Enrollment into the program.

Positive result from the written exam is necessary to enter the oral exam.

# Advanced chapters in mathematics for physicists

Basic concepts of group theory: Axioms of the group, basic examples of finite, discrete an continuous groups. Homomorphisms and isomorphisms, normal subgroups, factor groups and factor sets. Actions of groups, orbit spaces.

Basic concepts of theory of representations: Linear representations of finite groups, Irreducibility of representations. Characters. Irreducible representations of the groups U(1) and SU(2). Examples of application.

Basic concepts of manifold theory: Definition of the manifold by means of charts and atlases, basic examples. Fundamental notions of the topology of manifolds. Vector fields, differential forms and more general tensor fields. Gauges and gauge transformations.

J. Rotman, An introduction to the theory of groups; Springer, New York (1994)

J. P. Serre, Linear Representations of Finite Groups; Springer, Berlin (1977)

M. Stone, P. Goldbart, Mathematics for Physics; Cambridge University Press (2009)

W. S. Massey; Algebraic topology: An introduction; Springer, Berlin (1997)

Objectives:

Some advanced mathematical chapters which play an important role in physics and in particular in mathematical physics are covered.

Competences:

Students learn to solve problems in certain advanced mathematical chapters. They learn to apply relevant mathematical constructions in the study of certain physical situations.

Knowledge and understanding:

Students acquire the means for the understanding the mathematical aspects of certain topics they already encountered and of many topics that they will study in the future.

Application:

Students will use the knowledge acquired in the course in the study and understanding of most topics of mathematical physics that they will encounter in the future.

Reflection:

Students learn to appreciate certain advantages, offered by the use of advanced mathematical tools in the treatment of various problems in mathematical physics.

Transferable skills:

Students learn to study comprehensive fields of mathematics selectively. More precisely, they acquire the skill of selecting those parts of various extensive mathematical theories, which are needed in the rigorous treatment of physical problems.

Lectures, exercises, homework, consultations

Written exam in the form of a homework assignment

Oral exam

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

prof. dr. J. Mrčun:

• J. Mrčun: On isomorphisms of algebras of smooth functions. Proc. Amer. Math. Soc. 133 (2005), 3109-3113.

• I. Moerdijk, J. Mrčun: Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge (2003).

• I. Moerdijk, J. Mrčun: On the integrability of Lie subalgebroids. Adv. Math. 204 (2006), 101-115.

prof. dr. P. Saksida:

• P. Saksida: On the nonlinear Fourier transform associated with periodic AKNS_ZS systems and

its inverse, J. Phys. A: Math. Theor., 46 (2013), 22pp

• P. Saksida: On zero-curvature condition and Fourier analysis, , J. Phys. A: Math. Theor., 44 (2011), 19pp

• P. Saksida: , Integrable anharmonic oscillators on spheres and hyperbolic spaces, Nonlinearity, 14, (2001), 18pp

prof. dr. S. Strle:

• S. Strle: Bounds on genus and geometric intersections from cylindrical end moduli

spaces, Journal of differential geometry, 2003, vol. 65, str. 469-511.

• B. Owens, S. Strle: A characterization of the Z [sup] n [oplus] Z([delta]) lattice and definite nonunimodular intersection forms. American journal of mathematics, 2012, vol. 134, str. 891-913.

• D. Ruberman, S. Strle: . Concordance properties of parallel links, Indiana University mathematics

journal, 2013, vol. 62, 799-814