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## Top tautological group of $M_{g,n}$

Journal of the European Mathematical Society. 2016. Vol. 18. No. 12. P. 2925-2951.

We describe the structure of the top tautological group in the cohomology of the moduli space of smooth genus g curves with n marked points.

Buryak A., Mathematical Research Letters 2016 Vol. 23 No. 3 P. 675-683

In a previous paper we proved that after a simple transformation the generating series of the linear Hodge integrals on the moduli space of stable curves satisfies the hierarchy of the Intermediate Long Wave equation. In this paper we present a much shorter proof of this fact. Our new proof is based on an explicit ...

Added: September 28, 2020

Buryak A., Moscow Mathematical Journal 2017 Vol. 17 No. 1 P. 1-13

In this paper, using the formula for the integrals of the psi-classes over the double ramification cycles found by S. Shadrin, L. Spitz, D. Zvonkine and the author, we derive a new explicit formula for the n-point function of the intersection numbers on the moduli space of curves. ...

Added: September 27, 2020

Buryak A., Janda F., Pandharipande R., Pure and Applied Mathematics Quarterly 2015 Vol. 11 No. 4 P. 591-631

The relations in the tautological ring of the moduli space $M_g$ of nonsingular curves conjectured by Faber-Zagier in 2000 and extended to the moduli space $\overline{M}_{g,n}$ of stable curves by Pixton in 2012 are based upon two hypergeometric series $A$ and $B$. The question of the geometric origins of these series has been solved in at least two ways (via the Frobenius ...

Added: September 28, 2020

Chebochko N.G., / Cornell University. Series math "arxiv.org". 2017. No. 1712.01810.

The description of global deformations of Lie algebras is important since it is related to unsolved problem of the classification of simple Lie algebras over a field of small characteristic.
In this paper we study global deformations of Lie algebras of type ${D}_{l}$ over an algebraically closed field K of characteristic 2. It is proved that ...

Added: December 8, 2017

Fedor Bogomolov, Tschinkel Y., Communications on Pure and Applied Mathematics 2013 Vol. 66 No. 9 P. 1335-1359

We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture. ...

Added: December 27, 2013

Buryak A., Rossi P., Letters in Mathematical Physics 2016 Vol. 106 No. 3 P. 289-317

In this paper we define a quantization of the Double Ramification Hierarchies using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with ...

Added: September 28, 2020

Buryak A., Communications in Number Theory and Physics 2015 Vol. 9 No. 2 P. 239-271

In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the Dubrovin-Zhang theory of the hierarchies of the topological type. It occurs that our deformation of ...

Added: September 29, 2020

Buryak A., Shadrin S., Advances in Mathematics 2011 Vol. 228 P. 22-42

We give a new proof of Faber's intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves $\M_g$. The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles. ...

Added: October 1, 2020

Buryak A., Dubrovin B., Guere J. et al., Communications in Mathematical Physics 2018 Vol. 363 No. 1 P. 191-260

In this paper we continue the study of the double ramification hierarchy introduced by the first author. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree ...

Added: September 27, 2020

Брудный Ю. А., Зайденберг М. Г., Лин В. Я. et al., Успехи математических наук 2019 Т. 74 № 5 С. 170-180

A detailed review of the scientific activities of the remarkable domestic mathematician E. A. Gorin and his results ...

Added: March 17, 2020

В.А.Васильев, Известия РАН. Серия математическая 2016 Т. 80 № 4 С. 163-184

Rational homology groups of spaces of non-resultant (that is, having only trivial common zeros) systems of homogeneous quadratic polynomial systems in R^3 are calculated ...

Added: March 3, 2018

Podkopaev O., Вестник Санкт-Петербургского университета. Серия 1. Математика. Механика. Астрономия 2018 Т. 5(63) № 4 С. 631-636

The goal of this note is to give a proof of the following proposition. Let π be a profinite group and K∗ be a bounded complex of discrete Fp[π]-modules. Assume all Hi (K∗) are finite abelian groups. Then there exists a quasiisomorphism L∗ −→ K∗, where L∗ is a bounded complex of discrete Fp[π]-modules such ...

Added: April 18, 2021

Buryak A., Communications in Mathematical Physics 2015 Vol. 336 No. 3 P. 1085-1107

It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples. ...

Added: September 29, 2020

Buryak A., Shadrin S., Spitz L. et al., American Journal of Mathematics 2015 Vol. 137 No. 3 P. 699-737

DR-cycles are certain cycles on the moduli space of curves. Intuitively, they parametrize curves that allow a map to the complex projective line with some specified ramification profile over two points. They are known to be tautological classes, but in general there is no known expression in terms of standard tautological classes. In this paper, ...

Added: September 30, 2020

Kazaryan M., Zvonkine D., Lando S., International Mathematics Research Notices 2018 No. 22 P. 6817-6843

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 ...

Added: July 10, 2017

Angella D., Tomassini A., Verbitsky M., / Cornell University. Series arXiv "math". 2016.

We study cohomological properties of complex manifolds. In particular, we provide an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces, according to the dichotomy b1 even or odd. In higher dimension, a similar result is obtained at degree 1 under additional metric conditions (see Theorem 2.4). ...

Added: May 14, 2016

Kurnosov N., Soldatenkov A., Verbitsky M., Advances in Mathematics 2019 Vol. 351 P. 275-295

Let M be a simple hyperkähler manifold. Kuga-Satake
construction gives an embedding of H^2(M, C) into the
second cohomology of a torus, compatible with the Hodge
structure. We construct a torus T and an embedding of the
graded cohomology space H^•(M, C) → H^{•+l}(T, C) for some
l, which is compatible with the Hodge structures and the
Poincaré pairing. Moreover, this ...

Added: June 3, 2019

Przyjalkowski V., Shramov K., Collectanea Mathematica 2020 Vol. 71 P. 549-574

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge numbers in the same row of the Hodge diamond. This allows us to classify varieties whose Hodge numbers are like that of a projective space, ...

Added: November 13, 2020

Barannikov S., / hal.archives-ouvertes.fr (CNRS). Series HAL "math". 2018. No. 01804639.

The construction from [B06], see also [B10], of cohomology classes of compactified moduli spaces of Riemann surfaces, starting from a derivation of associative whose square is nonzero, is generalized to the case of A-infinity algebras. It is shown that the constructed cohomology classes define Cohomological Field Theory. ...

Added: October 25, 2018

V.A.Vassiliev, Doklady Mathematics 2018 Vol. 98 No. 1 P. 330-333

Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in R^n are calculated ...

Added: December 7, 2018

Buryak A., Rossi P., Shadrin S., Letters in Mathematical Physics 2021 Vol. 111 Article 13

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture. ...

Added: October 29, 2021

Buryak A., Hernandez Iglesias F., Shadrin S., Epijournal de Geometrie Algebrique 2022 Vol. 6 Article 8595

We propose a conjectural formula for DR_g(a,−a)\lambda_g and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Guéré and Rossi, and we prove that the two conjectures are in fact equivalent, though in a quite non-trivial way. ...

Added: September 14, 2022

Guere J., Rossi P., Buryak A., Geometry and Topology 2019 Vol. 23 No. 7 P. 3537-3600

We present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin–Zhang equivalence conjecture introduced by the authors with Dubrovin. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ...

Added: April 21, 2020

Khoroshkin A., Willwacher T., / Cornell University. Серия "Working papers by Cornell University". 2019. № 1905.04499.

We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use this model to find different Hopf models of the algebraic operad of Chains and ...

Added: October 30, 2019