Topics in geometry

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

The content consists of a selection of standard graduate topics in geometry, such as the theories of schemes, sheaves, real (semi)algebraic sets, symplectic and contact manifolds, foliations, symmetries of nonlinear differential equations, Stein surfaces, gauge theories, homologies of Floer type etc. The choice depends on students' research interests.

Readings

[1] T. Broecker, T. tom Dieck, Representations of Compact Lie Groups, Springer, 1985.
[2] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley, 1978.
[3] R. Hartshorne. Algebraic Geometry, GTM 52, Springer-Verlag, 1977.
[4] D. Husemoller, Fibre bundles, Springer, 1994.
[5] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Wiley, 1963.[6] J. Milnor, J. D. Stasheff, Characteristic classes, Princeton University Press, 1974.
[7] I. Moerdijk, J. Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University press, 2003.[8] F. W. Warner, Foundations of differentiable manifolds and Lie groups, Springer, 1983.

Objectives and competences

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

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) (according to the Statute of UL)

Lecturer's references

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]
FORSTNERIČ, Franc. A complex surface admitting a strongly plurisubharmonic function but no holomorphic functions. The Journal of geometric analysis, ISSN 1050-6926, 2015, vol. 25, iss. 1, str. 329-335. [COBISS-SI-ID 17136217]
FORSTNERIČ, Franc, LÁRUSSON, Finnur. Holomorphic flexibility properties of compact complex surfaces. International mathematics research notices, ISSN 1073-7928, 2014, vol. 2014, no. 13, str. 3714-3734. [COBISS-SI-ID 17143641]
FORSTNERIČ, Franc. Stein manifolds and holomorphic mappings : the homotopy principle in complex analysis, (Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, vol. 56). Heidelberg [etc.]: Springer, cop. 2011. X, 489 str., ilustr. ISBN 978-3-642-22249-8. ISBN 978-3-642-22250-4. [COBISS-SI-ID 16008025]
BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]
KOŠIR, Tomaž, OBLAK, Polona. On pairs of commuting nilpotent matrices. Transformation groups, ISSN 1083-4362, 2009, vol. 14, no. 1, str. 175-182. [COBISS-SI-ID 15077977]
KOŠIR, Tomaž, SETHURAMAN, B. A. Determinantal varieties over truncated polynomial rings. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2005, vol. 195, no. 1, str. 75-95. [COBISS-SI-ID 13266265]
MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0. [COBISS-SI-ID 12683097]
MOERDIJK, Ieke, MRČUN, Janez. On integrability of infinitesimal actions. American journal of mathematics, ISSN 0002-9327, 2002, vol. 124, no. 3, str. 567-593. [COBISS-SI-ID 11700057]
MOERDIJK, Ieke, MRČUN, Janez. On the integrability of Lie subalgebroids. Advances in mathematics, ISSN 0001-8708, 2006, vol. 204, iss. 1, str.101-115. [COBISS-SI-ID 14074201]
KALIŠNIK, Jure, MRČUN, Janez. A Cartier-Gabriel-Kostant structure theorem for Hopf algebroids. Advances in mathematics, ISSN 0001-8708, 2013, vol. 232, iss. 1, str. 295-310. [COBISS-SI-ID 16432473]
SAKSIDA, Pavle. On the nonlinear Fourier transform associated with periodic AKNS-ZS systems and its inverse. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2013, vol. 46, no. 46, 465204 (22 str.). [COBISS-SI-ID 16833369]
SAKSIDA, Pavle. On the generalized Maxwell-Bloch equations. Symmetry, integrability and geometry: methods and applications, ISSN 1815-0659, 2006, vol. 2, paper 038 (14 str.). [COBISS-SI-ID 13932633]
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]