Descripción del proyecto
Resolución de conjeturas que relacionan invariantes en diversas áreas de las matemáticas
Ciertos invariantes que aparecen en la topología de baja dimensión y la geometría algebraica admiten refinamiento. Mientras que los invariantes clásicos generan polinomios en una variable (q), y a menudo tienen expresiones que involucran polinomios de Schur, los invariantes refinados generan polinomios en dos variables (q y t) y a menudo tienen expresiones que involucran polinomios de Macdonald. El objetivo del proyecto REFINV, financiado con fondos europeos, es desarrollar una teoría completa para resolver varias conjeturas arraigadas que relacionan invariantes en geometría algebraica, combinatoria y topología de baja dimensión. Para ello, sus investigadores emplearán métodos desarrollados en trabajos anteriores con respecto a soluciones de conjetura aleatoria, cálculos de homología de nudos de toro y polinomios de Poincaré de variedades de caracteres, y la demostración de la peculiar conjetura compleja de Lefschetz.
Objetivo
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.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-COG - Consolidator GrantInstitución de acogida
1010 Wien
Austria