Advanced seminar on Geometry, Topology and Physics to be offered in 2016-17 academic year

May 16, 2016 - CHTM

Arash Mafi

Dr. Arash Mafi

Arash Mafi, PhD, General Chair, Optical Science and Engineering; Associate Professor, Department of Physics & Astronomy; and Faculty Member, Center for High Technology Materials, together with Rouzbeh Allahverdi, PhD, Associate Professor, Department of Physics & Astronomy, said the seminar should be viewed as a forum for collaborative study of the subject, and everyone is most welcome to participate and contribute.

The seminar emerged from their discussions on mutual research interests. The topic was chosen for its broad applications in topics such as general relativity, the Aharonov-Bohm effect and Berry's phase, quantum Hall effect, instantons and monopoles, cosmic strings, liquid crystals, topological insulators, topological photonics and optical isolators, and others.

The main textbook will be: "Geometry, Topology, and Physics, 2nd Ed." by Mikio Nakahara. It is available at: http://www.amazon.com/Geometry-Topology-Physics-Graduate-Student/dp/0750306068

Dr. Rouzbeh Allahverdi

Dr. Rouzbeh Allahverdi

This is a highly regarded book and covers the mathematical foundations without overburdening the reader with unnecessary formalism. Because the book is intended for physicists, the flow provides a familiar tone that can be readily grasped by advanced undergraduate and beginning graduate students. Please see below for a tentative syllabus for two semesters. This will likely be a fast-paced seminar to cover the proposed material and will require considerable independent reading through the text.

First semester (6 chapters)

1 Quantum Physics (We will go through this chapter rather quickly as a large part of it is also covered in the QFT I course in the fall)
1.1 Analytical mechanics
1.2 Canonical quantization
1.3 Path integral quantization of a Bose particle
1.4 Harmonic oscillator
1.5 Path integral quantization of a Fermi particle
1.6 Quantization of a scalar field
1.7 Quantization of a Dirac field
1.8 Gauge theories
1.9 Magnetic monopoles
1.10 Instantons

2 Mathematical Preliminaries
2.1 Maps
2.2 Vector spaces
2.3 Topological spaces
2.4 Homeomorphisms and topological invariants

3 Homology Groups
3.1 Abelian groups
3.2 Simplexes and simplicial complexes
3.3 Homology groups of simplicial complexes
3.4 General properties of homology groups

4 Homotopy Groups
4.1 Fundamental groups
4.2 General properties of fundamental groups
4.3 Examples of fundamental groups
4.4 Fundamental groups of polyhedra
4.5 Higher homotopy groups
4.6 General properties of higher homotopy groups
4.7 Examples of higher homotopy groups
4.8 Orders in condensed matter systems
4.9 Defects in nematic liquid crystals
4.10 Textures in superfluid 3He-A

5 Manifolds
5.1 Manifolds
5.2 The calculus on manifolds
5.3 Flows and Lie derivatives
5.4 Differential forms
5.5 Integration of differential forms
5.6 Lie groups and Lie algebras
5.7 The action of Lie groups on manifolds

6 de Rham Cohomology Groups
6.1 Stokes’ theorem
6.2 de Rham cohomology groups
6.3 Poincare’s lemma
6.4 Structure of de Rham cohomology groups

Second semester: (6 chapters)

7 Riemannian Geometry
8 Complex Manifolds (We will likely skip this chapter)
9 Fibre Bundles
10 Connections on Fibre Bundles
11 Characteristic Classes
12 Index Theorems
13 Anomalies in Gauge Field Theories
14 Bosonic String theory (We will likely skip this chapter)

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