# Math/Physics Learning Seminar

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## Overview

MATH/PHYSICS LEARNING SEMINAR

Thursdays 2pm, Bahen 6183.

Organizers: Daniel Rowe (daniel.rowe@utoronto.ca) Jonathan Fisher (jmfisher@math.utoronto.edu). If you are interested in giving a talk, please let us know.

Synopsis: This will be an informal and introductory seminar series with the aim of studying modern mathematical structures in the context of physical systems. Hopefully this should lead to a deeper understanding and motivation for those structures. No physics background is supposed. The outline of the seminar is roughly as follows: symplectic geometry and classical mechanics; various quantization schemes and basic quantum field theory; Dirac structures; G-torsors and gauge theory, the moduli space of G-connections modulo gauge equivalence; K-theory, spin-geometry and the index theorem(s); representation theory and the space-algebra correspondence; groupoids and the categorification program; and noncommutative geometry. Of course, this is not a complete list of potential topics that suit the theme of this seminar.

For background information and details not presented during the talks, please see our page of references.

## Talks

JANUARY 19 2010, Brent Pym, University of Toronto, The Lagrangian formalism of classical mechanics.

JANUARY 26 2010, Jonathan Fisher, University of Toronto, Symplectic/Hamiltonian formalism of classical mechanics and Noether's theorem.

FEBRUARY 2 2010, Joseph Johns, Courant Institute, Fukaya categories of Lefschetz fibrations and quiver representations. (special geometric representation theory seminar)

FEBRUARY 9 2010, Jonathan Fisher, University of Toronto, Symplectic/Hamiltonian formalism of classical mechanics and Noether's theorem (continuation).

FEBRUARY 16 2010, Reading week--no seminar.

FEBRUARY 23 2010 David Spivak, University of Oregon, Derived Differential Geometry. (special symplectic seminar)

MARCH 2, 2010 Brent Pym, University of Toronto, Quantum Mechanics: The Five 'W's for Mathematicians.

MARCH 9, 2010, Travis Li, University of Toronto, Geometric quantization.

MARCH 16 2010 Daniel Rowe, University of Toronto, Representations and Special Relativity.

MARCH 23 2010 Jonathan Fisher, University of Toronto, The Dirac Equation.

APRIL 29 2010 Jonathan Fisher, University of Toronto, Review of the Lagrangian and Hamiltonian formulations.

## Outline of potential topics to be covered

Topics in classical mechanics and field theory:

• Lagrangian mechanics
• Hamiltonian mechanics and symplectic geometry
• Symmetry and group actions and the relationship to conservation laws
• Special relativity
• General relativity
• Classical electrodynamics and Yang-Mills
• $\sigma$-models
• topological field theories

Topics in quantum mechanics and field theory:

• Quantum mechanics
• Group theory and quantum mechanics
• The not-so-rigid rotor and molecular spectroscopy
• Quantization of constrained systems
• Free field QFT
• Nonrigorous interacting QFT (S-matrix, Dyson series expansion, Feynman diagrams and all that)
• Renormalization group

Topics in mathematics:

• Yang-Mills over Riemann surfaces (moduli spaces)
• Anti-self dual Yang-Mills(-Higgs) and hyperkahler geometry (moduli spaces redux)
• Superalgebra, supersymmetry, supergeometry
• Yang-Mills flow (infinite dimensional Morse theory)
• Equivariant cohomology and localization
• Poisson geometry
• Dirac structures
• TQFTs
• Geometric Quantization
• Deformation quantization
• Microlocal analysis and pseudodifferential operators
• Spin geometry and index theorems
• Noncommutative geometry
• The Landau-Lifshitz equation and Schroedinger maps

It should be emphasized that many, if not most, of these topics should have an entire course (or lifetime!) devoted to them--our aim is to give just hint of the flavor of the subject and to (hopefully) get you interested in learning more. This list of topics is large enough to keep us busy indefinitely. If you have any comments or suggestions, or are interested in giving a talk, please let us know.