arXiv subjects

Ivan Cherednik

Publications and source records attributed to Ivan Cherednik.

At least 19 recordsLinked to original sources

arXiv2026

Instanton slices and their superpolynomials

The key theorem is a connection between motivic superpolynomials of plane curve singularities in any ranks with superpolynomials of the corresponding instanton slices, Nekrasov-type instanton sums with conductors. In this case, instanton slices are related to compactified Jacobians, but they form a much wider class and, generally, have no connection to plane curve singularities. This development is expected to impact theory of affine Springer fibers (at least, in type $A$), and related fields. For instance, we obtain a motivic interpretation (counting $\mathbb{F}_q$-points of some stacks) of superpolynomials for hyperbolic knots K12n242, K12n725, among many others. New formulas for instanton sums in any ranks are obtained using that they are inductive limits of superpolynomials of proper families of plane curve singularities. The main conjecture states that instanton superpolynomials for arbitrary Young diagrams (almost all are not from plane curve singularities) can be transformed to the corresponding DAHA superpolynomials if and only if the latter are positive or (equivalently) the former are superdual. Our DAHA superpolynomials are parallel to Galashin-Lam EHA ones, but we used the nonsymmetric approach. The connection with reduced Khovanov-Rozansky polynomials of the corresponding Coxeter knots is expected. Among applications, formulas for Nekrasov's instanton sums with an impressive connection to nonsymmetric Macdonald polynomials, topological invariance of unibranch motivic superpolynomials for valuation semigroups with 3-4 generators, some mixed-characteristic conjecture, and an upgrade of Weak Riemann Hypothesis.

math.AG↗

arXiv2024

Superpolynomials of algebraic links

Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and recurrences for iterated torus links. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. Applications include affine Springer fibers of type $A_n$ and compactified Jacobians in the most general case (for arbitrary characteristic polynomials) and extended rho-invariants of algebraic knots. The 2nd connection conjecture relates the superpolynomials to the Galkin-St\"ohr $L$-functions, which is some counterpart of the ORS conjecture. The corresponding theory of plane curve singularities is systematically exposed and developed, which can be seen in the case of Hopf links as a generalized version of Schubert Calculus.

math.QA↗

arXiv2023

Zeta-polynomials, superpolynomials, DAHA and plane curve singularities

We begin with modular form periods, a focal point of several Yuri Manin's works. The similarity is discussed between the corresponding zeta-polynomials and superpolynomials of algebraic links, closely related to Khovanov-Rozansky polynomials. We focus on DAHA superpolynomials and motivic ones, defined via compactified Jacobians of plane curve singularities and their counterparts in arbitrary ranks; the non-unibranch construction is new. They conjecturally coincide with the corresponding generalizations of L-functions and satisfy the Riemann Hypothesis in some sectors of the parameters. Presumably, the motivic ones an be interpreted as certain partition functions of Landau-Ginzburg models associated with plane curve singularities; RH for them is remarkably similar to the Lee-Yang circle theorem for Ising models. A q,t-deformation of the Witten index is obtained as an application. General perspectives of the motivic theory of isolated curve and surface singularities are discussed, including possible implications in number theory. Also, we introduce super-analogs of $\rho_{ab}$-invariants and discuss super-deformations of the Riemann's zeta. Among other topics: Verlinde algebras, and the topological vertex.

math.QA↗

arXiv2022

Integral formulas for DAHA inner products

The main aim is to obtain integral formulas for DAHA coinvariants and the corresponding inner products for any values of the DAHA parameters. In the compact case, our approach is similar to the procedure of ``picking up residues" due to Arthur, Heckman, Opdam and others; the resulting formula is a sum of integrals over double affine residual subtori. A single real integral provides the required formula in the noncompact case. As q tends to 0, our integral formulas result in the trace formulas for the corresponding AHA, which calculate the Plancherel measures for the spherical parts of the regular AHA modules. The paper contains a systematic theory of DAHA coinvariants, including various results on the affine symmetrizers and induced DAHA modules.

math.QA↗

arXiv2022

Discrete Poisson hardcore 1D model and reinfections

We suggest a new hardcore Poisson-type distribution for Young diagrams with the row lengths from some finite list. A discrete variant of the time-ordered Matérn II process in 1D is employed. This approach is related to that based on the interlacing sequences due to Kerov and others, but we restrict the number of rows. The basic lengths are assumed comparable with the total order of the diagram in the quasi-classical limit, which results in new methods and new formulas. An interesting application is to random walks where the steps are at the points satisfying the classical Poisson distribution or our truncatedone. In the simplest case, one obtains the distribution in terms of Bessel I-functions, which provides some probabilistic interpretation of its many properties. An immediate application of our truncated Poisson distributions is to modeling reinfections in epidemics.

math.CO↗

arXiv2021

Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gröbner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role in the study of Hilbert schemes of smooth surfaces. They are generally singular, but their Gröbner cells are affine spaces; they admit an explicit parametrization due to Conca and Valla. We use this to obtain the Gröbner decomposition of compactified Jacobians of plane curve singularities, which is non-trivial even for the generalized Jacobians (principal ideals only). One of the application is the topological invariance of certain variants of compactified Jacobians and the corresponding motivic superpolynomials for analytic deformations of quasi-homogenous plane curve singularities and some similar families.

math.AG↗

arXiv2020

Momentum managing epidemic spread and Bessel functions

Starting with the power law for the total number of detected infections, we propose differential equations describing the effect of momentum epidemic management. Our 2-phase formula matches very well the curves of the total numbers of the Covid-19 infections in many countries; the first phase is described by Bessel functions. It provides projections for the saturation, assuming that the management is steady. We discuss Austria, Brazil, Germany, Japan, India, Israel, Italy, the Netherlands, Sweden, Switzerland, UK, and the USA, including some analysis of the second waves.

q-bio.PE↗

arXiv2020

Artificial intelligence approach to momentum risk-taking

We propose a mathematical model of momentum risk-taking, which is essentially real-time risk management focused on short-term volatility of stock markets. Its implementation, our fully automated momentum equity trading system presented systematically, proved to be successful in extensive historical and real-time experiments. Momentum risk-taking is one of the key components of general decision-making, a challenge for artificial intelligence and machine learning with deep roots in cognitive science; its variants beyond stock markets are discussed. We begin with a new algebraic-type theory of news impact on share-prices, which describes well their power growth, periodicity, and the market phenomena like price targets and profit-taking. This theory generally requires Bessel and hypergeometric functions. Its discretization results in some tables of bids, which are basically expected returns for main investment horizons, the key in our trading system. The ML procedures we use are similar to those in neural networking. A preimage of our approach is the new contract card game provided at the end, a combination of bridge and poker. Relations to random processes and the fractional Brownian motion are outlined.

q-fin.RM↗

arXiv2018

Modules over plane curve singularities in any ranks and DAHA

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible bundles to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials of algebraic knots colored by columns.

math.QA↗

arXiv2018

Riemann Hypothesis for DAHA superpolynomials and plane curve singularities

Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all $a$-coefficients of the DAHA superpolynomials upon the substitution $q\mapsto qt$ satisfy the Riemann Hypothesis for sufficiently small $q$ for uncolored algebraic knots, presumably for $q\le 1/2$ as $a=0$. This can be partially extended to algebraic links at least for $a=0$. Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stöhr zeta-functions are discussed.

math.QA↗

arXiv2018

Affine Hecke Algebras via DAHA

A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of "picking up residues" due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for $A_1$ in the spherical case.

math.QA↗

arXiv2018

Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules

We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at $t=\infty$ divided by their norms. The latter are identified with the graded characters of Demazure slices, some canonical quotients of thick (upper) level-one Demazure modules, directly related to recent theory of generalized (nonsymmetric) global Weyl modules. The symmetric Rogers-Ramanujan-type series considered by Cherednik-Feigin were expected to have some interpretation of this kind; the nonsymmetric setting appeared necessary to achieve this. As an application, the coefficients of the nonsymmetric Rogers-Ramanujan series provide formulas for the multiplicities of the expansions of tensor products of level-one Kac-Moody representations in terms of Demazure slices.

math.RT↗

arXiv2017

DAHA and plane curve singularities

We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik-Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky-Mazin for their constant term. The paper mainly focuses on non-torus algebraic knots. A connection with the conjecture due to Oblomkov-Rasmussen-Shende is possible, but our approach is different. A motivic version of our conjecture is related to p-adic orbital A-type integrals for anisotropic centralizers.

math.QA↗

arXiv2017

DAHA approach to iterated torus links

We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the so-called splice diagrams. The specialization t=q of our superpolynomials conjecturally results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the corresponding plane curve singularities is expected for algebraic links in the uncolored case. These 2 connections are less certain, since the Khovanov-Rozansky theory for links is not sufficiently developed and the ORS polynomials are quite involved. However we provide some confirmations. For Hopf links, our construction produces the DAHA-vertex, similar to the refined topological vertex, which is an important part of our paper.

math.QA↗

arXiv2015

Refined composite invariants of torus knots via DAHA

We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology; such a connection is a challenge in the theory of the annulus. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E and closely related to the exceptional Deligne-Gross series of root systems; this theme is of experimental nature.

math.QA↗

arXiv2014

DAHA and iterated torus knots

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities.

math.QA↗

arXiv2014

On Harish-Chandra theory of global nonsymmetric functions

This paper is devoted to the Harish-Chandra-type decomposition of the global nonsymmetric spherical functions in terms of their asymptotic expansions and the q,t-generalization of the celebrated c-function. This is for any reduced root systems in the q,t-setting; we pay special attention to the case of A1, where this decomposition is very explicit.

math.QA↗

arXiv2014

DAHA-Jones polynomials of torus knots

DAHA-Jones polynomials of torus knots $T(r,s)$ are studied systematically for reduced root systems and in the case of $C^\vee C_1$. We prove the polynomiality and evaluation conjectures from the author's previous paper on torus knots and extend the theory by the color exchange and further symmetries. DAHA-Jones polynomials for $C^\vee C_1$ depend on $5$ parameters. Their surprising connection to the DAHA-superpolynomials (type $A$) for the knots $T(2p+1,2)$ is obtained, a remarkable combination of the color exchange conditions and the author's duality conjecture (justified by Gorsky and Negut). The DAHA-superpolynomials for symmetric and wedge powers (and torus knots) conjecturally coincide with the Khovanov-Rozansky stable polynomials, those originated in the theory of BPS states and the superpolynomials defined via rational DAHA in connection with certain Hilbert schemes, though not much is known about such connections beyond the HOMFLYPT and Kauffman polynomials. We also define certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group instead of torus knots in the case of $C^\vee C_1$.

math.QA↗