Projects

Relates to [18,19,21,30,31,32].

The Hermitian Tomei manifold of isospectral tridiagonal matrices is a quasitoric manifold over a permutohedron. Other shapes of matrices determine other manifolds with torus actions. 

At first, we studied the particular cases: arrowhead matrices (related to lots of cool physics), periodic tridiagonal matrices (related to periodic discrete Schrödinger operator and Toda flow), and staircase matrices (related to Hessenberg varieties and a whole bunch of applied linear algebra). In these cases we studied the structure of face posets, orbit spaces, cohomology and equivariant cohomology of such manifolds.

I noticed that manifolds of staircase matrices are equivariantly formal, while periodic tridiagonal and arrowhead cases are not. This somehow resembles the fact that staircase matrices can be easily diagonalized by QR-type algorithm. I looked for a bridge between toric topology and applied algorithms of linear algebra and finally proved that there exists an asymptotical diagonalization algorithm on a class of symmetric sparse matrices if and only if this class is a Hessenberg class. 

This result required application of Morse theory, the classification of proper interval graphs, some recent results about discrete torus actions, graphicahedra, matroids, and in the end - the computation of homology of large posets on a high-performance cluster. 

Relates to [17,22,23,26,28,29,35,36,38]. 

When k-dim torus acts on 2n-dim manifold effectively with isolated fixed points, the number n-k is called the complexity. Torus manifolds, toric varieties, quasitoric manifolds all have complexity 0, - these are well studied in toric topology.

Buchstaber-Terzic studied the canonical action of 3-torus on 8-dim Grassmann manifold of 2-planes in C^4. They proved that the orbit space of this action is (surprize!) homeomorphic to the 5-sphere. They also posed the question "How can we apply ideas of toric topology to study actions of positive complexity?" 

I am working in this direction. I started by proving  the local result that the orbit space of complexity 1 action in general position is a topological manifold. After that we proved that whenever such action is equivariantly formal, its orbit space is a homology sphere (if things are simply connected, the Poincare conjecture tells this is an actual sphere). So Grassmann manifold is not something special - we get spheres in many other examples. This includes the torus action on quaternionic proj. plane, - some related orbit spaces were studied by Vladimir Arnold, - in this case we may use the notion of spectrahedron to prove sphericity, avoiding Poincare conjecture.

After that, we noticed that our methods to prove sphericity generalize to actions of higher complexities. On this way, we noticed the relation between faces of toric actions and theory of geometrical lattices and matroids. 

Further on, we proved that whenever the action is equivariantly formal, and its tangent weights are in some general position, then both the orbit space and the poset of faces have some degree of acyclicity. This, in particular, gives an obstruction for a GKM graph to be a GKM graph of an equivariantly formal GKM manifold. 

Relates to [7,8,16]. 

Together with Masuda, Park, and Zeng, we proved that there exists 6-dim quasitoric manifolds which are not toric origami. Surprisingly, some coloring and isoperimetric properties of 3-dimensional polytopes were involved in the proof. In fact, most 3-dim polytopes support q-toric manifolds which are not toric origami - this can be seen from the Brownian map properties.

Another question is whether any combinatorial simple polytope support a smooth projective toric variety. The answer is given by Delaunay: if P is the moment polytope of 3-dim proj.nonsing. toric variety, then P has either triangular or quadrangular face. Hence there is no toric variety over dodecahedron. She uses Reid's results on minimal algebraic models in the proof, and her arguments were not clear for me. So I rewrote the proof in more toric-theoretical terms.

Relates to [13,20]. 

Timorin, following the remark of Pukhlikov and Khovanskii, introduced the geometrical model of the polytope algebra (cohomology of toric variety) based on the notion of the volume polynomial. Multi-fans and multi-polytopes were introduced by Hattori and Masuda as generalizations of fans and polytopes used to study torus manifolds (instead of toric varieties). 

Together with Mikiya Masuda, we developed the theory of volume polynomials and corresponding duality algebras of multi-fans. We related them to h''-numbers in combinatorics, matroids, and proved that the analogue of g-conjecture does not hold for multi-fans. Later on, I made several computer experiments and introduced an exotic invariant of 3-dimensional pseudomanifolds, based on convex geometry.

Done while working in OCU with Mikiya Masuda. Relates to [10,11,12,14,15]. Assume an action on a torus manifold is locally standard. Then its orbit space is a manifold with corners. If the orbit space is a simple polytope (or a homology polytope meaning that all its faces are acyclic), then lot of things are known about the homology of a manifold itself. This standard theory is related to face algebras of spheres.

I proved a generalization, when the orbit space is a manifold with corners in which all PROPER faces are acyclic. In this case face algebras of triangulated manifolds are used, however the answers are more complicated. In particular we described homology of a class of toric origami manifolds.

Partly my specialist diploma. Relates to [3,6,9]. I viewed Buchstaber number as generalized chromatic invariant and, based on this understanding, proved several things: (1) computed Buch.inv. of graphs, (2) constructed example that Buch.inv. is not additive with respect to join operation, (3) constructed example when ordinary and real Buch.inv. are different, (4) constructed example that Buch.inv. is not determined by bigraded Betti numbers. I used Taylor resolutions and some cool results of Kolya Erokhovets to prove the latter fact. 

My cand.sc. thesis. Relates to [1,2,4,5]. I introduced the notion of nerve-complex of a (non-simple) convex polytope and used it to study several notions of toric topology generalized to such polytopes: moment-angle spaces, Stanley-Reisner algebras, iterated wedges. Moment-angle space is a certain quadrics' intersection, it may be degenerate (unlike the case of simple polytopes). However its homotopy type can be described using homotopy colimits.

I generalized iterated wedges, and defined "substitutions" which make the collection of all simplicial complexes an operad. Some categorical constructions utilizing simplicial complexes (polyhedral products, Stanley-Reisner algebras) can be viewed as modules over this operad. 

Relates to [37]. 

The classical notion of Euler's Tonnetz describes harmonical relations of pitch classes modulo octave in the form of a specific triangulation of a 2-torus. We introduced a model of tonal space which takes different octaves into consideration. Vector embeddings based on our tonal space can be used as an input to transformer networks for MIR tasks. The homeomorphisms of the tonal space can be used for automatic music transformation.

Relates to [33]. 

This is the project of my PhD student German Magai. We are interested in topological processes inside deep networks. One looks at the data flowing across layers of a network, and compute some topological invariants. The rate of change of invariants across the layers seem to correlate with the efficiency of a network architectures.

Relates to [24,34]. 

Up to moment, we only have one mathematical preprint on this subject. My PhD student Kostya Sorokin is doing empirical research on this subject.

There exist place cells in a mammal's brain which activate whenever an animal moves within a specific location. There is a wonderful interdisciplinary research connecting topology to brain study. Nerve theorem is applied to tell the topology of the maze, where a mouse moves freely, if one only looks at the activation patterns of particular neurons in its brain. 

We plan to extend this research theoretically by adding functional hierarchies between neurons and collections of neurons into account. The perfect candidate for the method is formal concepts' analysis (FCA) which aims to extract  implementations in databases in theoretical and applied computer science. We combined topology and FCA. In the end, we want to construct topological FCA: a theory which describes both the shape and logical implementations in the brain data. 

Kostya also studies experimental data on real mice. Mice are amazing!

Relates to [25]. 

We found that non-free actions of groups on manifolds have an unexpected application in crystallography and material science. Given a polycrystall material consisting of two lattices, we have a misorientation: the position of one lattice relative to another lattice. One can ask a question, what is the space parametrizing all misorientations? This is the two-sided quotient G1\SO(3)/G2, where G1,G2 are point crystallography groups of the lattices.

Mathematically, misorientation spaces are 3-dim elliptic orbifolds. We classified their underlying manifolds: it happens that most of them are homeomorphic to S^3 due to Poincare conjecture.

One particular case D2\SO(3)/D2, where D2 is the symmetry group of the matchbox, is of particular interest. It appears in toric topology as the real analogue of the homeomorphism F_3/T^2=S^4 proved for the full flag manifold F_3 by Buchstaber and Terzic.

Several of my students write their term papers, generalizing recent results in toric topology to the actions of discrete torus.

Here is a web app that I wrote in Unity+C# to demonstrate basics of TDA. It visualizes and computes persistent homology of 3d point-clouds with up to 40 points. You can initialize the cloud by simply copying a Python list, choose the type of filtration, look at persistent bars and corresponding simplices, choose euclidean minimal spanning tree. You can also set up the pallete to produce nice looking screenshots.

There was an idea to develop it into a full-fledged game by adding puzzles and fights with homological bosses, but currently I gave up on this.

I noticed that topological data analysis behaves not as expected in some cases: there are higher homology in low dimensional data. See some examples on github for details.

My dream is to make a reasonable simulator of non-euclidean geometries in 3d graphics (better in VR).

Together with my students, I am testing different approaches. My idea is that raymarching rendering should be well suited for this task: it is quite mathematical and only requires a metric, while the metric is the only thing which makes Lobachevsky different from Euclid.