Topics in analysis

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

The lecturer selects some important topics in analysis, such as: several complex variables, minimal surface theory, spectral theory and other topics chosen by the lecturer.

Readings

T. H. Colding, W. P. Minicozzi, A course in minimal surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011
U. Dierkes, S .Hildebrandt, F. Sauvigny, Minimal surfaces. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 339. Springer, Heidelberg, 2010.
J. Lebl, Tasty Bits of Several Complex Variables, A whirlwind tour of the subject, https://www.jirka.org/scv/scv.pdf 2018.

B. Simon: A Comprehensive Course in Analysis, Part 4: Operator Theory, AMS, 2015

Objectives and competences

Students encounter some of the important areas of analysis, such as: several complex variables, minimal surface theory, spectral theory and other topics chosen by the lecturer.

With individual presentations and team work interactions within seminar/project activities students acquire communication and social competences for successful team work and knowledge transfer.

Intended learning outcomes

Knowledge and understanding: Students get acquainted with the subject matter, the methods, and the main results of various areas of analysis.

Application: Students will be able to use their knowledge in different mathematical and other contexts.

Reflection: Students comprehend the interplay and mutual enrichment of various areas of analysis.

Transferable skills: Students learn methods which are useful in construction and analysis of mathematical models.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):

homework, seminar paper, exam

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Miran Černe:
– 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]

– ČERNE, Miran. Nonlinear Riemann-Hilbert problems for quasilinear ¯∂-equations on the unit disc. Complex variables and elliptic equations, ISSN 1747-6933, 2018, vol. 63, iss. 2, str. 278-291. [COBISS-SI-ID 18052185]

Barbara Drinovec Drnovšek:

– DRINOVEC-DRNOVŠEK, Barbara, KUZMAN, Uroš. Lelong functional on almost complex manifolds. Complex variables and elliptic equations, ISSN 1747-6933, 2015, vol. 60, iss. 2, str. 168-180 [COBISS-SI-ID 16979289]

–DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Minimal hulls of compact sets in R3. Transactions of the American Mathematical Society, ISSN 0002-9947, 2016, vol. 368, no. 10, str. 7477-7506. [COBISS-SI-ID 17543769]

– ALARCÓN, Antonio, DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc, LÓPEZ, Francisco J. Minimal surfaces in minimally convex domains. Transactions of the American Mathematical Society, ISSN 0002-9947, Feb. 2019, vol. 371, no. 3, str. 1735-1770. [COBISS-SI-ID 18379865]

Franc Forstnerič:

– ALARCÓN, Antonio, FORSTNERIČ, Franc. Darboux charts around holomorphic Legendrian curves and applications. International mathematics research notices, ISSN 1073-7928, Feb. 2019, vol. 2019, iss.3, str. 893-922 [COBISS-SI-ID 18086233

– FORSTNERIČ, Franc. A properly embedded holomorphic disc in the ball with finite area and dense boundary. Mathematische Annalen, ISSN 0025-5831, Feb. 2019, vol. 373, iss. 1-2, str. 719-742. [COBISS-SI-ID 18364505]

– ALARCÓN, Antonio, FORSTNERIČ, Franc. Every conformal minimal surface in R3 is isotopic to the real part of a holomorphic null curve. Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, 2018, vol. 2018, iss. 740, str. 77-109. [COBISS-SI-ID 17540953]

Oliver Dragičević:

  • CARBONARO, Andrea, DRAGIČEVIĆ, Oliver. Functional calculus for generators of symmetric contraction semigroups. Duke mathematical journal, ISSN 0012-7094, 2017, vol. 166, no. 5, str. 937-974 [COBISS-SI-ID 17897305]

  • DRAGIČEVIĆ, Oliver, VOLBERG, Alexander. Bilinear embedding for real elliptic differential operators in divergence form with potentials. Journal of functional analysis, ISSN 0022-1236, 2011, vol. 261, iss. 10, str. 2816-2828. [COBISS-SI-ID 16051545]

  • DRAGIČEVIĆ, Oliver. Weighted estimates for powers of the Ahlfors-Beurling operator. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 6, str. 2113-2120. [COBISS-SI-ID 15876697].

Aleksey Kostenko:

  • KOORNWINDER, Tom H., KOSTENKO, Aleksej Sergejevič, TESCHL, Gerald. Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Advances in mathematics, ISSN 0001-8708, July 2018, vol. 333, str. 796-821 [COBISS-SI-ID 18416473]

  • ECKHARDT, Jonathan, KOSTENKO, Aleksej Sergejevič, TESCHL, Gerald. Spectral asymptotics for canonical systems. Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, 2018, vol. 2018, iss. 736, str. 285-315 [COBISS-SI-ID 18344537]

  • KOSTENKO, Aleksej Sergejevič, NICOLUSSI, Noema. On the Hamiltonian-Krein index for a non-self-adjoint spectral problem. Proceedings of the American Mathematical Society, ISSN 0002-9939, Sep. 2018, vol. 146, no. 9, 11 str. 3907-3921. [COBISS-SI-ID 18460249]