Analysis on manifolds

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M3
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

The notion of a smooth manifold and map. Basic constructions and examples. The differential. The tangent bundle and the tangent map. Manifolds with boundary. Group actions on manifolds. Covering and quotient manifolds. Fiber bundles and vector bundles.
Immersions and submersions. Submanifolds. Embedding manifolds to Euclidean spaces.
Vector fields as dynamical systems Flows. Commutator of vector fields. The theorem of Frobenius. The tubular neighborhood theorem. Index of a critical point of a vector field. The Poincaré-Hopf theorem.
Lie groups. The exponential map. Invariant vector fields. The Lie algebra of a Lie group. The adjoint representation.
Sard's theorem. The Thom transversality theorem. The intersection number of submanifolds. Morse functions.
Other possible topics:
Differential forms and integration. Stokes' theorem. De Rham cohomology. Poincaréjeva dualnost. Eulerjev and Thomov class.
Riemannian manifolds. Volume form and integration. The Hodge *-operator. Laplace operator. Harmonic forms. Hodge decomposition.

Readings

W. M. Boothby: An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, Orlando, 1986.
V. Guillemin, A. Pollack: Differential Topology, Prentice Hall, Englewood Cliffs, 1974.
M. W. Hirsch: Differential Topology, Springer, New York, 1997.
M. Spivak: Calculus on Manifolds, W. A. Benjamin, New York-Amsterdam, 1965.
F. W. Warner: Foundations of Differentiable Manifolds and Lie Groups, Springer, New York-Berlin, 1983.

Objectives and competences

Students learns some of the main basic concepts and methods of the theory of smooth manifolds and its connection to related fields of mathematics such as analytic and algebraic geometry, the theory of Lie groups, the theory of Riemann surfaces, etc. Basic methods of analysis, algebra and topology are applied in the course.

Intended learning outcomes

Knowledge and understanding: Methods of mathematical analysis, algebra and topology are applied and further developed in the context of smooth manifolds.
Application: The theory of smooth manifolds is one of the most interdisciplinary areas of modern mathematics. It is a basis of a number of areas such as analytic, algebraic and differential geometry, the theory of Lie groups, the theory of Riemann surfaces, dynamics, etc. Manifolds are a major tool in natural and technical sciences.
Reflection: Understanding the theory on the basis of examples. Acquiring skills in applying the theory to diverse scientific problems.
Transferable skills: The ability to identify, formulate and solve scientific problems using methods of smooth manifolds. Developing skills of using the domestic and foreign literature.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (homework, written exam, oral exam):
Homework and/or written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Franc Forstnerič:
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]
FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]
FORSTNERIČ, Franc. Manifolds of holomorphic mappings from strongly pseudoconvex domains. The Asian journal of mathematics, ISSN 1093-6106, 2007, vol. 11, no. 1, str. 113-126. [COBISS-SI-ID 14352473]
Janez Mrčun:
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]
MRČUN, Janez. Functoriality of the bimodule associated to a Hilsum-Skandalis map. K-theory, ISSN 0920-3036, 1999, let. 18, št. 3, str. 235-253. [COBISS-SI-ID 9163353]
Pavle Saksida:
SAKSIDA, Pavle. Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity, ISSN 0951-7715, 2001, vol. 14, no. 5, str. 977-994. [COBISS-SI-ID 10942809]
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]
SAKSIDA, Pavle. On zero-curvature condition and Fourier analysis. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2011, vol. 44, no. 8, 085203 (19 str.). [COBISS-SI-ID 15909465]