We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bialynicki-Birula's results of 1966-67. We prove ``fusion theorems'' for n-dimensional tori in the affine and in the special affine Cremona groups of rank n. In the final section we introduce and discuss the notions of Jordan decomposition and torsion prime numbers for the Cremona groups.

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Working paper

We prove the polynomiality of the bigraded ring J_{*,*}^{w, O}(F_4) of weak Jacobi forms for the root system F_4 which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root systems of Dn type for 2\le n\le 8 was studied.

Added: Sep 24, 2020

Working paper

We consider sets in the real line that have Littlewood--Paley properties LP(p) and LP and study the following question: How thick can these sets be?

Added: Feb 16, 2014

Working paper

An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.

Added: Oct 15, 2018

Working paper

Added: Oct 13, 2019

Working paper

The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.

Added: Oct 2, 2018

Working paper

Added: Oct 30, 2015

Working paper

We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). By using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion for Banach algebras. As an application, we completely characterize topologically amenable Köthe co-echelon algebras.

Added: Dec 17, 2020

Working paper

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to Ω-stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.

Added: Sep 11, 2017

Working paper

Provability logic concerns the study of modality $\Box$ as provability in formal systems such as Peano arithmetic. Natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970's by Harold Simmons and Leo Esakia. They have observed that the dual $\Diamond$ modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere. More recently, a new impetus came from the study of polymodal provability logic GLP that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that GLP was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged. We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also included a few results that have not been published so far, most notably the results of Section 6 (due the second author) and Sections 10, 11 (due to the first author).

Added: Feb 12, 2013

Working paper

Beilinson A., Kings G.,

. arxiv.org. math. Cornell University, 2014
We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for GL_n(Z) and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of L-functions of totally real fields and a construction of the associated p-adic L-function. Using a result of Graf, we also apply this to prove some integrality and p-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.

Added: Feb 5, 2015

Working paper

Added: Jan 9, 2013

Working paper

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.

Added: Aug 24, 2021

Working paper

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety. We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

Added: Oct 5, 2015

Working paper

Braverman A.,

, Nakajima H. arxiv.org. math. Cornell University, 2016
Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact Lie group Gc and its quaternionic representation M. Physicists study its Coulomb branch, which is a noncompact hyper-K¨ahler manifold with an SU(2)-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with C ×-action when M is of a form N ⊕ N∗ , as the second step of the proposal given in [Nak15].

Added: Jan 21, 2016

Working paper

We show how one can twist the definition of Hochschild homology of an algebra or a DG algebra by inserting a possibly non-additive trace functor. We then prove that many of the usual properties of Hochschild homology survive such a generalization. In some cases this even includes Keller's Localization Theorem.

Added: Oct 14, 2013

Working paper

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in p-th cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus T admitting a symplectic embedding to a hyperkahler manifold M. If M is generic in a d-dimensional family of deformations, then dimT≥2^[(d+1)/2].

Added: Dec 5, 2015

Working paper

Triangulated endofunctors of the derived category of coherent sheaves which do not admit DG liftings

Recently, Rizzardo and Van den Bergh constructed an example of a triangulated functor between the derived categories of coherent sheaves on smooth projective varieties over a field $k$ of characteristic $0$ which is not of the Fourier-Mukai type. The purpose of this note is to show that if $char \, k =p$ then there are very simple examples of such functors. Namely, for a smooth projective $Y$ over $\mathbb Z_p$ with the special fiber $i: X\hookrightarrow Y$, we consider the functor $L i^* \circ i_*: D^b(X) \to D^b(X)$ from the derived categories of coherent sheaves on $X$ to itself. We show that if $Y$ is a flag variety which is not isomorphic to $\mathbb P^1$ then $L i^* \circ i_*$ is not of the Fourier-Mukai type. Note that by a theorem of Toen (\cite{t}, Theorem 8.15) the latter assertion is equivalent to saying that $L i^* \circ i_*$ does not admit a lifting to a $\mathbb F_p$-linear DG quasi-functor $D^b_{dg}(X) \to D^b_{dg}(X)$, where $D^b_{dg}(X)$ is a (unique) DG enhancement of $D^b(X)$. However, essentially by definition, $L i^* \circ i_*$ lifts to a $\mathbb Z_p$-linear DG quasi-functor.

Added: Nov 8, 2017

Working paper

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface Y with a certain moduli space of twisted cubic curves on Y. These relations are generalizations of the "beautiful" Y-F(Y) relation by Galkin and Shinder which connects Y with the Hilbert scheme of two points on Y and the Fano variety F(Y) of lines on Y. We concentrate mostly on the case of cubic surfaces. The symmetries of 27 lines on a smooth cubic surface give a lot of restrictions on possible forms of the relations.

Added: Oct 23, 2018

Working paper

Braverman A.,

. arxiv.org. math. Cornell University, 2014
In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces.

Added: Feb 5, 2015

Working paper

We prove that the quartic threefolds defined by
∑_{i=0}^5 x_i=∑_{i=0}^5 x^4_i−t(∑_{i=0}^5 x^2_i)^2=0
in ℙ5 are rational for t=1/6 and t=7/10.

Added: Nov 25, 2015

Working paper

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure.

Added: May 28, 2021