Periodic Reporting for period 2 - REFINV (Refined invariants in combinatorics, low-dimensional topology and geometry of moduli spaces)
Reporting period: 2022-12-01 to 2024-05-31
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.
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.
1. Proof of the P=W conjecture
2. Computation of the zero-dimensional COHA of arbitrary algebraic surface with pure cohomology
3. Proof of the Segre-Verlinde correspondence and computation of Segre-Verlinde integrals
4. Proof of the Dunfield-Gukov-Rasmussen conjecture
5. Proof of the Morton-Samuelson conjecture
2. Computation of the zero-dimensional COHA of arbitrary algebraic surface with pure cohomology
3. Proof of the Segre-Verlinde correspondence and computation of Segre-Verlinde integrals
4. Proof of the Dunfield-Gukov-Rasmussen conjecture
5. Proof of the Morton-Samuelson conjecture
Until the end of the project we expect to completely understand refined invariants of algebraic surfaces on one side, and relate them to the topologically invariants on the other side. This will imply several long-standing conjectures that relate invariants of different kind, relating character varieties of curves, moduli spaces of Higgs bundles, plane curve singularities, Hilbert schemes.