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## Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology

Cornell University
,
2016.

Gorsky E., Negut A., Rasmussen J.

We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials d_N on these dg algebras and conjecture that their homology matches that of the gl_N projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende.

Publication based on the results of:

Eugene Gorsky, Oblomkov A., Rasmussen J. et al., Duke Mathematical Journal 2014 Vol. 163 No. 14 P. 2709-2794

We conjecturally extract the triply graded Khovanov–Rozansky homology of the (m,n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n-1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the ...

Added: December 9, 2014

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Gorsky E., Geometry and Topology 2018 Vol. 22 P. 645-691

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the ...

Added: August 21, 2018

Kuznetsov A., Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2016.

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...

Added: May 16, 2016

Gorsky E., Contemporary Mathematics Series 2012 Vol. 566 P. 212-232

We propose an algebraic model of the conjectural triply graded homology of S. Gukov, N. Dunfield and J. Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of A. Garsia and M. Haiman. ...

Added: December 9, 2014

Gorsky Evgeny, Mazin M., Journal of Combinatorial Theory, Series A 2013 Vol. 120 No. 1 P. 49-63

J. Piontkowski described the homology of the Jacobi factor of a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his answer, in particular, relate it to the bigraded deformation of Catalan numbers introduced by A. Garsia and M. Haiman. ...

Added: December 9, 2014

Gorsky E., Stosic M., Gukov S., Fundamenta Mathematicae 2018 Vol. 243 P. 209-299

We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a simple representation-theoretic model for the HOMFLY homology of ...

Added: December 28, 2017

Gorsky Eugene, Rasmussen J., Oblomkov A., Experimental Mathematics 2013 Vol. 22 No. 3 P. 265-281

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity. ...

Added: December 9, 2014

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2014. No. 1405.5154.

We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...

Added: May 21, 2014

Buryak A., Feigin B. L., Nakajima H., International Mathematical Research Notices 2015 Vol. 2015 No. 13 P. 4708-4715

In a recent paper the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper we give a very short geometrical proof of that formula. ...

Added: September 29, 2020

Gorsky E., Hogancamp M., / Cornell University. Series arXiv "math". 2017.

We define a deformation of the triply graded Khovanov-Rozansky homology of a link L depending on a choice of parameters for each component of L. We conjecture that this invariant restores the missing symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points ...

Added: December 28, 2017

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Gorsky E., Hogancamp M., Mellit A. et al., / Cornell University. Series arXiv "math". 2019.

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of Kálmán. ...

Added: September 3, 2019

Gorsky E., Negut A., Journal de Mathématiques Pures and Appliquées 2015 Vol. 104 No. 3 P. 403-435

We consider the construction of refined Chern-Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We prove Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants with the Hilbert scheme of points on the ...

Added: February 14, 2015

Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109-185

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...

Added: November 22, 2017

Galkin S., Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.07001.

Let X(n) denote n-th symmetric power of a cubic surface X. We show that X(4)×X is stably birational to X(3)×X, despite examples when X(4) is not stably birational to X(3). ...

Added: October 19, 2018

Popov P., / Cornell University. Series math "arxiv.org". 2018. No. 1810.04563.

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface Y with a certain moduli space of twisted cubic curves on Y. These relations are generalizations of the "beautiful" Y-F(Y) relation by Galkin and Shinder which connects Y with the Hilbert scheme of two points on Y and the Fano variety F(Y) of lines on Y. We ...

Added: October 23, 2018

Bogomolov F. A., Kulikov V. S., Central European Journal of Mathematics 2013 Vol. 11 No. 2 P. 254-263

The article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙ m+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof ...

Added: November 21, 2012

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624

Added: February 27, 2013

Ilyashenko Y., Яковенко С. Ю., М. : МЦНМО, 2013

Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...

Added: February 5, 2014

Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70

A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...

Added: July 19, 2014

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83

We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...

Added: November 1, 2019