Title of the course: Introduction to Topological Invariants with Applications to Gauge Theory
Instructor’s name: Assoc. Prof. Özgür Kelekçi
Institution: University of Turkish Aeronautical Association
Dates: 31 August – 6 September 2026
Prerequisites: Linear Algebra; Multivariable Calculus
Level: Graduate and advanced undergraduate
Abstract: Topological invariants are quantities that remain unchanged under continuous deformations and provide powerful tools for detecting global properties of spaces and fields. In gauge theory, such invariants arise naturally in the study of holonomy, flux quantization, monopoles, and topological action functionals. They explain how global features of the underlying space may lead to physically observable effects even when the local differential equations appear trivial.
In these lectures we aim to provide an introductory presentation of some basic topological invariants with emphasis on their applications to gauge theory. We will discuss homotopy and the fundamental group, introduce differential forms and de Rham cohomology, and explain how closed and exact forms encode global obstructions. We will then study classical examples from gauge theory including the Aharonov–Bohm effect, the Dirac monopole, the first Chern number, winding number and degree, and the abelian Chern–Simons functional together with its relation to linking number and helicity.
The course is intended as a first systematic introduction for advanced undergraduate and graduate students who wish to acquire both mathematical background and physical intuition for topological structures arising in gauge theory.
Language: English
Textbook:
1. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer, New York (1982).
2. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, vol. 94. Springer, New York (1983).
3. N. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press, London (1983).