Annotation of the results obtained in 2022
Peterson varieties are well-known examples of projective algebraic varieties given by explicit equations. These varieties were previously studied from the point of view of algebraic geometry: they are known to be singular, but their cohomology rings have Poincaré duality. The conjecture arose that these varieties are homology manifolds. Within the framework of the project, the real points of the Peterson manifold in the manifold of complete three-dimensional flags were investigated. It was proved that this real algebraic variety is a closed homology manifold, moreover, it is homeomorphic to a two-dimensional torus. Obtained in 2021, an explicit hypothetical formula expressing the cobordism class of the nth symmetric power [Sym^n(V_g)] of a compact Riemann surface V_g of arbitrary genus g as an integer polynomial in the classes [CP^1], [CP^2], [ CP^3], ..., is proved for all n<6, as well as for g=0 and g=1 and all n. An action of 2m commuting involutions on the compact torus T^{2m+1}, which are automorphisms of this torus as a Lie group, is constructed such that the corresponding orbit space is homeomorphic to RP^{2m+1}. As a corollary, it is proved that the manifold RP^{2m+1} is a 2^{2m}-valued abelian coset topological group. The action of 2m commuting projective involutions on RP^{2m+1} whose orbit space is homeomorphic to the (2m+1)-sphere is explicitly constructed. It is proved that for any effective orientation-preserving action of m commuting involutions on an orientable closed connected topological manifold X of dimension greater than 2, the rational cohomological lengths of the quotient space Y=X/Z_2^{m} and the manifold X itself are related by the inequality L(Y)>=L( X)/(m+1). The results were accepted for publication in the journal Mathematical Notes. It is proved that for any finite subgroup G in SO(3), the manifold X_G obtained from SO(3) by the action of conjugations by elements from G is homeomorphic to RP^3 in the case of an odd order of the group G, and homeomorphic to S^3 in the case of an even order of the group G. Also, for any finite subgroup G in SO(3), when it is lifted to the group 2G under the canonical projection Sp(1)->SO(3), the manifold Y_G obtained from Sp(1)=S^3 by the action of conjugations by elements from 2G is always homeomorphic to S^3. Both of these theorems give new series of examples of coset n-valued topological groups on S^3 and RP^3. The results has been accepted for publication in the journal Functional Analysis and its Applications. A conjecture that strengthens the previously known assertion that the crystalline core in any Delaunay set X on the plane is a Delaunay subset is proved. It is proved that among the vertices of any cell of the Delaunay tiling, at least one is crystalline. And since the radius of the circumscribed circle around the cell does not exceed R, the radius R_{cr} does not exceed 2R. This theorem is the maximum generalization of the classical theorem on the impossibility of a fifth-order axis in a lattice. A complete proof of the estimate 4R for the regularity radius in the plane is completed. This result means that a 4R-isometric Delaunay set, that is, a set in the plane with equivalent 4R-clusters, is a regular system (a set with a transitive symmetry group). We prove that the order n of any local rotation in a 2R-isometric set in R^4 that has no fixed subspaces does not exceed 25. This is an important step towards obtaining a universal estimate for the regularity radius in R^4. The results are published in Proceedings of the Steklov Institute of Mathematics https://link.springer.com/article/10.1134/S0081543822040071 Consider an oriented closed surface S of genus 3, that is, a sphere with three glued handles. The isotopy classes of orientation-preserving homeomorphisms of the surface S onto itself form a group called the mapping class group. This group has two very important subgroups called the Torelli group and the Johnson kernel. Consider on a surface S the union of three homotopically nontrivial pairwise disjoint simple closed curves M such that the complement S-M is connected. We study the abelizations of the stabilizers of a multicurve M in the Torelli group I, in the Johnson kernel K, and in some special intermediate group G. In the case of the Torelli group I and the group G, a complete description of the abelization of the stabilizer is obtained; in the case of the Johnson kernel K, a partial result is obtained. It is proved that for any genus g, the Klein hyperelliptic function corresponding to a curve of this genus defines a solution to the parametric Korteweg-de Vries hierarchy. Hierarchy parameters are given as polynomials in hyperelliptic function parameters. On the basis of this result, the relation between the Lie algebra of derivations of hyperelliptic functions and the higher stationary Korteweg-de Vries equations is described. The results are published in the journal Funkts. analysis and its applications. https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid= faa&paperid=4020&option_lang=rus It is proved that there exists a one-parameter family of three-dimensional hyperbolic polytopes of finite volume connecting an ideal right-angled polytope and a polytope obtained from it by an edge-twist operation. For polytopes of the family, all dihedral angles, except for one, are right, and one angle changes from a straight angle to zero. It is proved that if the family is smooth, then Schläfli's differential formula is applicable to this family and gives the monotonicity of the volume change under the edge-twist operation. The question of smoothness is still open. It is shown that the lamplighter's Lie algebra over the field of rational numbers, introduced in the works of S.O. Ivanov, R.V. Mikhailov and A.A. Zaikovsky, is isomorphic to an infinite-dimensional naturally graded Lie algebra of maximal class. The structure of the nilsoliton metric for all finite-dimensional factors of the positive part of the current algebra of the Lie algebra sl(2) is found. The results are published in Proceedings of the Steklov Institute of Mathematics https://link.springer.com/article/10.1134/S0081543822040101 On the basis of polyhedral products and dga-models, equivariant cohomologies of the moment-angle complex Z_K with respect to the actions of the coordinate subtori T_I in T^m are calculated. The results are published in Proceedings of the Steklov Institute of Mathematics https://link.springer.com/article/10.1134/S0081543822020079 A description of the space of sheets of a real or holomorphic foliation on the complement to a set of real or complex coordinate planes given by a configuration of vectors is obtained as a set of minima of norms on sheets of the foliation in various metrics. A complete description of the basis of the 4-th graded component of the adjoint Lie algebras for right-angled Coxeter groups corresponding to simplicial complexes on 4 points is obtained. An algorithm that writes out the basis of the 4-th graded component of the adjoint Lie algebra for a right-angled Coxeter group in the general case is developed.

 

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Annotation of the results obtained in 2020
We say that a manifold M^n has a partially framed (with defect 1) U-structure if there exists a one-dimensional complex vector bundle ξ → M and an isomorphism (T M) ⊕ξ⊕ (2N - n − 2) → (2N), where T M is the tangent bundle and (k) is the trivial real bundle. We calculate the image of the canonical homomorphism of the group of partially framed (with defect 1) complex bordisms into the group of complex bordisms. We describe generators of this image. As a consequence, we obtain a solution to the problem posed 50 years ago about geometric representatives of the coefficients of the universal Chern character. The results are published in https://iopscience.iop.org/article/10.1070/RM9955 In the work by Ayzenberg-Buchstaber, the notion of twin-manifolds in the manifold of complete flags was introduced. During the reporting period, a detailed study of the structure of these manifolds was carried out. The construction of twins gives examples of pairs of manifolds with similar properties: a compact torus acts on the twins, and many of their invariants coincide (equivariant cohomology rings are isomorphic, orbit spaces are homeomorphic, and the Betti numbers often coincide). However, earlier in dimension 4 a specific example when the twins are not diffeomorphic was given. Main general examples of twins are the pairs: a semisimple regular Hessenberg variety and a manifold of isospectral Hermitian matrices in row echelon form. During the reporting period, it was proved that, with the exception of the trivial case, twins from this class are not diffeomorphic. The local theory of regular systems studies Delaunay sets with pairwise congruent clusters of a certain radius. Progress in this theory suggested a new direction in the study of Delaunay sets. It was shown that even for an arbitrary Delaunay set one can obtain results that have important applications in the theory under discussion. In the framework of the new direction, we formulated conjectures and obtained new results. We proved that in an arbitrary Delaunay set on the plane, the subset of points at which the local group is crystallographic is a Delaunay subset. This result (an article has been prepared) and the formulated conjecture about its three-dimensional analogue maximally generalize the famous statement about the absence of 5th-order global axes in two- and three-dimensional lattices. It is proved that in locally isometric sets in three-dimensional space there are no local 5th-order axes, including local groups D_5. This result is also a significant and an important generalization of the classical theorem on 5th order axes in three-dimensional lattices. A detailed article with a technically very difficult proof of this theorem is being prepared. We proved the 6R estimate for the radius of regularity of Delaunay sets in the hyperbolic plane. The well-known Torelli group is a subgroup of the mapping class group of an oriented surface of genus g (a sphere with g handles), which consists of all mapping classes that act trivially on the homology of the surface. For the Torelli group of genus 3, a detailed study of the spectral sequence was carried out - an algebraic structure arising from the action of the Torelli group on some special cellular complex - a complex of cycles, built by Bestvina, Bux and Margalit in 2007. A number of results are obtained on this spectral sequence, which are an essential advance towards the conjecture that the Torelli group of genus 3 cannot be presented by a finite number of generators and defining relations. A series of Lie algebras acting on functions of the total space of the universal bundle of Jacobians of hyperelliptic curves of genus g is constructed in an explicit form. The generators of these algebras correspond to multidimensional Schrödinger operators in magnetic fields with quadratic potentials. These operators define a system of 2g equations defining the sigma function of a universal hyperelliptic curve of genus g. For any genus g, an explicit form is obtained for the Schrödinger operators of grading 0, 2, and 4, and recurrent formulas expressing the remaining Schrödinger operators as elements of a polynomial Lie algebra using the Lie brackets of the operators of grading 0, 2, and 4. The result is published in the paper http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=3837&option_lang=rus Thurston's geometrization conjecture (proved by Perelman) states that each closed orientable three-dimensional manifold can be canonically cut into pieces so that each piece will have a geometric structure of one of 8 types. In the paper by Davis-Januszkiewicz (1991), the notion of a small cover over a simple polytope was introduced. It is a smooth manifold glued from copies of a polytope. At the same time, in a series of works (1986-2017) A.Yu. Vesnin and A.D. Mednykh proposed a construction of manifolds obtained from three-dimensional right-angled polytopes in the geometries H^3, H^2xR, R^3, S^3 and S^2xR (three other geometries were realized using orbifolds). For a compact polytope, the manifold coincides with a small cover. During the reporting period, the canonical Thurston decomposition of any small cover was found in an explicit form. A simple three-dimensional flag polytope different from the cube (having Euclidean geometry) can be canonically cut along 4-belts so that each part is homeomorphic to a non-compact right-angled polytope of finite volume with geometry H^2xR or H^3. It is shown that for an orientable small cover, this decomposition corresponds to the well-known JSJ decomposition of an irreducible manifold along incompressible tori, as well as to its canonical minimal decomposition into geometric pieces of finite volume along incompressible tori and Klein bottles. The parts have finite volume geometries of H^2xR or H^3. The result is published at https://arxiv.org/abs/2011.11628 The structure of an abelian 2^(m-1)-valued Lie group on RP^m for all odd m is constructed. It is proved that on the product of k spheres of arbitrary dimensions, k>=3, there is no action of (k-2) commuting involutions such that the quotient space is a rational homological sphere. The estimate (k-2) is sharp, which follows from the existence of examples constructed by D.V. Gugnin in 2019. A 2^(m-1)-sheeted topological ramified covering CP^m over S^(2m), m>=3, is constructed, which is a generalization of the classical Kuiper-Massey theorem, which states that CP^2 / conj and S^4 are homeomorphic. Moreover, for all m>=3, this construction has given the degree of a ramified covering CP^m over the 2m-sphere less than the previously known lower bound (2m)!. The results were presented at the international conference https://cs.hse.ru/mirror/pubs/share/418556206.pdf Lie algebras and superalgebras play an important role in the apparatus of modern theoretical and mathematical physics. Some of them, such as the Virasoro algebra, play a special, distinguished role. The same special role is played by its super-analogs, there are two of them, the algebras of Ramond and Neveu-Schwartz. Therefore, each new result about their properties and applications always becomes the focus of specialists' attention. During the reporting period, we managed to describe important structural properties of the so-called positive components of these algebras - maximal nilpotent subalgebras. It is shown that from the point of view of presentation by generators and relations, the positive parts of the Ramond and Neveu-Schwarz algebras have a completely different nature, which does not contradict the connection to the Virasoro algebra. The results were presented at the international conference: https://lomonosov-msu.ru/file/event/6054/eid6054_attach_9bc6d41fe4caa424589bc7d6d5529e53fbcc87a7.pdf The algebra of basic Dolbeault cohomology of the canonical foliation on the class of non-Kähler complex manifolds with toric symmetry is described, in particular, on complex moment-angle manifolds. A complex moment-angle-manifold with a canonical holomorphic foliation can be regarded as a model of an "non-rational" toric variety (in the rational case, a holomorphic foliation turns into a fibration with a base, a toric variety and a complex torus fiber). Calculation of the basic Dolbeault cohomology of the canonical foliation is an important step in the construction of a non-rational analogue of toric geometry.

 

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Annotation of the results obtained in 2021
In the well-known work of I. Macdonald in 1962, the ring of integer cohomology and the Chern classes of the tangent bundle of the n-th symmetric power W^n_g = Sym ^n (V_g) of a compact Riemannian surface V_g of genus g were calculated. It turned out that this ring, despite the absence of torsion, has a rather complicated description. The formulas for the Chern classes of the tangent bundle are also quite complicated. In the project, we found explicit for the Milnor s-numbers of the tangent bundle of manifolds W^n_g. It is known that the complex cobordism class of a closed complex manifold is uniquely determined by its Milnor s-numbers. Moreover, any element of the complex cobordism ring is a polynomial with rational coefficients in the cobordism classes of complex projective spaces. Within the framework of the project, we presented and checked for many examples an explicit formula for the cobordism class [W^n_g] in the form of a polynomial with integer coefficients from [CP ^ 1], [CP ^ 2], [CP ^ 3], .... A pair of abstract three-dimensional crystal lattices are associated with some topological space: the space of misorientations. It encodes all possible ways of the mutual arrangement of this pair of lattices and arises naturally in problems of crystallography and materials science when describing the physical properties of polycrystalline materials. It is known that the space of misorientations is a three-dimensional orbifold: from the topological point of view, it is just a closed three-dimensional manifold, but it contains a distinguished trivalent graph, the so-called graph of orbifold singularities. At the previous stage, the topological type of misorientation spaces was described for all pairs of point groups of crystals. The goal at this stage was to describe the graphs of orbifold singularities of spaces of misorientations. A computational approach based on the theory of invariants of representations of finite groups is proposed. A Python script based on this approach showed the correct result for the misorientation space of a pair of lattices with the point symmetry group D_2 (the symmetry group of a matchbox). In this particular case the space of misorientations has been defined and studied within the framework of toric topology. It is proved that the graph of its singularities is the graph K_ {3,3} ("3-houses-3-wells") arising in Kuratowski's criterion for graph planarity. The nonplanarity of the graph K_ {3,3} shows that the space of complete flags in R ^ 3 is not a small cover over a polytope. In an arbitrary Delaunay set, local groups, that is, groups of 2R-clusters, can be arbitrarily complicated. However, as follows from our results, in any Delaunay set X in R^3, the subset of all points at which the order of local axes does not exceed 6 is a Delaunay set with a covering radius R'<3R. This theorem significantly improves the previous estimate for R' in terms of R. It is an important step towards proving the hypothesis that the crystal core of an arbitrary Delaunay set, that is, the subset of all points whose local groups contain rotations of only crystallographic orders 2, 3, 4, or 6 is also a Delaunay set. This result is also important in the local theory of regular systems. The regular system, that is, the Delaunay set with a transitive group, is the basic component of the atomic structure of an ideal crystal. The radius of regularity means such a size of clusters, the pairwise congruence of which for a given Delaunay set ensures the regularity of this set. The performed study of 2R-isometric Delaunay sets in R^3 with local groups whose tower heights are 4 and higher, by virtue of the local criterion of regular systems, implies that the congruence radius does not exceed 8R. The results are published in the journal Russian Mathematical Surveys in the section "Communications of the Moscow Mathematical Society" http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=10037&option_lang=rus The mapping class group of a closed surface of genus g is the quotient group of the group of orientation-preserving homeomorphisms of this surface by the connected component of the identity. The most important subgroup of the mapping class group is the Torelli group, which consists of all mapping classes that act trivially on the homology of a surface. Another important subgroup is the so-called Johnson kernel; its definition is more complicated. The following problems were posed long ago and are still open: is the genus 3 Torelli group finitely defined and whether the genus 3 Johnson kernel is finitely generated. The approach used to these problems is based on the study of spectral sequences associated with the actions of these groups on the complex of cycles - a special cell complex built by Bestvina, Buks and Margalit in 2007. At this stage of the project, results have been obtained on the infinite generation of a number of members of these spectral sequences, which give progress in the direction of the conjecture that the Torelli group of genus 3 is not finitely determined, and the Johnson kernel of genus 3 is not finitely generated. The results were published in the journal Izvestia: Mathematics. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=9116&option_lang=eng On the basis of the theory of hyperelliptic functions, results are obtained on the connection between the solutions of the g-th stationary Korteweg-de Vries equation and the structures defining the differential geometry of the total spaces of universal bundles of Jacobians of algebraic curves in the case of hyperelliptic curves of genus 2 and 3. Thurston's geometrization conjecture (proved by Perelman) states that each closed orientable three-dimensional manifold can be canonically cut into pieces so that each piece will have a geometric structure of one of 8 types: S^3, R^3, H^3, S^2xR, H^2xR, universal cover of Sl (2, R), Nil and Sol. At the previous stage of the project, the canonical Thurston decomposition of any three-dimensional orientable small cover was found in an explicit form. At this stage, the result was generalized to orientable 3-manifolds corresponding to vector-colorings of polytopes of any rank. The article was accepted for publication in the journal Sbornik: Mathematics http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=9665&option_lang=eng We constructed explicitly 2n commuting involutions on RP^{2n + 1}, whose joint orbit space is homeomorphic to the (2n + 1)-sphere. A new combinatorial description of the symmetry algebra of the Liouville differential equation is obtained. The connection between the lamplighter Lie algebra, introduced in the works of Ivanov, Mikhailov, and Zaikovsky, and the simplest graded Lie algebra of the maximal class m_0, introduced by Vergne in the early 70s, has been discovered. We give a negative answer to the question of Murillo and Felix about whether a finitely generated infinite-dimensional pro-nilpotent Lie algebra can have a finite-dimensional second homology group. It is proved that the ring of SU-linear operations in the theory of complex cobordism MU is generated by the well-known geometric operations \partial_i of Conner-Floyd-Novikov. For the theory of c_1-spherical bordisms W, all SU-linear multiplications in the theory W and the projections MU onto W are described. The complex orientations of the theory W and the corresponding formal groups F_W are described. In the work of V.M. Buchstaber, 1972, the localizations of the ring of coefficients of the formal group F_W were described, at which the localization of the ring of scalars \Omega_W of the theory W is obtained. The following generalization of the result of V.M. Buchstaber is obtained: for any SU-linear multiplication and complex orientation of the theory W, the coefficients of the corresponding formal group F_W without localization do not generate the ring \Omega_W. The article was submitted to the journal. We study real and holomorphic foliations on the complement U (K) to the set of coordinate subspaces in R^m and C^m, given by the configuration of the vectors Γ. It is proved that the space of sheets of such a foliation is Hausdorff if and only if the configuration of vectors Γ and the simplicial complex K defining the complement of a set of subspaces are Gale dual to a simplicial fan. A criterion for the compactness of the space of sheets is obtained in terms of the completeness of the fan. It is proved that in this case the space of sheets is diffeomorphic to the moment-angle manifold. The case of a normal fan of a convex polytope or a (possibly unbounded) polyhedron is considered separately. It is proved that in this case the space of sheets of the foliation is given by the complete intersection of real or Hermitian quadrics. A criterion is obtained for a Gale dual configuration to Γ and a simplicial complex K to define a simplicial fan, as well as a criterion for this fan to be a normal fan of a convex polytope. We proved an isomorphism of the first three graded components of the adjoint Lie algebra of the right-angled Coxeter group of a discrete set of N points and the algebra FL <m_1, ..., m_N> / ([[m_i, m_j], m_j] = [[m_i, m_j], m_i] for any i <j), and also found common additional relations in the fourth graded component of these two algebras.

 

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