About

Ivan Cherednik is the Austin M. Carr Distinguished Professor at the Department of Mathematics at the University of North Carolina at Chapel Hill. His research interests encompass representation theory, mathematical physics, combinatorics, number theory, algebraic geometry, low-dimensional topology, mathematical biology, and quantitative finance. Cherednik's academic background includes a PhD in arithmetic geometry, where he obtained the p-adic uniformization of Shimura curves associated with quaternion algebras, a breakthrough used in the proof of Fermat's Last Theorem. He later shifted focus to representation theory and integrable models in physics, leading to the development of Double Affine Hecke Algebras (DAHA), the proof of the Macdonald conjectures in q-combinatorics, and advancements in hypergeometric functions, DAHA invariants of algebraic knots and links, and Rogers-Ramanujan identities. His recent work explores applications of DAHA knot invariants in number theory, including classical zeta and L-functions, as well as contributions to financial mathematics and modeling the spread of COVID-19 using Bessel functions.

Research topics

• Waste management
• Chemistry
• Chemical engineering
• Nuclear engineering
• Organic chemistry
• Materials science
• Environmental science
• Geology
• Composite material
• Engineering

Selected publications

• Modeling the Waves of Covid-19

  UNC Libraries · 2025-06-20

  articleOpen access1st authorCorresponding
  PublisherDOI

• Discrete Poisson hardcore 1D model and applications

  Contemporary mathematics - American Mathematical Society · 2025-01-01

  other1st authorCorresponding

  We suggest and study new hardcore Poisson-type distributions originated in sequences of non-overlapping subsegments in a given one of the lengths from some finite list, which are governed by a variant of the Matérn II process in 1D. The boundary effects are important: the last subsegment is allowed to go beyond the endpoint of the initial one. This is somewhat analogous to the interlacing sequences due to Kerov and others, but the corresponding Young diagrams are finite and the basic point process is different in our setup. When the lengths of subsegments are comparable with that of the initial segment, the classical limit becomes a new “truncated” Poisson-type distribution. The normal-type classical limit is of interest too. A variant is a random walk when the number of steps satisfies the classical Poisson distribution or our truncated one. Then the Bessel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functions naturally occur, which provides an interesting probabilistic interpretation of their basic properties. Among applications, our truncated Poisson distribution can be used to model reinfections in epidemics.

  PublisherDOI

• Superpolynomials of Algebraic Links

  The Quarterly Journal of Mathematics · 2025-08-25

  article1st authorCorresponding

  Abstract Theory of motivic superpolynomials is developed, including its extension to algebraic links coloured 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 double affine Hecke algebra ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. Applications include affine Springer fibres of type $A_n$ and compactified Jacobians in the most general case (for arbitrary characteristic polynomials) and extended rho-invariants of algebraic knots. The second connection conjecture relates the superpolynomials to the Galkin–Stöhr $L$-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.

  PublisherDOI

• On R-Matrix Quantization of Formal Loop Groups

  2024-10-31 · 5 citations

  book-chapter1st authorCorresponding

  1. The general plan of quantization. We choose a basis {Ia} in the associative algebra MN of NxN - matrices with unit Io and for any X∈ MN define Xα ∈ C from the expansion https://www.w3.org/1998/Math/MathML" display="inline"> X = ∑ α X α I α https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003580850/6547cd04-0b1f-4948-b2f4-c67f2028b35d/content/p161-1.tif"/> in terms of Iα https://www.w3.org/1998/Math/MathML" display="block"> 1 I α = I α ⊕ I o ⊕ . . . ⊕ I o , 2 I α = I o ⊕ I α ⊕ I o ⊕ . . . ⊕ I o , etc . , ij X = ∑ α , β x αβ i I α j I β for X = ∑ α , β x αβ I α ⊕ I β ∈ ∈ M N ⊕ M N https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003580850/6547cd04-0b1f-4948-b2f4-c67f2028b35d/content/dmath_p161-1.tif"/>

  PublisherDOI

• Zeta-Polynomials, Superpolynomials, DAHA, and Plane Curve Singularities

  Simons symposia · 2024-01-01 · 1 citations

  book-chapter1st authorCorresponding
  PublisherDOI

• Zeta-polynomials, superpolynomials, DAHA and plane curve singularities

  arXiv (Cornell University) · 2023-04-05

  preprintOpen access1st authorCorresponding

  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 $ρ_{ab}$-invariants and discuss super-deformations of the Riemann's zeta. Among other topics: Verlinde algebras, and the topological vertex.

  PublisherOA PDFDOI

• Discrete Poisson hardcore 1D model and reinfections

  arXiv (Cornell University) · 2022-02-18

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Gröbner cells of punctual Hilbert schemes in dimension two

  Journal of Algebra · 2022-01-29 · 1 citations

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Integral formulas for DAHA inner products

  arXiv (Cornell University) · 2022-11-02

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Combinatorics, Modeling, Elementary Number Theory

  WORLD SCIENTIFIC eBooks · 2022-08-09 · 1 citations

  book1st authorCorresponding
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Recent grants

• Double Affine Hecke Algebras

  NSF · $575k · 2019–2025

• DOUBLE AFFINE HECKE ALGEBRAS

  NSF · $170k · 2011–2015

• Double Affine Hecke Algebras

  NSF · $228k · 2008–2012

• Double Affine Hecke Algebras

  NSF · $145k · 2005–2009

• Double Affine Hecke Algebras

  NSF · $525k · 2014–2020

Frequent coauthors

• S. A. Yankovsky

  Tomsk Polytechnic University

  36 shared

• Г. В. Кузнецов

  National Research Tomsk State University

  36 shared

• Anton Tolokolnikov

  National Research Tomsk State University

  28 shared

• Daniel L. Orr

  10 shared

• Boris Feigin

  10 shared

• V. A. Vysloukh

  Universidad de las Américas Puebla

  9 shared

• А. V. Zenkov

  Tomsk Polytechnic University

  8 shared

• Ian Philipp

  University of North Carolina Health Care

  5 shared