This page contains some information and supplementary materials for courses I currently teach at FNSPE of CTU in Prague. All lecture notes are work in progress and they should be shared with care. If you find some errors, please feel free to contact me. In the worst case scenario I will just ignore you and pretend to be always right.

Geometrical Methods in Physics 2

Abstract: A theory of gauge fields forms the foundation of contemporary particle physics, namely of the Standard Model. The main goal of this course to to acquaint students with the mathematical apparautus required for its geometric description. We will focus on theory of principal fiber bundles and the interpretation of gauge fields as connection forms on principal fiber bundles. All theoretical concepts are demonstrated on particular examples, e.g. frame bundle, Hopf fibration and Yang-Mills field.

Outline of the lecture:

  • Recapitulation of elementary differential geometry
  • Maxwell equations in the language of differential forms, gauge invariant action, local gauge invariance, minimal interaction with the complex scalar field
  • Lie group actions and their properties, fiber bundles, principal fiber bundles, fundamental vector fields and the vertical subspace
  • Forms valued in vector spaces, forms of affine connections, Cartan structure equations, connection forms on principal fiber bundles
  • Smooth distributions and their integrability, horizontal distributions, horizontal lift and parallel transport
  • Exterior covariant derivative, curvature form, integrability of parallel transport, holonomy
  • Local connection and curvature forms, gauge transformation
  • Gauge invariant action, equations of motion of gauge theory, Yang-Mills field as an example
  • Reduction of vector bundles, associated fibration, mass fields in gauge theories.
Recommended literature::
  1. M. Fecko: Differential geometry and Lie groups for physicists, Cambridge University Press, 2006.
  2. S. B. Sontz: Principal Bundles: The Classical Case, Springer, 2015.
  3. J. Lee: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer, 2012.
  4. M. Nakahara: Geometry, topology and physics, CRC Press, 2003.

Documents: lecture notes (in Czech)

Algebraic topology

Abstract: A study of modern mathematical and theoretical physics requires one to acquire an ever increasing knowledge of mathematical apparautus. The main goal of this course is to acquaint students with basic methods used in algebraic topology, namely elements of category theory, homototopies, homological algebra and cohomology. An important objective is to enhance the mathematical language by concepts appearing universally across disciplines like differential geometry and abstract algebra. During excercise sessions, students will try practical calculations of introduced mathematical structures.

Outline of the lecture:

  • Homotopy relation
  • Fundamental group
  • Categories and functors
  • Cellular and simplicial complexes
  • Simplicial and singular homology and their relation
  • de Rham cohomology, Poincaré lemma and duality
  • Sheaves and associated Čech cohomology
  • Čech - de Rham cohomology
  • Cohomology of Lie algebras.
Recommended literature::
  1. Hatcher, Allen: Algebraic Topology. Cambridge University Press, 2002.
  2. Bott, Raoul, and Tu, Loring W.: Differential forms in algebraic topology. Vol. 82. Springer Science & Business Media, 2013.
  3. Tu, Loring W.: Differential geometry: connections, curvature, and characteristic classes. Vol. 275. Springer, 2017.
  4. Spanier, Edwin H.: Algebraic topology. Vol. 55. No. 1. Springer Science & Business Media, 1989.
  5. Knapp, Elias, and Knapp Anthony W.: Lie groups, Lie algebras, and cohomology. Vol. 34. Princeton University Press, 1988.

Documents: lecture notes (in Czech)

Lie groupoids and algebroids

Abstract: Lie groups and algebras are undoubtedly the cornerstone of modern theoretical physics and differential geometry. It turns out that in certain situations (point-dependent symmetries, non-transitive actions of Lie groups), it is appropriate to extend this concept. Groupoids are a natural generalization of groups. In simplified terms, group elements are replaced by "arrows" that can be associatively multiplied only when they "connect" in a certain sense. An illustrative example is the transition from the group of homotopy classes of loops at a given point to the groupoid of homotopy classes of general curves. If all the involved sets are manifolds and the corresponding operations are smooth, we speak of Lie groupoids. As the name suggests, Lie algebroids are the corresponding "infinitesimal objects." Mathematically, these are vector bundles, whose module of smooth sections forms a Lie algebra. In this lecture, we will explore basic concepts and constructions, with a focus on various examples of Lie groupoids/algebroids across differential geometry. We will also discuss their significance in Poisson and symplectic geometry. Students are expected to have a good understanding of basic concepts in differential geometry, especially in Lie theory and the theory of principal bundles.

Outline of the lecture:

  • Lie groupoids
  • Transitivity and local triviality
  • Bisections and actions
  • Algebraic constructions with Lie groupoids
  • Lie algebroids
  • Lie functor
  • Exponential maps, adjoint representations
  • Algebraic constructions with Lie algebroids
  • Poisson structures and Lie algebroids
  • Poisson and symplectic Lie groupoids
Recommended literature::
  1. Mackenzie, Kirill: General theory of Lie groupoids and Lie algebroids. Cambridge University Press, 2005.
  2. Moerdijk I., Mrcun J.: Introduction to Foliations and Lie groupoids. Cambridge University Press, 2003.
  3. Pradines J.: In Ehresmann‘s footsteps: from Group Geometries to Groupoid Geometries. Banach Center Publications vol. 76, 87-157, 2007.
  4. Weinstein A.: Poisson geometry. Differential geometry and its applications 9(1-2), 213-238, 2007.

Documents: lecture notes (in English)