REFINV
Financiado
Refined invariants in combinatorics low dimensional topology and geometry of mo...
Refined invariants in combinatorics low dimensional topology and geometry of moduli spaces
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...
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.
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Duración del proyecto: 63 meses
Fecha Inicio: 2021-02-26
Fecha Fin: 2026-05-31
Fecha Fin: 2026-05-31
Línea de financiación: concedida
El organismo H2020 notifico la concesión del proyecto el día 2021-02-26
Línea de financiación objetivo
El proyecto se financió a través de la siguiente ayuda:
ERC-2020-COG:
ERC CONSOLIDATOR GRANTS
Cerrada
hace 4 años
Presupuesto
El presupuesto total del proyecto asciende a
2M€
Líder del proyecto
UNIVERSITAT WIEN
No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico
TRL
4-5
Perfil tecnológico
TRL
4-5
1.85M€ |
Participante