Course  IIT  Madras 

January-February 2015

Combinatorics and Physics

Abstract

The interaction between combinatorics and theoretical physics started in the 60's with a combinatorial resolution of the Ising model  for ferromagnetism by means of  enumeration of dimers tilings on a planar lattice (the Pfaffian method). In recent years, this interaction has been very active and fruitful not only for statistical mechanics, but also to other parts of physics (quantum field theory, dynamical systems, ...). A new domain called "combinatorial physics" is emerging. At the same time, there is a spectacular renaissance of combinatorics, especially for enumerative, algebraic and bijective combinatorics, in interaction with computer science and other parts of pure and applied mathematics. New tools are appearing in combinatorics such as heaps of pieces, the LGV lemma or bijections between planar maps and some decorated trees, having fruitful applications both in mathematics and in physics.

Here is a list of topics for this course:

- introduction to enumerative and bijective combinatorics

- non-crossing paths, tilings, determinants and Young tableaux. The LGV Lemma.

- introduction to the theory of  heaps of pieces (interpreting partial commutation of variables) : the 3 basics lemma

- heaps of pieces and statistical mechanics: directed animals, gas models, q-Bessel functions in physics

- heaps of pieces and 2D Lorentzian quantum gravity

- combinatorics of the PASEP (partially asymmetric exclusion model in the physics of dynamical systems), relation with orthogonal polynomials

- alternating sign matrices, Fully packed loop model (FPL) and the (ex)-Razumov-Stroganov conjecture

No preliminary knowledge in physics is needed. This course can be followed by both physicists and mathematicians. 

Monday-Wednesday 3-4:30 pm

Ch 0     Overview of the course

              14  january 2015

part 1     slides         (pdf  11,7  Mo,  48 p.)

part 2     slides         (pdf  12,6 Mo,  40 p.)

part 3      slides         (pdf  18,4  Mo,  34 p.)

part 4     slides          (pdf  14,1  Mo,  39 p.)

part 5     slides          (pdf  14,8  Mo,  40 p.)