in algebra. Possible themes include group theory, commutative algebra, noncommutative algebra, linear algebra, Lie and other nonassociative algebras, homological algebra, ordered algebraic structures etc.

Readings

I. M. Isaacs, Algebra: A Graduate Course, AMS, 1994.

T. W. Hungerford, Algebra, Springer, 1974.

S. Lang, Algebra, Springer, 2002.

L. H. Rowen: Graduate Algebra: Noncommutative View, AMS, 2008.

J. J. Rotman, Advanced modern algebra, AMS, 2010.

Objectives and competences

The main goal of the course is to provide students with some important topics in algebra.

Intended learning outcomes

Knowledge and comprehension of presented concepts.
Ability to use acquired knowledge and skills.

Learning and teaching methods

Lectures, consultations, problem sessions

Assessment

Writen exam (homeworks), oral exam

Grading: 5 (fail), 6-10 (pass)

Lecturer's references

Matej Brešar:

– 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]
– BREŠAR, Matej. Introduction to noncommutative algebra, (Universitext). Cham [etc.]: Springer, cop. 2014. XXXVII, 199 str. ISBN 978-3-319-08692-7. ISBN 978-3-319-08693-4 [COBISS-SI-ID 17143897]
– 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]

Jakob Cimprič:
– CIMPRIČ, Jaka. A Real Nullstellensatz for free modules. Journal of algebra, ISSN 0021-8693, 2013, vol. 396, str. 143-150 [COBISS-SI-ID 16912729]
– CIMPRIČ, Jaka, SAVCHUK, Yurii, SCHMÜDGEN, Konrad. On q-normal operators and the quantum complex plane. Transactions of the American Mathematical Society, ISSN 0002-9947, 2014, vol. 366, no. 1, str. 135-158 [COBISS-SI-ID 16921177]
Igor Klep:
HELTON, J. William, KLEP, Igor, VOLČIČ, Jurij. Geometry of free loci and factorization of noncommutative polynomials. Advances in mathematics, ISSN 0001-8708, June 2018, vol. 331, str. 589-626.https://doi.org/10.1016/j.aim.2018.04.007, doi: 10.1016/j.aim.2018.04.007. [COBISS-SI-ID 18416217]
KLEP, Igor, ŠPENKO, Špela, VOLČIČ, Jurij. Positive trace polynomials and the universal Procesi-Schacher conjecture. Proceedings of the London Mathematical Society, ISSN 0024-6115, Dec. 2018, vol. 117, iss. 6, str. 1101-1134.https://doi.org/10.1112/plms.12156, doi: 10.1112/plms.12156. [COBISS-SI-ID 18564185]
HELTON, J. William, KLEP, Igor, MCCULLOUGH, Scott. The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. Journal of the European Mathematical Society, ISSN 1435-9855, 2017, vol. 19, iss. 6, str. 1845-1897. http://dx.doi.org/10.4171/JEMS/707, doi: 10.4171/JEMS/707. [COBISS-SI-ID 18057817]

Primož Moravec:
– MORAVEC, Primož. Unramified Brauer groups of finite and infinite groups. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 6, str. 1679-1704 [COBISS-SI-ID 16521305]
– DELIZIA, Constantino, MORAVEC, Primož, NICOTERA, Chiara. Groups with all centralizers subnormal of defect at most two. Journal of algebra, ISSN 0021-8693, 2013, vol. 374, str. 132-140 [COBISS-SI-ID 16556889]
– JEZERNIK, Urban, MORAVEC, Primož. Bogomolov multipliers of groups of order 128. Experimental mathematics, ISSN 1058-6458, 2014, vol. 23, iss. 2, str. 174-180 [COBISS-SI-ID 17109593]

Peter Šemrl:
– ŠEMRL, Peter. The optimal version of Hua's fundamental theorem of geometry of rectangular matrices. Memoirs of the American Mathematical Society, ISSN 0065-9266, 2014, vol. 232, no. 1089, str. 1-74 [COBISS-SI-ID 16947545]
– ŠEMRL, Peter. Invertibility preservers on central simple algebras. Journal of algebra, ISSN 0021-8693, 2014, vol. 408, str. 42-60 [COBISS-SI-ID 16962649]