Obiettivo
The purpose of this proposal is to investigate new connections which
have emerged in the recent years between problems from statistical
mechanics, namely two-dimensional exactly solvable models, and a variety
of combinatorial problems, among which: the enumeration of plane partitions,
alternating sign matrices and related objects;
combinatorial properties of certain
algebro-geometric objects such as orbital varieties or the Brauer loop scheme;
or finally certain problems in free probability. One of the key methods
that emerged in recent years is the use
of quantum integrability and more precisely the quantum Knizhnik--Zamolodchikov
equation, which itself is related to many deep results in representation theory.
The fruitful interaction between all these ideas has led to many advances
in the last few years, including proofs of some old conjectures but
also completely new results. More specifically, loop models
are a class of statistical models where the PI has made
significant progress, in particular in relation to the so-called
Razumov--Stroganov conjecture (now Cantini--Sportiello theorem).
New directions that should be pursued include:
further applications to enumerative combinatorics such as proofs of various
open conjectures relating Alternating Sign Matrices, Plane Partitions
and their symmetry classes;
a full understanding of the quantum integrability of the
Fully Packed Loop model,
a specific loop model at the heart of the Razumov--Stroganov correspondence;
a complete description of the Brauer loop scheme, including its
defining equations, and of the underlying poset; the extension
of the work on Di Francesco and Zinn-Justin on the loop model/6-vertex vertex
relation to the case of the 8-vertex model
(corresponding to elliptic solutions of the Yang--Baxter equation);
the study of solvable tilings models, in relation to
generalizations of the Littlewood--Richardson rule, and the determination
of their limiting shapes.
Campo scientifico
Invito a presentare proposte
ERC-2011-StG_20101014
Vedi altri progetti per questo bando
Meccanismo di finanziamento
ERC-SG - ERC Starting GrantIstituzione ospitante
75794 Paris
Francia