Descrizione del progetto
Risolvere le congetture relative agli invarianti in diverse aree matematiche
Alcuni invarianti che compaiono nella topologia a bassa dimensionalità e nella geometria algebrica ammettono il cosiddetto perfezionamento. Mentre i classici invarianti producono polinomi a una sola variabile (q) e spesso hanno espressioni che prevedono i polinomi di Schur, gli invarianti perfezionati producono polinomi a due variabili (q e t) e spesso hanno espressioni che prevedono polinomi di Macdonald. Il progetto REFINV, finanziato dall’UE, intende sviluppare una teoria comprensiva per risolvere diverse congetture di lungo corso relative agli invarianti nella geometria algebrica, alla combinatoria e alla topologia a bassa dimensionalità. A tal scopo, i ricercatori si baseranno sui metodi sviluppati nel lavoro precedente relativo alle soluzioni della congettura di Gilbreath, i calcoli omologici dei nodi torici e i polinomi di Poincare delle varietà di carattere, nonché la dimostrazione della curiosa congettura di Lefschetz.
Obiettivo
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.
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-COG - Consolidator GrantIstituzione ospitante
1010 Wien
Austria