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Working paper

Bauer and Catanese \cite{bauercat} have found 4 families of surfaces of general type with p_g=q=0 which are quotients of the product of curves by the action of finite abelian group. We compute integral homology groups of these surfaces.

Added: Feb 18, 2014

Working paper

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary resolution of singularities. We believe that homotopy finiteness holds also over perfect fields of finite characteristic.
We also prove the analogous result for $\Z/2$-graded DG categories of coherent matrix factorizations on such schemes.
In both cases, we represent our DG category as a DG quotient of a smooth and proper DG category by a subcategory generated by one object (we call this a smooth categorical compactification).

Added: Oct 31, 2013

Working paper

We consider a dynamical system on a metric graph, that corresponds to a semiclassical solution of a time-dependent Schrodinger equation. We omit all details concerning mathematical physics and work with a purely discrete problem. We find a weak inequality representation for the number of points coming out of the vertex of an arbitrary tree graph. We apply this construction to an "H-junction" graph. We calculate the difference between numbers of moving points corresponding to the permutation of edges. Then we find a symmetrical difference of the number of points moving along the edges of a metric graph.

Added: Oct 21, 2014

Working paper

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.

Added: Aug 29, 2012

Working paper

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.

Added: Nov 13, 2019

Working paper

We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare–Bendixson theory. In the
case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.

Added: Nov 13, 2019

Working paper

Braverman A.,

, Nakajima H. arxiv.org. math. Cornell University, 2014. No. 2381.
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras").

Added: Oct 2, 2014

Working paper

Braverman A.,

, Nakajima H. arxiv.org. math. Cornell University, 2014
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincar\'e pairing) in terms of representation theory of some vertex operator algebras ("W-algebras").

Added: Jan 30, 2015

Working paper

Frenkel E.,

, Nekrasov N. arxiv.org. math. Cornell University, 2008
Added: Feb 27, 2013

Working paper

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero–Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. This allows computing the IC stalks of the latter.

Added: Nov 15, 2015

Working paper

We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers up to two, one associates an adelic group. We show that this operation commutes with taking intersections if the surface is defined over an uncountable field and we provide a counterexample otherwise.

Added: Oct 31, 2013

Working paper

The paper is devoted to some applications of Stepanov method. In the first part of the paper we obtain the estimate of the cardinality of the set, which is obtained as an intersection of additive shifts of some different subgroups of F_p^*. In the second part we prove a new upper bound for Heilbron's exponential sum and obtain a series of applications of our result to distribution of Fermat quotients.

Added: Sep 2, 2014

Working paper

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider certain sum of Bessel functions and prove the upper bound for its weighted mean value. This bound provides estimates for the \gamma-th moments of gaps for all \gamma \leq 2.

Added: Oct 19, 2017

Working paper

The minimum number of NOT gates in a logic circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that $\lceil\log_{2}(d(f)+1)\rceil$ NOT gates are necessary and sufficient to compute any Boolean function f (where d(f) is the maximum number of value changes from greater to smaller one over all increasing chains of tuples of variables values). This result is extended on k-valued functions computing in the paper. Thereupon one can use monotone functions ``for free'' like in Boolean case. It is shown that the minimal sufficient for a realization of the arbitrary k-valued logic function f number of non-monotone gates is equal to [log2(d(f)+1)], if Post negation is used in NOT nodes and is also equal to [logk(d(f)+1)], if {\L}ukasiewicz negation is used in NOT nodes. Similar extension for another classical result of A. A. Markov for the inversion complexity of system of Boolean functions to k-valued logic functions has been obtained.

Added: Oct 20, 2015

Working paper

Added: Feb 6, 2013

Working paper

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix is an expanded version of my notes on open problems posted on the site of this workshop. It contains formulations of some open problems and the relevant comments.

Added: Jul 21, 2013

Working paper

Added: Feb 6, 2013

Working paper

We study the problem of existence of K\"ahler--Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree 22 that admit a faithful action of the multiplicative group ℂ∗. We prove that, except possibly two explicitly described cases, all such smooth Fano threefolds are K\"ahler--Einstein.

Added: Oct 21, 2018

Working paper

Added: Feb 2, 2018

Working paper

Klein foams are analogues of Riemann surfaces for surfaces with one-dimensional singularities. They first appeared in mathematical physics (string theory etc.). By definition a Klein foam is constructed from Klein surfaces by gluing segments on their boundaries. We show that, a Klein foam is equivalent to a family of real forms of a complex algebraic curve with some structures. This correspondence reduces investigations of Klein foams to investigations of real forms of Riemann surfaces. We use known properties of real forms of Riemann surfaces to describe some topological and analytic properties of Klein foams.

Added: Sep 22, 2016

Working paper

Kamenova L., Lu S.,

arxiv.org. math. Cornell University, 2013
The Kobayashi pseudometric on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkaehler manifold if it admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. The SYZ conjecture claims that any parabolic nef line bundle on a deformation of a given hyperkaehler manifold is semi-ample. We prove that the Kobayashi pseudometric vanishes for all hyperkaehler manifolds satisfying the SYZ property. This proves the Kobayashi conjecture for K3 surfaces and their Hilbert schemes.

Added: Aug 28, 2013