arXiv subjects

Maxim Kazarian

Publications and source records attributed to Maxim Kazarian.

30 records · Page 2Linked to original sources

arXiv2020

Topological Recursion for the extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots

We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).

math-ph↗

arXiv2020

Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a closed simple expression.

math.DG↗

arXiv2019

Recursion for Masur-Veech volumes of moduli spaces of quadratic differentials

We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langleτ_2^nλ_k\rangle$. These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller, and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

math-ph↗

arXiv2019

Double Hurwitz numbers and multisingularity loci in genus 0

In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincare dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.

math.AG↗

arXiv2019

On framed simple purely real Hurwitz numbers

We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions for these numbers. We also construct a topological field theory for them.

math.AG↗

arXiv2018

Polynomial graph invariants and the KP hierarchy

We prove that the generating function for the symmetric chromatic polynomial of all connected graphs satisfies (after appropriate scaling change of variables) the Kadomtsev--Petviashvili integrable hierarchy of mathematical physics. Moreover, we describe a large family of polynomial graph invariants giving the same solution of the KP. In particular, we introduce the Abel polynomial for graphs and show this for its generating function. The key point here is a Hopf algebra structure on the space spanned by graphs and the behavior of the invariants on its primitive space.

math.CO↗

arXiv2017

Non-associative Hilbert scheme and Thom polynomials

Thom polynomial describes the cohomology class Poincaré dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is based on a new construction of the embedding space of punctual Hibert scheme that we call the non-associative Hilbert scheme. The efficiency of the method is demonstrated on explicit computation of a number of Thom polynomials, including those associated with singularities of Thom-Boardman types~$Σ^{i,j}$, $Σ^{2,1,1}$, $Σ^{2,2,1}$, and~$Σ^{2,2,2}$.

math.AG↗

arXiv2017

Discriminant of the ordinary transversal singularity type. The global equivalence class

Consider a space X with the singular locus of positive dimension, Z=Sing(X). Suppose both Z and X are locally complete intersections at each point. The transversal type of X along Z is generically constant but at some points of Z it degenerates. In the previous work we have introduced (locally) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is "ordinary". We have established the basic local properties of the discriminant. In the current paper we consider the global case. We compute the equivalence class of the discriminant in the Picard group, Pic(Z). If $X$ is a hypersurface, the discriminant is naturally stratified by the singularities of fibres in the projectivized normal cone $\mathbb{P}$N$_{X/Z}$. In this case (under some additional assumptions) we compute the classes of low codimension strata in the Chow group, A$^2$(Z). As immediate applications, we (re)derive the multi-degrees of the classical discriminant of projective complete intersections and bound the jumps of multiplicity of X along Z (when the singular locus is one-dimensional).

math.AG↗

arXiv2015

Virasoro constraints and topological recursion for Grothendieck's dessin counting

We compute the number of coverings of ${\mathbb{C}}P^1\setminus\{0, 1, \infty\}$ with a given monodromy type over $\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.

math.CO↗

arXiv2015

Universal cohomological expressions for singularities in families of genus 0 stable maps

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincaré dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the above properties.

math.AG↗

arXiv2013

Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of the Bouchard-Mariño conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard-Mariño conjecture and the ELSV formula (it is a refinement of an argument by Eynard).

math.AG↗

arXiv1997

Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.

dg-ga↗