Advanced mathematics for physicists

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November 15 - 19, 2021 :

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Faculty + Tutor: Amir-Kian Kashani-Poor
ECTS credits: 3
Language of instruction: English

 Description

Physics and mathematics have advanced in lockstep ever since the former graduated from being a subfield of philosophy to a full-blown science.
As physicists, we can turn to mathematics in search for conceptual clarity - e.g., the notions of intrinsic geometry as defined by Gauss were essential ingredients in Einstein’s passage from Special to General Relativity - or computational tools - such as the methods of group theory in computing spectra of quantum mechanical Hamiltonians. Often of course, we are not so lucky as to find off-the-shelf methods ready for immediate implementation, and we must prod and collaborate with mathematicians to advance existing fields or develop entirely new ones. The interaction between physics and mathematics is thus a two-way street. The more mathematical literacy we as physicists acquire, the better equipped we will be to embark on this fruitful interchange.

This course is meant as a continuation of sorts of the L3 Mathematics for Physicists course (or any other basic maths methods course). We will be focussing on two advanced topics:

- representation theory of finite groups,
- selected topics in differential manifolds.

In accord with the introductory words above, the goals of the course are twofold: to introduce mathematical tools relevant in many fields of physics such as quantum mechanics, solid state physics, particle physics, and general relativity, as well as to familiarize the student with basic mathematical terminology and concepts (manifolds, bundles, forms, group actions, pullback/pushforward, …) which will facilitate his or her future confrontation with mathematics literature. Exercises will be scattered throughout the lectures to clarify the material and illustrate it with examples.

 Bibliography

The representation theory part of the course is based on various references; a particularly lucid treatment of the subject, including exercises and their solutions, is the book

- Representations And Characters of Groups, by Gordon James and Martin Liebeck.

The differentiable manifolds part of the course follows the books

- Foundations of Differentiable Manifolds and Lie Groups, by Frank W. Warner
- An Introduction to Manifolds, by Loring W. Tu.

 Exercises and exams from previous years

You will find below exercise sheets and solutions from previous years, as well as some exams from previous years (the latter to be treated with care as exam preparation: the material treated in class has varied from year to year).

Attached documents

Quick links

Next student seminar :
Access to the program

Here you can find information about your internships:
Experimental Internship - Undergraduate program
Master ICFP first year Internship

News : ICFP Research seminars
November 15 - 19, 2021 :

All information about the program

Contact us - Student support and Graduate School office :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr