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Working paper

In this paper we investigate the connection between the Mac Lane (co)homology and Wieferich primes in finite localizations of global number rings. Following the ideas of Polishchuk-Positselski [PP], we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute this invariants for finite localizations of global number rings with an element w and obtain that the result is closely related with the Wieferich primes to the base w. In particular, for a given non-zero integer w, the infiniteness of Wieferich primes to the base w turns out to be equivalent to the following: for any positive integer n, we have HMLII,0 (Z[ 1 n! ], w) 6= Q. As an application of our technique, we identify the ring structure on the Mac Lane cohomology of a global number ring and compute the Adams operations (introduced in this case by McCarthy [McC]) on its Mac Lane homology.

Added: Oct 5, 2015

Working paper

The aim of this note is to define for any $e_n$-algebra $A$ and a compact parallelizable n-manifold $M$ without borders a morphism from the homology of homotopy Lie algebra $A[n-1]$ to the topological chiral homology of $M$ with coefficients in $A$. This map plays a crucial role in the perturbative Chern-Simons theory.

Added: Apr 23, 2013

Working paper

These are notes for a mini-course of 3 lectures given at the St. Petersburg
School in Probability and Statistical Physics (June 2012). My aim was to
explain, on the example of a particular model, how ideas from the
representation theory of big groups can be applied in probabilistic problems.
The material is based on the joint paper arXiv:1009.2029 by Alexei
Borodin and myself; a broader range of topics is surveyed in the lecture notes
by Alexei Borodin and Vadim Gorin arXiv:1212.3351.

Added: Nov 16, 2013

Working paper

We present a construction of an integrable model as a projective type limit of spin Calogero-Sutherland model with N fermionic particles, where N tends to infinity. It is implemented in the multicomponent fermionic Fock space. Explicit formulas for limits of Dunkl operators and the Yangian generators are presented by means of fermionic fields.

Added: Oct 24, 2019

Working paper

Altmann K.,

, Petersen L. arxiv.org. math. Cornell University, 2013. No. 1210.4523v2.
Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans. The torus in question is obtained as the identity component of the quotient group N/H, where N is the normalizer of H in G. The resulting Chow quotient is equal to (a blowup of) the simple toroidal compactification of G/(H N^0). In the horospherical case, for example, it is equal to a flag variety, and the slices (coefficients) of the divisorial fan are merely shifts of the colored fan along the colors.

Added: Feb 6, 2013

Working paper

We consider real and complex Clifford algebras of arbitrary even and odd dimensions and prove generalizations of Pauli's theorem for two sets of Clifford algebra elements that satisfy the main anticommutative conditions. In our proof we use some special operators - generalized Reynolds operators. This method allows us to obtain an algorithm to compute elements that connect two different sets of Clifford algebra elements.

Added: Oct 9, 2016

Working paper

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between the Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves.

Added: Jan 30, 2015

Working paper

A minifold is a smooth projective $n$-dimensional variety such that its bounded derived category of coherent sheaves $\D^b(X)$ admits a semi-orthogonal decomposition into an exceptional collection of $n+1$ exceptional objects. In this paper we classify minifolds of dimension $n \leq 4$. We discuss the structure of the derived category of fake projective spaces and conjecture that under some conditions they admit a quasi-phantom subcategory.

Added: May 27, 2013

Working paper

Akhtar M.,

, et al. arxiv.org. math. Cornell University, 2012. No. 1212.1785.
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

Added: Sep 14, 2013

Working paper

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

Added: Sep 14, 2013

Working paper

In this article we will introduce the way to extend the mitosis algorithm for Schubert polynomials, developed by Ezra Miller, to the case of Grothendieck polynomials.

Added: Nov 30, 2017

Working paper

We first establish several general properties of modality of al-
gebraic group actions. In particular, we introduce the notion of a modali-
ty-regular action and prove that every visible action is modality-regular.
Then, using these results, we classify irreducible linear representations of
connected simple algebraic groups of every fixed modality < 3. Next, ex-
ploring a finer geometric structure of linear actions, we generalize to the
case of any cyclically graded semisimple Lie algebra the notion of a packet
(or a Jordan/decomposition class) and establish the properties of packets.

Added: Jul 26, 2017

Working paper

Added: Dec 3, 2019

Working paper

In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$
The function $\Xi_N(z_1, z_2)$ is used in two cases and for two different purposes. Firstly, we prove generalization of the Zagier theorem (\cite{La}, \cite{Za3}) for the Hecke subgroups $\Gamma_0(N)$ of genus $g>0$. Namely, we obtain a kind of ``kernel function'' for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for $\Gamma_0(N)$, which is the analogue of the Zagier series $\omega_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, ~$J_{\Gamma_0(N)}(z_1)-J_{\Gamma_0(N)}(z_2)$, for genus zero congruence subgroup $\Gamma_0(N)$.

Added: Feb 23, 2018

Working paper

In this paper we construct the modular Cauchy kernel on the Hilbert modular surface ΞHil,m(z)(z2−z2¯), i.e. the function of two variables, (z1,z2)∈H×H, which is invariant under the action of the Hilbert modular group, with the first order pole on the Hirzebruch-Zagier divisors. The derivative of this function with respect to z2¯ is the function ωm(z1,z2) introduced by Don Zagier in \cite{Za1}. We consider the question of the convergence and the Fourier expansion of the kernel function. The paper generalizes the first part of the results obtained in the preprint \cite{Sa}

Added: Feb 27, 2018

Working paper

For each k greater than or equal to 6, we construct a modular operad of "k-log-canonically embedded" curves.

Added: Dec 11, 2014

Working paper

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many birational equivalence classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure.

Added: Feb 20, 2015

Working paper

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and $\{S_i\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.

Added: Sep 5, 2014

Working paper

We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems. Moreover, if some $N^n_i$ contains a single saddle, then $N^n_i$ is projective-like (in particular, $n\in\{4,8,16\}$, and $N^n_i$ is a simply-connected and orientable manifold). Given input dynamical data, one constructs a supporting manifold $M^n$. We give a formula relating the number of sinks, sources and saddle periodic points to the connected sum for $M^n$. As a consequence, we obtain conditions for the existence of heteroclinic intersections for Morse-Smale diffeomorphisms and a periodic trajectory for Morse-Smale flows.

Added: Oct 22, 2018

Working paper

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z×, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G_a-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].

Added: Oct 13, 2015

Working paper

We show existence of a natural rational structure on periodic cyclic homology, conjectured by L. Katzarkov, M. Kontsevich, T. Pantev, for several classes of dg-categories, including proper connective C-dg-algebras and dg-categories of local systems. The main ingredient is derived nilpotent invariance of A. Blanc's semi-topological K-theory, which we establish along the way.

Added: Feb 17, 2021