Description du projet
Résoudre les conjectures qui relient les invariants dans divers domaines mathématiques
Certains invariants apparaissant en topologie et en géométrie algébrique de basse dimension admettent ce que l’on appelle un raffinement. Alors que les invariants classiques produisent des polynômes à une variable (q), et ont souvent des expressions impliquant des polynômes de Schur, les invariants raffinés produisent des polynômes à deux variables (q et t) et ont souvent des expressions impliquant des polynômes de Macdonald. Le projet REFINV, financé par l’UE, entend élaborer une théorie complète destinée à résoudre plusieurs conjectures de longue date qui concernent les invariants en géométrie algébrique, en combinatoire et en topologie en basses dimensions. Pour ce faire, les chercheurs s’appuieront sur les méthodes développées dans des travaux antérieurs portant sur les solutions de la conjecture Shuffle, les calculs d’homologie des nœuds de tore et des polynômes de Poincaré des variétés de caractères, et la preuve de la curieuse conjecture de Lefschetz.
Objectif
Some invariants appearing in low-dimensional topology and algebraic geometry admit a so-called refinement: a much stronger, but more complicated invariant. For instance, in low-dimensional topology we study the Poincare polynomial of the triply-graded Khovanov-Rozansky homology of a knot, which is a refined version of the HOMFLY-PT polynomial. The usual Poincare polynomial and the E-polynomial of a character variety are refined by the full mixed Hodge polynomial. On the cohomology of a moduli space of Higgs bundles, we need an extra filtration, the so-called perverse filtration, to define the refinement. Where classical invariants produce polynomials in one variable q, and often have expressions involving Schur polynomials, refined invariants produce polynomials in two variables q and t, and often have expressions involving Macdonald polynomials. The connection conjectures and the P=W conjecture relate refined invariants appearing in the three contexts above.
We propose to develop a comprehensive theory connecting these notions, and as main applications, to solve the P=W conjecture for character varieties, the Gorsky-Negut-Rasmussen conjectures relating knot invariants and sheaves on the Hilbert scheme, Cherednik's conjectures computing homologies of algebraic links via DAHA, the Hausel-Letellier-Rodriguez-Villegas conjectures computing mixed Hodge polynomials of character varieties, nabla positivity, and the Stanley-Stembridge positivity conjectures.
To achieve our goal, we will build on methods developed in our previous work on the solution of the shuffle conjectures, the computations of homology of torus knots and Poincare polynomials of character varieties, and the proof of the curious hard Lefschetz conjecture. These methods include combinatorics of Dyck paths, symmetric functions and Macdonald theory, the A(q,t) algebra, cell decompositions of character varieties, natural actions on cohomology and K-theory, counting geometric objects over finite fields.
Champ scientifique
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
1010 Wien
Autriche