In this work, we realize the quotient for the Stanley-Reisner ring by a certain ideal by a continuous map of topological spaces. It is not hard to produce the spaces themselves. Namely, these are the Davis-Januszkiewicz space which is well known in toric topology. The other space is the Deligne-Mumford compactification for the moduli space for rational stable curves with marked points, or equivalently, the wonderful compactification for the braid arrangement. (Notice that the cohomology rings of the latter were described in the original work of de Concini and Procesi.) My input was in providing with the map. Namely, I proposed to use the c-plexification operation for arrangements (known to von Neumann, that is, almost 100 years old construction). It clearly induces the correct map in cohomology after taking colimit, by virtue of Milnor's lim1 theorem. I also showed that in the colimit such a stabilization space becomes the DJ space up to homotopy. The is based on a surprising lemma describing the homotopy type for the colimit for a blow-up, whose normal bundle has rank tending to infinity. (Then to compare two spaces in question applying this lemma to the iterated blow-up for dCP and iterated link decomposition for DJ spaces.)

This map is compatible with the operadic structures on both spaces (i.e. it is a morphism of topological operads). This problem was asked by V. Dotsenko and served as a starting problem for this project. (The operads on DJ space constructed by T. Bahri et al., and by A. Ayzenberg, do not induce the correct map in cohomology.) Homology of  these topological operads are known as 2-HyCom and hypercommutative operads. E. Hoefel proposed to identify the intermediate spaces (dCP compactifications of c-plexifications) with quotients of configuration spaces by a certain affine transformation group. Following his conjecture, I provided with Chen-Gibney-Krashen construction for the intermediate spaces. This took my collaborators to absolutely wonderful new results in the fields of (homotopy) Koszul duality, generators and relations descriptions for operads, Givental actions, etc.

This joint work with O. Goertsches follows the problem of torus acton extension on "intermediately" independent GKM manifolds (see [8] below), that was posed by M. Masuda. We resolved the combinatorial part of the problem in two interesting cases. Namely, we proved that any 4-independent (a.k.a. GKM_4) GKM-graph of a GKM manifold (of dimension 2n with acting torus having dimension k) is extendible to a complexity 0 GKM graph (a.k.a. torus graph), provided that the manifold is either Hamiltonian or has complexity n-k=1.

My input was first in providing with a coordinate-free (global sections of a certain sheaf on graph) description for S. Kuroki obstruction to GKM graph extensions (an axial function group), and second in reducing the computation for this group to that for subgraphs in the GKM graph. (The first part was presented in my talk on 2019 "Toric Topology in Okayama" conference; thanks to Oliver and his collaborators, I was able to apply these ideas successsfully much later.) 

An important data for GKM graph extension problem is the connection on the GKM graph. It provides with the notion of parallel transport on the graph, and with i-faces (i.e. i-valent connected subgraphs that are invariant with respect to the parallel transport). The reduction is possible if the fundamental group of the graph is generated by 2-faces. Establishing this property is open in general (and has something to do with shellings of the following regular CW complex: the 2-skeleton for the respective GKM manifold orbit space). We were able to show this property in the Hamiltonian case (using convexity of the moment polytope by Atiyah, Guillemin and Sternberg), and in complexity 1 case (using universal covers of GKM graphs and cohomology of covers as invariants for the deck transformation group). Our results seem to fill the gaps in several preceeding works. 

In this overview of applications for sheaf theory to modern Computer Science and Topological Data Analysis, my input was merely in writing certain parts of mathematical addendums (C, in particular). I explained to my co-authors (who know much more about the applied part of the project than me) Grothendieck's theorem on universal delta functor (a well known result for algebraic geometers), pointed out that any presheaf is a sheaf on a finite topological space, a reference to the original work of Roos for cochain complexes computing right derived limits, an earlier result of Everitt and Turner on isomorphism for cellular and sheaf homology on finite spaces, provided details on general sheaf theory, and discovered many interesting works of Husainov.

In this paper, I used several spectral sequences (discussed below) to describe cohomology groups (with any coefficients) for a wide family of topological spaces. In the focus of this work is the Bousfield-Kan spectral sequence converging to cohomology of hocolim for a toric diagram (or its composition with classifying functor). This class of spaces includes those of all toric varieties, split T-CW complexes with contractible faces and their Borel spaces. The second page for BKSS has a well known desciption in terms of cohomology of presheaf on a certain poset. This provides with a nice computational tool for cohomology of singular toric varieties, for instance. 

I wrote a proof for the collapse for the BKSS on page 2 (with trivial additive extension) based on "natural formality argument". In the case of a complete toric variety, this spectral sequence is Poincaré dual to the well-known "orbit spectral sequence". I show this using Zeeman-McCrory spectral sequence collapse, a certain Poincaré-Lefschetz duality for sheaves on finite topological spaces (studied in tropical geometry, for instance). The OSS collapse for toric varieties was previously established (for Q coefficients) in works of B. Totaro, and V. Danilov (using purity argument from Hodge theory). For Z coefficients the OSS collapse was conjectured for any toric variety by M. Franz in 2006 paper. Caution: Unfortunately, my "natural formality" argument holds only over Q (also see [6] below).

I relate BKSS to Eilenberg-MacLane SS by a Grtothendieck-type 3-graded spectral sequence. Using this relation and certain long exact sequences for sheaf cohomology, I computed all bigraded Betti numbers for skeletons of any smooth projective toric variety X. This result is somewhat similar to broken toric varieties studied by E. Sundbo (but the studied topological spaces in these two works, and formulas for Betti numbers are completely different.) My formulas (for ordinary Betti numbers) agree with those following from Xin Fu's homotopy decomposition in the particular case when X is a complex projective space. The resulting comparison of our results involves a particular identity for the generalized hypergeometric function. 

In this joint work with S. Kuroki, we obtained a graph-theoretical proof of the well-known formula for the equivariant cohomology of a projectivization for a complex toric vector bundle over a GKM manifold. Passage from topological space to combinatorial objects (graphs with labels) is given by GKM theory, provided that the projectivization satisfies pairwise independence of weights at fixed points (i.e. is a GKM manifold). The main tool is the notion of leg bundle corresponding to the above vector bundle (which is an invention of my co-author, following ideas from his earlier joint paper with V. Uma). My input in this project was in proving that any projective GKM bundle (in sense of Guillemin, Sabatini and Zara) is a projectivization of a leg bundle if one allows rational labels of the legs. This, together with the BH type formula, provided with graph equivariant cohomology of any projective GKM bundle. (Namely, with Q coefficients; with Z coefficients this seems to be an open problem for abstract GKM projective bundles). Notice that GSZ proved only free module structure for the respective ring (i.e. Leray-Hirsch type theorem), but did not describe the multiplicative relation. Also note that in general a projective GKM bundle may not have a horizontal leaf isomorphic to the base, so it is totally unclear how to represent it as a projectivization. I also explained that projectives are projectivizations in the category of topological fiber bundles, which is based on an old result of J.C. Su.

This work takes homology vanishing of orbit spaces for GKM manifolds (by Ayzenberg and Masuda) further using homotopy colimits.

For any GKM manifold with a sufficiently generic (j-independent, i.e. any i<=j weights at the same fixed point are linearly independent) fixed point data the respective orbit space has vanishing (reduced with Z coefficients) homology up to degree j+1, by Ayzenberg and Masuda. An interesting phenomenon is that all known GKM manifolds (of dimension 2n, with compact k-dimensional torus action) that are j-independent (and not extendible to a higher dimensional effective torus action) satisfy either j<4 or j=n (i.e. torus manifold), which was observed by M. Masuda. I constructed GKM graphs that are k-independent (for any n>k>2). The first step of this construction is to take a periodic (with respect to a certain subgroup of translations) GKM graph in R^n (this was considered earlier by S. Kuroki, unpublished). The second step adds edges to this graph, preserving periodicity and increasing "complexity" n-k. The third step is to take quotient of this infinite GKM graph, resulting in a finite GKM graph embedded to a torus. I show that this GKM graph cannot be realized by a (equivariantly formal) GKM manifold, using the generalization of mentioned above homology vanishing for orbit space, see [9]. The argument is based on the computation for the Euler characteristic of certain face posets of GKM graph using P. Hall formula. (The possibly nontrivial homology is only in one degree, which is a condition on the sign of the Euler characteristic, which is not satisfied for my examples.) An interesting open problem is to find the graph equivariant cohomology ring for these examples.

Milnor hypersurfaces H_{i,j}, i<=j, are rational projective nonsingular varieties known (in algebraic topology) to be additive generators for the complex bordism ring by Hirzebruch.  The hypersurface is invariant with respect to a subtorus in the naturally acting torus on the ambient toric variety (product of complex projective spaces). It is well known that this subtorus does not extend to a dense open orbit on Milnor surfaces (except i=0 or the case studied in [5] below) by looking at the respective cohomology ring. I proved that this torus action cannot be extended to any higher dimensional torus algebraic action, by explicitly describing the respective automorphism group. (The result is probably known to experts in algebraic geometry by using very ample embeddings.) I linked Milnor hypersurfaces with the hypersurfaces of Ray, and Buchstaber and Ray by iterated blow-up constructions. The latter two hypersurfaces are degenerate, therefore the cohomology ring description is not known in general (but there is a formula in terms of annihilator). Then I showed that these hypersufraces do not have a structure of a toric variety in general. This justified my previous work [4], namely, there exist no short proof using their hypersurfaces. The proof involved a certain generalization of GKM graph notion to hypergraphs (because the isotropy representation for the torus action at fixed points had multiplicities >1, in general), and a study of monodromy with respect to a "connection" on these hypergraphs.

M. Franz showed that any toric variety has a homotopy decomposition into a homotopy colimit of a certain diagram (whose objects are compact tori, arrows are group homomorphisms, and the indexing category corresponds to the cone poset of the fan). In this joint work with I. Limonchenko, I proved a slight generalization of this result in the category of moment-angle complexes and their quotients. Another input of mine was to prove the Eilenberg-MacLane spectral sequence collapse (on page 2 with Z coefficients) for the associated Borel fibration. (This is another result in a series of EMSS collapses obtained before by Franz and several others.) The idea was to generalize the "natural formality" argument from a paper by Notbohm and Ray. Caution: although this result seems to be true, the argument holds only over Q. (The diagram (26) is not commutative, in general; this can be fixed by choosing a different natural quasi-isomorphism.) This will be addressed in my later work.

In the appendix to a paper by S..P. Novikov (written joint with A. Mischenko) it was observed that certain Minor hypersurfaces are complex bordant to Cartesian product of complex projective spaces. (This was used in their proof for the computation for the logarithm of the universal formal group law for complex cobordism theory.) The proof was implicit (by comparing the respective Chern numbers). Since 1967, a problem remained open to construct explicitly a complex bordism between these two complex manifolds (which gathered attention since 2000). I was first to solve this problem (in particular, in two different ways). The first construction relied on a "well known construction", a gluing procedure based on hexagon, from a paper by B. Totaro. The second construction was based on a general construction for bordisms in the category of stably complex orbifolds with quasitoric boundary by S. Sarkar. Notice that both constructions lead to a null-bordant connected component. The construction of bounding manifold for it remains an open problem.

A compact manifold with stably complex structure is equivalent to a sum of line bundles is called stably tangentially split manifold, or a TTS manifold. The definition of a TNS manifold is given in a similar way by replacing the stably tangential stably complex structure with the complementary vector bundle (i.e. such that the Whitney sum of both fiber bundles is a trivial complex vector bundle). These two classes appeared in the works of Arthan and Bullet, and Ochanine. N. Ray proved that every complex bordism class (of degree >2) has a manifold that is TNS and TTS at the same time. Buchstaber and Ray proved s similar result, but for quasitoric manifolds. I prove a unification for these two results, namely, in the class of quasitoric TNS manifolds. (Any such manifold is TTS automatically.) The proofs followed my previous work with Y. Ustinovsky (in particular, computations for Chern numbers), with an increment in number-theoretical difficulty related to divisibility properties of binomial coefficients (based on known generalizations of Lucas theorem in number theory).

The class of compact TNS manifolds of dimension 4 (see [4] above) was characterized by J. Lannes in terms of non-semidefiniteness for the intersection form. I took this further in the class of quasitoric manifolds of arbitrary dimension, by replacing intersection form with degree k polynomial forms (defined by pairing the cup-product of any fixed x and k-th power of the argument y, with the fundamental class of the manifold). My criterion of TNS property for quasitoric manifolds was in terms of non-semidefiniteness of these forms, or in terms of K-theory, or in terms of the volume polynomial (by virtue of Khovanskii and Pukhlikov description for the cohomology). The proof relied on the study of certain cones (over particular convex bodies) in the complex K-theory (as a vector space over Q) of the quasitoric manifold. (The passage from cohomology to K-theory tensor Q is by Chern character.) Algebraically, this corresponds to Hilbert's 17th problem, Pfister theory etc (e.g. see nice book "Sums of even powers of real linear forms" by Bruce Reznick). Even though the obtained criterion involves infinitely many forms, it is algorithmically verifiable by Tarski's theorem (which was suggested to me by A. Ayzenberg). In (complex) dimension 3 I characterized explicitly (nonsingular projective) TNS toric varieties in terms of the combinatorics for the fan. In higher dimension I proposed a conjecture, potentially related to Hodge-Riemann relations and Horrocks theorem in algebraic geometry.

This work completes a partial series of generators built by A. Wilfong (among nonsingular projective toric varieties) for the complex bordism ringin the remaining dimensions 2n (where n is even and n+1 is not a power of a prime). By a theorem of Milnor and Novikov, this problem reduces to divisibility propoerties of the s-number, a certain Chern number for toric varieties. I found a series of two stage blow-up operations B_k defined for any toric variety that changes the s-number in a way that depends only on its dimension (using Sage). Then I computed these changes precisely, which turned out to be a rather long task with cohomology of toric varieties and their Chern numbers. An additional task was to prove that the required divisibility property can be achived by sufficiently many compositions of these operations. This step relied on the classica; Lucas theorem, which was achieved by joint effort with Y. Ustinovsky. (He also explained a classical Frobenius result on conductors for semigroups to me, and suggested studying this problem using my operations, for which I am grateful to him.)

The Poincaré series of a multi-indexed filtration on the ring of germs for a germ of plane cruve coincides with the Alexander polynomial of the corresponding link, which was proved by Ebeling and Gusein-Zade. The obtained expression had an elegant A'Campo type formula (ratio of product for cyclotomic polynomials). It was conjectured (motivated by HOMFLY polynomial interpretations) that a similarly looking multi-index filtration on the same ring provides with another topological invariant of the singularity, potentially described by a similar rational function formula. The first part turned out not to be the case, which is my observation. I provided with a very simple example of two (nondegenerate with respect to the same Newton diagram) curves with different Poincaré series, by an explicit computation. (It required writing a rather long C++ code to find the first coefficients of the Poincaré series.) In general, computation of the Poincaré series for this filtration seems to be an open problem.