## PHY 313: Topology and Geometry

3 credits | Prerequisites: PHY 221, 222

### Course rationale

Geometry is an inseparable element of modern physics. With the advent of String Theory, ideas from topology are entering into physics. Not only in String Theory, topological ideas are also used in Condensed Matter Physics. Therefore, to master modern ideas in theoretical physics this is quite important to be adept at Topology and Geometry. This elective course will introduce some basic and advanced concepts in these two areas of mathematics.

### Course content

Linear Algebra and Topology: review of vector space, subspace, linear operators, image, kernels, dual vector space, inner product, tensors, topological space, continuous map, neighborhood, connectedness, compactness, homeomorphism and topological invariants, Euler characteristics; Homology Group: definition of group, Abelian and non-Abelian groups, cyclic group, simplexes and simplicial complexes, oriented simplexes, chain, cycle, and boundary group, homology group, general properties, Betti numbers, Euler-Poincare theorem; Homotopy Group: paths, loops, homotopy, fundamental group, general properties, examples, fundamental groups of polyhedral, higher homotopy group, example in physics; Manifolds: definition, example, differentiable maps, vectors, oneforms, tensors, flows and Lie derivatives, differential forms, exterior derivative, integration, Lie group and Lie algebra, orbits, isotropy group, adjoint representation; de Rham Cohomology Groups: Stokes theorem, definition of de Rham cohjomology group, duality, Poincare Lemma, Poincare duality, Kunneth formula; Riemannian Geometry: Riemannian and pseudo-Riemannian manifolds, metric tensor, parallel transport, connection, covariant derivative, the metric connection, the Riemann tensor, Ricci tensor and curvature, LeviCivita connection, holonomy, isometries, Killing vectors, conformal Killing vectors, non-coordinate basis, Cartan’s structure equation, differential forms and Hodge theory, harmonic form and de Rham cohomology group; Complex Manifolds: definition, holomorphic maps, complexifications, almost complex structures, complex differential forms, Hermitian metric, Kahler form, Kahler manifold, Kahler differential geometry, Harmonic form, almost complex manifold, orbifolds; Fiber Bundles: tangent bundles, definition of fiber bundles, bundle maps, pullback, vector bundles, sections, cotangent and dual bundles, principal bundles, connection on principal bundles, holonomy, curvature, Ambrose-Singer theorem, covariant derivative on associated bundles, gauge theories; Characteristic Class: Chern-Weil homomorphism, invariant polynomials, Chern classes, properties, Chern character, Pontrjagin and Euler classes, Chern-Simons forms, Stiefel-Whitney classes.

### Course objectives

- Understand the advanced concepts of geometry and topology.
- Understand different geometrical entities in mathematics and physics.
- Use appropriate mathematical formalism to solve different physical problems.
- Understand the basic mathematical concepts required for Quantum Field Theory and String Theory.
- Familiarize yourself with the notions of homology, co-homology, and different kinds of geometry.

### References

- Geometry, Topology, and Physics by Mikio Nakaha
- The Geometry of Physics: An Introduction (3rd edition) by Theodore Frankel
- Topology and Geometry for Physicists by Charles Nash, and Siddhartha Sen
- An Introduction to Manifolds (2nd edition) by Loring Tu
- Differential Geometry: Connections, Curvature, and Characteristic Classes (1st edition) by Loring Tu
- Introduction to Riemannian Manifolds (2nd edition) by John M. Lee
- Riemannian Geometry (3rd edition) by Peter Petersen
- Topology and Geometry by Glen E. Bredon