## Details of PH4105 (Autumn 2019)

Level: 4 |
Type: Theory |
Credits: 4.0 |

Course Code | Course Name | Instructor(s) |
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PH4105 |
Advanced Mathematical Methods of Physics |
Ananda Dasgupta |

Preamble |
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This is an advanced course on the mathematical methods of physics. This will introduce students to advanced mathematical techniques useful for students of theoretical physics. The emphasis will be more on techniques rather than on rigour. |

Syllabus |
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Calculus on $R^n$: the differential, chain rule, partial derivatives, directional derivatives, pullback, the inverse function theorem.
Differential forms : exterior algebra, differential forms, pullback of for forms, densities and integration, the exterior derivative , closed and exact forms, the Hodge duality operator, Stokes theorem, applications to electrodynamics. Topology for physics: Introduction to metric and topological spaces. Continuity. Neighbourhoods. Hausdorff spaces. Compactness. Connectedness. Homomorphisms and topological invariants. The Euler chracteristic as an example. Homology : Finitely generated Abelian groups. Simplexes and Simplicial complexes. Oriented simplexes. Triangulation and computation of homology groups. Betti numbers and the Euler-Poincare theorem. Homotopy : Paths, loops and homotopy equivalence of paths. Fundamental groups. Homotopy invariance of fundamental groups. Deformation retracts, contractibility. Computation of fundamental group for polyhedra. Triangulation of general spaces and their fundamental groups. Higher homotopy groups. The exact homotopy sequence. Application to the theory of defects. Cohomology : Stokes' theorem. Exact and closed differential forms. de Rham cohomology groups. Duality of homology and cohomology groups de Rham's theorem. Poincare's lemma. Poincare duality. Fiber bundles : Trivial and nontrivial bundles. Sections of bundles. Bundle maps. Pullback on bundles. Vector bundles. Frames. Cotangent bundles and dual bundles. Principal bundles. Connection on fiber bundles and gauge theories. The Aharonov-Bohm effect. Berry's phase. Instantons. Differential geometry : Differentiable manifolds. Differentiability on manifolds. Vectors connection between the the geometric, the algebraic and the physicist's view. 1-forms. Pull-backs ans push-forwards. Tensors and tensor fields. Flows and Lie derivatives. Differentiable forms on a manifold. Inner products and Lie derivative of differentiable forms. Integration on a manifold. Applications to thermodynamics and classical mechanics. Basic Riemannian geometry and applications to General Theory of Relativity. |

Prerequisite |
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Mathematical methods courses upto and including Level 3 |

References |
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1 P. Szekeres, A course in modern mathematical physics: Groups, Hilbert spaces and differential geometry, Cambridge University Press.
2 T. Frankel, The geometry of physics, Cambridge University Press (1997). 3 R. Geroch, Mathematical Physics, The university of Chicago press (1985). 4 C. Nash and S. Sen, Topology and geometry for physicists, Academic press (1983). 5 M. Nakahara, Geometry, topology and physics, Adam Hilger (1989). |

#### Course Credit Options

Sl. No. | Programme | Semester No | Course Choice |
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1 | IP | 1 | Not Allowed |

2 | IP | 3 | Elective |

3 | IP | 5 | Not Allowed |

4 | MR | 1 | Elective |

5 | MR | 3 | Not Allowed |

6 | MS | 7 | Elective |

7 | RS | 1 | Elective |

8 | RS | 2 | Elective |