# Seminars

## 15.07, 18:10 Spectral clustering of combinatorial fullerene isomers based on their facet graph structure

**Meeting ID:**998 175 7123

**Speaker: **Artur Bille (PhD student at Ulm University, Germany)

After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene's stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface. With regard to real-world applications, calculations mostly were performed on all isomers of $C_{40}, C_{60}$ and $C_{80}$. We extend this range, and analyse the spectra of $C_n$ for all feasible $n\leq 150$, and the limit case when $n$ goes to infinity. We suggest a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by constructing a graph invariant using the eigenvalues of matrices representing fullerene graphs. This invariant leads to a linear order within the set of all fullerenes.

## 20.05, 13:10 Mathematical AI for molecular data analysis

**Meeting ID:**998 175 7123

**Speaker:** Kelin Xia (Assistant Professor, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore)

Artificial intelligence (AI) based molecular data analysis has begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics based molecular representations and featurization (or feature engineering). Molecular structures and their interactions are represented as various simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex), hypergraphs, and Tor-algebra-based models. Molecular descriptors are systematically generated from various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models, including random forest, CNN, RNN, Transformer, BERT, and others. They have demonstrated great advantage over traditional models in drug design, material informatics and chemical informatics.

## 15.04,18:10 Parameterized Vietoris-Rips Filtrations via Covers

**Link to Zoom**

**Meeting ID:** 860 7775 1824

**Speaker:** Bradley Nelson (William H. Kruskal Instructor in the Department of Statistics at the University of Chicago)

A challenge in computational topology is to deal with large filtered geometric complexes built from point cloud data such as Vietoris-Rips filtrations. This has led to the development of schemes for parallel computation and compression which restrict simplices to lie in open sets in a cover of the data. We extend the method of acyclic carriers to the setting of persistent homology to give interleavings between the persistent homology of Vietoris-Rips filtrations restricted to covers and the full construction. We show how these filtrations can be used to study data over a base space and use our results to guide the selection of covers of data.

## 08.04, 13:00 Configuration space of Moebius Kaleidocycle

**Link to Zoom**

**Meeting ID:** 869 2093 0113

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password

**Speaker: **Shizuo Kaji (Professor at the Institute of Mathematics for Industry, Kyushu University, Japan)

The configuration of points in the Euclidean space has long been a research topic in geometry and topology, sometimes in relation to the analysis of mechanical linkages. In this talk, we consider the configuration of lines in the Euclidean space, which provides a model for a certain type of mechanical linkage. The linkage is also popular as an origami toy and called Kaleidocycle. We see how geometry and topology help to analyse Kaleidocycles; in particular, we construct

- a rare example of a single degree-of-freedom underconstrained linkage, which we have named the Moebius Kaleidocycle

- a motion of a Kaleidocycle governed by some integrable systems, and show it preserves a discretised total elastic energy.

We will discuss a wide range of open problems.

## 25.03, 18:10 Simplicial complexes of polyomino tilings

**Link to Zoom**

**Meeting ID:** 976 1355 2747

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker:** Djordje Baralić (Deputy Communications Director at Mathematical Institute of the Serbian Academy of Sciences and Arts)

A polyomino is a collection of squares of equal size arranged with coincident sides. Firstly introduced by Martin Gardner in 50's they quickly attracted wide interest beside recreational mathematics. We assign a simplicial complex to the problem of regular tiling of a finite region in a square by polyominoes from a given finite set of shapes. It turns out that both combinatorics and topology of these complexes are interesting. Particularly we showed that in some cases they have homotopy type of the wedge of spheres. F vectors of the complexes reveal us some information on partial tilings.

## 18.03, 18:10 Path homology and join of digraphs

**Meeting ID: **974 3950 5315

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker:** A.A.Grigor'yan (Professor of Mathematics, Bielefeld University, Germany)

In this talk I'll introduce the path homology theory of digraphs and present Künneth-like formulas for homology groups of joins of digraphs.

## 17.02, 18:10 Wasserstein Stability for Persistence

**Meeting ID: **962 3218 9709

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker: **Primoz Skraba (Queen Mary University of London)

In this talk, I will introduce a result on Wasserstein stability for persistence. Wasserstein or optimal transport distances have been used in applications of topological data analysis for some amount of time but have been missing a stability proof. I will present an elementary proof in the case of finite complexes, which also provides a simple proof of bottleneck stability, followed by a discussion of applications and generalizations. In particular, I will discuss the algebraic version of stability as well as discuss open questions. This is joint work with Katharine Turner.

## 22.10, 18:10 1/x power-law in a close proximity of the Bak–Tang–Wiesenfeld sandpile

**Meeting ID: **899 0332 5272

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker: **Sasha Shapoval, professor, Big Data and Information Retrieval School, Faculty of Computer Science, HSE

A cellular automaton constructed by Bak, Tang, and Wiesenfeld (BTW) in 1987 to explain the 1/f noise was recognized by the community for the theoretical foundations of self-organized criticality (SOC). Their conceptual work gave rise to various scientific areas in statistical physics, mathematics, and applied fields. The BTW core principles are based on steady slow loading and an instant huge stress-release. Advanced models, extensively developed far beyond the foundations for 34 years to successfully explain SOC in real-life processes, still failed to generate truncated 1/x probability distributions. This is done here through returning to the original BTW model and establishing its larger potential than the state-of-the-art expects. We establish that clustering of the events in space and time together with the core principles revealed by BTW lead to approximately 1/x power-law in the size-frequency distribution of model events.

## 04.06, 18:10 Metric and Topological Approaches to Network Data Analysis

**Meeting ID:**** **:839 6788 5298

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker:** Samir Chowdhury (Stanford University)

Statistical and mechanistic models are crucial in studying dynamic neurobiological processes. Often these models can be translated into (possibly directed) graphs, and being able to efficiently perform statistical learning on sets of these graphs may be of crucial interest to clinicians and researchers alike. While network science approaches have been immensely useful in this regard, we expect that novel methodologies will provide answers to inference and prediction problems beyond those that have been introduced in the literature. To this end, we develop a framework that associates generalized metric space structures to graphs, and further embeds these structures in a global pseudometric space equipped with rich theoretical and practical properties. As an application, we show how to incorporate path homology--an algebro-topological descriptor of directed graphs--into a persistence framework that inherits stability and convergence properties from this pseudometric. We conclude by describing how this pseudometric further admits geometry-aware, practically-computable relaxations via the theory of optimal transport.

## 28.05, 18:10 Hodge Laplacian via Random Walks on Simplicial Complexes

**Meeting ID:**** **: 892 5830 5301

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker:** Gabor Lippner (Department of Mathematics, Northeastern University)

It is well-known that the simple random walk on a graph is closely related to the normalized Laplace operator, and that this connection yields interesting applications to data analysis - PageRank type methods, for instance. I will explain a way to define a random walk on simplicial complexes that relates to the higher order (aka Hodge) Laplacians in a similar — albeit somewhat more complicated — manner. This, in turn, can be used to devise methods to analyze data that is naturally represented as a simplicial complex.

## 13.05, 18:10 A Higher-Dimensional Generalization of the Heawood Theorem on Embeddings of Graphs into Surfaces

11 Pokrovsky Bulvar, Pokrovka Complex, room R 408

Please contact Anastasiya Kamyshanova (akamyshanova@hse.ru) to recieve the password.

**Speaker: **Eugene Kogan (HSE) and Arkadiy Skopenkov (MIPT and IUM)

We present a short well-structured exposition of the 2019 Patak-Tancer higher-dimensional generalization of the Heawood inequality for embeddings of graphs into surfaces. This exposition clarifies the relation of the Patak-Tancer proof to earlier known results. This exposition is accessible to non-specialists in the field.

A simplified version of the Patak-Tancer result is as follows.

Theorem. If the union of k-dimensional faces of the n-dimensional simplex PL embeds into the connected sum of g copies of the Cartesian product S^k \times S^k of two k-dimensional spheres, then g is at least (n-2k-1)/(k+2).

## 25.02, 18:00 Topological Autoencoders

**Speaker: **Maxim Beketov, PhD student, Intern researcher, International Laboratory of Algebraic Topology and Its Applications, HSE

Topological Autoencoders, TopoAE-s, are an architecture of autoencoders (a family of generative models of machine learning) proposed quite recently by M. Moor et al. (arxiv.org/abs/1906.00722), that successfully attempts to preserve some topological features of the learned dataset. This is achieved by penalising the autoencoder by the difference between the persisnent homologies of the dataset and that of it mapped onto the autoencoder's latent space (encoded). We will see the results on several datasets authors have experimented with, and discuss the limitations of this approach. No prior knowledge of autoencoders or persistent homology is required, there will be a quick summary.

## 24.12, 18:10 Towards Dynamic-Point Systems on Metric Graphs with Longest Stabilization Time

**Meeting ID:**** **870 0136 6827 **Password:**** **dynamics

**Speaker: **L.W. Dworzanski

Dynamical system of points moving along the edges of a metric graph is a discrete dynamical system. For the set of such systems that can be constructed from a given set of commensurable edges with fixed lengths, it is shown that there always exists a system consisting of a bead graph with vertex degrees not greater than three that demonstrates the longest stabilization time in such a set.

## 26.11, 18:00 - 19:00 Topological data analysis of eye movements

**Meeting ID: ** 854 0155 0858 **Password:**** **tda

**Speaker - **Arseniy Onuchin (4-th grade student, Faculty of Psychology, MSU)

The increasing use of eye-tracking in modern cognitive and clinical psychology, neuroscience and ophthalmology requires new methods of objective quantitative analysis of complex eye movements data. In their work, the authors use topological data analysis (TDA) to extract a new type of features of eye movements to differentiate between two eye movements groups, obtained upon the presentation of two different stimuli images – a human face, shown straight and rotated for 180 degrees, which corresponds to the processing of the normal and unusual visual information respectively. Experimental evidence shows that the proposed topology-based features have more discriminative power over the generally accepted features of eye movements, allowing to separate provided different stimuli with good accuracy. Moreover, the concatenation of the topology-based and region of interest fixation ratios features further improves the performance of the classification task, showing the complementariness of the proposed topological features to the existing ones. The authors believe that the new class of features is able to serve as a valuable addition for the eye-tracking data-based medical diagnosis of mental and neurological disorders and ophthalmological diseases.

## 23.11, 15:00 The coalgebra of chains and the fundamental group

**Meeting ID: **984 9557 1246 **Password:** The fundamental group of a circle, in words

**Speaker - **Manuel Rivera (Purdue)

Rational homotopy theory tells us that simply connected spaces, up to rational homotopy equivalence, may be classified algebraically by means of rational cocommutative coalgebras (Quillen) or in the finite type case by rational dg commutative algebras (Sullivan). Goerss and Mandell proved versions of these results for fields of arbitrary characteristic by means of simplicial cocommutative coalgebras and E-infinity algebras, respectively. The algebraic structures in these settings are considered up to quasi-isomorphism.

In this talk, I will describe how to extend these results to spaces with arbitrary fundamental group. The key new observation is that the homotopy cocommutative coalgebraic structure of the chains on a space determines the fundamental group in complete generality. The corresponding algebraic notion of weak equivalence between coalgebras is drawn from Koszul duality. The end goal of this program is to completely understand homotopy types in terms of algebraic “chain level” structure. This is joint work with M. Zeinalian and F. Wierstra.

## 4.11, Topological data analysis applied to text processing

**Identifier:** 870 1996 6775 **Password:** manifold

**Speaker -** Laida Kushnareva (MSU, Huawei)

We will tell about modern approaches to study the topology of data manifolds, and present our work on the construction of informative representations of texts, based on neural networks’ embeddings. The neural model BERT allows, to each small text, to associate a high dimensional vector, which carries the meaning of the text. It happens that, instead of this vector, one can use a smaller amount of information extracted from the topology of BERT neural network, and the classification will become even more effective. A series of new problems arises, which may be interesting to undergraduate and PhD students.

## 29.10 Extendability of simplicial maps is undecidable

**Speaker - **Arkadiy Skopenkov, Moscow Institute of Physics and Technology, and Independent University of Moscow

We explain why the problem of extendability of simplicial maps is interesting to computer scientists. In particular, we illustrate the relation to polygonal lines in the plane, to words in finite alphabets, and to realizability of hypergraphs in higher-dimensional space. We present a short proof of the Čadek-Krčál-Matoušek-Vokřínek-Wagner 2013 result from the title (in the following form due to Filakovský-Wagner-Zhechev, 2020). For any fixed integer l>1 there is no algorithm recognizing the extendability of the identity map of the wedge Y of two l-dimensional spheres to a PL map from X to Y for a given 2l-dimensional simplicial complex X containing a subdivision of Y as a given subcomplex. In this talk topological concepts are exposed in the way interesting and accessible to non-specialists, in particular, to computer science students.

## 22.10 Cyclic space-filling curves and their clustering property

**Speaker -** Igor Netay (Kryptonite company, IITP)

In this work we propose an algorithm to construct cyclic space-filling curves. One of such constructions leads to a family of space-filling curves of any dimension (H-curves). They are compared with the Hilbert curves in the sense of their clusterization property, and it happens that they are close, and sometimes better than the Hilbert curves. At the same time, their construction is simpler, which allows to perform faster computations.

## 15.10 Calculation of topological characteristics via combinatorial laplacians

**Speaker - **Polina Borisova, Math. Faculty HSE.

**Title: **Calculation of topological characteristics via combinatorial laplacians

Most of us know the notion of Laplace operator on the space of smooth functions. The notion of a similar operator on a chain complex is less known. Such operator appears quite often when we try to understand the structure of a simplicial complex. For example, calculation of Betti numbers can be reduced to the problem of finding certain eigenvalues of the Laplace operator. Similarly, we may treat the generating cycles in homology using the Laplacian.

**Speaker - **Nikita Kalinin, FCS HSE.

**Title: **Classification of quiver representations and representations with commutativity restrictions

Theory of quivers arises unexpectedly in many areas of mathematics, from the representation theory to matrix problems of persistence theory. In this talk Nikita will describe the motivation for using quivers in topological data analysis. He will present the classical results in the classification of representations as well as some modern results about classification with restrictions. Speaking of latter, he will consider the quivers of special type, the so called “ladders”.

## 8.10 Estimation of covariance matrix, decomposing into tensor product

**Speaker -** Dmitry Trushin, associate professor FCS HSE

The basic problem is the following: assume there is a random vector xi in euclidean space V with zero mean and covariance matrix s (a positive-definite matrix). We want to estimate s by an independent sample x_1,...,x_d from V. This estimate is usually found by minimization of some target function. There are two classical examples:

1) Gaussian distribution.

2) Elliptic distribution..

The most important parameter is the minimal sample size for the existence and uniqueness of the minimum. In the first case it equals n, in the second - n + 1.

Usually d is strictly less than n and we have to consider additional requirements on the distribution. For example, xi lives in the tensor product V\otimes U (dim V = n and dim U = m), centered, and its covariance matrix is s = p\otimes q. In this case the elements x_1,...,x_d can be considered as matrices. It happens that in both cases the estimate for the sample size can be improved; it is equal to m/n + n/m. These estimates cannot be improved.

The question about nontrivial estimate in case of a tensor product was open for 20 yers. The review and further information can be found in the paper

I. Soloveychik, D. Trushin, Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness, Journal of Multivariate Analysis 149 (2016), 92-113. In the talk we will tell about the minimization task and other methods applied for its solution.

## 01.10 The mergegram of a dendrogram and its stability

**Speaker -** Yury Elkin, University of Liverpool, PhD Student.

TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.

## 24.09 Neuron activation data processing in the rodent’s hippocampus; Computational optimal transport and Wasserstein barycenters’ search; Description of the complex cobordism class of a toric variety.

**Speaker - **Constantine Sorokin, Math Faculty HSE.

**Title:** Neuron activation data processing in the rodent’s hippocampus

The discovery of place cells encoding the space memory in the hippocampus was an important moment for neurobiology and generated a lot of research in mathematics and biology. The speaker will tell about his work on the preprocessing of neural data, extraction of place cells and results of topology reconstruction based on the data thus obtained.

**Speaker - **Daniil Tyapkin, FCS HSE.

**Title:** Computational optimal transport and Wasserstein barycenters’ search

Computational optimal transport is a popular sphere in application, in particular, in applications to machine learning. The speaker will tell about the approach to finding Wasserstein barycenters, which reduces the problem to a (seemingly difficult) class of saddle problems.

**Speaker -** Vladimir Smurygin, FCS HSE..

**Title:** Description of the class of complex cobordism of a toric variety.

There is an old problem in algebraic topology - the description of the coefficients in the exponent of the complex cobordism formal group law. Victor Buchstaber proposed a conjecture about the explicit form of these coefficients. We want to check (or disprove) this conjecture numerically by writing a package for the computation of characteristic numbers of toric varieties, based on the differentiation of the volume polynomial.

## 24.07 Complexes of Tournaments in Directed Networks

**Speaker -** Ran Levi, professor at University of Aberdeen.

Clique graphs whose edges are oriented are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial sets, that we refer to as “tournaplexes", whose simplices are tournaments. In particular, given a directed graph G, we associate with it a “flag tournaplex" which is a tournaplex containing the directed flag complex of G, but also the geometric realisation of cliques that are not directed. We define several types of filtration on tournaplexes, and exploiting persistent homology, we observe that filtered flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of those ideas by applying them to graph data arising from the Blue Brain Project’s digital reconstruction of a rat’s neocortex.

## 3.07 Image processing and controlled linear algebra

**Speaker - **Ilyas Bayramov, Math Faculty HSE.

In 1996, Michael Freedman, famous for the results in the classification of 4-manifolds, decided to apply its methods in image processing. He and his co-author employed a curious method of altering the matrix of the district wavelet transform, instead of introducing the usual window, to reduce Gibbs (edge) effects. This method is based on a deep theory behind the classification result, specifically, an enhancement of the famous Kirby torus trick. I would like to explain these methods, and their extension by Freedman to quantum computing.

## 12.06 Persistent toric topology in the combinatorial graph theory

**Speaker-** Elizaveta Streltsova, MSU.

Methods of persistent topology appeared in the connection with the development of the machinery to analyze big data. They are directed towards the calculation of classical invariants of algebraic topology for a sequence of embedded simplicial complexes. In the first part of the talk, the speaker will present her results about invariants from toric topology, which arise in this field. The second part of the talk will be devoted to the construction of characteristics for certain series of graphs.

## 5.06 Collapsing and free deformation retraction

**Speaker - **Alexey Gorelov, Math.Faculty HSE.

A free deformation retraction is a strong deformation retraction which satisfies $f_t f_s = f_\max(t,s)$. Isbell in 1964 had shown that for 2-dimensional compact polyhedra the existence of free deformation retraction to a point is equivalent to collapsibility to a point. The construction of Berstein, Cohen, Connelly, 1978, shows that for polyhedra of dimension greater than 3 this is not the case: there exists noncollapsible polyhedra, which free deformation retract to a point. In the joint paper with Sergey Melikhov the speaker had proved that, if we require piecewise linearity from a free deformation retraction, then its existence is equivalent to collapsibility.

## 29.05 Disease progression models

**Conference ID:** 812 6896 9014 **Password: **brain

**Speaker - **Boris A. Gutman, PhD, Assistant Professor of Biomedical Engineering, Armour College of Engineering, Illinois Institute of Technology

As humans, we like to predict all manner of things, not least of them being our physical health. And while there is much excitement in the world of artificial intelligence about the ever-improving accuracy with which we can predict things, we are often less concerned with domain-specific relevance and utility of the prediction. Simple binary questions such as “does this patient have disease X?” or even “will the patient acquire disease X in Y years?” have proven less interesting to basic researchers and health professionals than “how quickly will the patient’s health deteriorate?”, “when and in what order will future symptoms appear?” and “what are the connections among the observable biomarkers and between biomarkers and symptom onset?”

Disease progression models (DPMs) attempt to answer the more interesting questions. In this talk, we will focus on applications of DPMs to brain imaging and neurodegenerative disease. I will go over some recent imaging-DPM developments from simple Bayesian models describing temporal biomarker order to differential equation models linking brain structure, connectivity and prior neurobiological knowledge to dynamically predict the course of an individual’s neurodegeneration.

## 15.05 Quillen-McCord theorem and its variations Link to Zoom

**Identifier: **883 7845 5118 **Password: **008694

**Speaker - **Vitaliy Guzeev, Math. Faculty HSE.

The particular case of Quillen’s Theorem A (also known as Quillen-McCord theorem) allows to prove homotopy equivalence of partially ordered sets under certain assumptions. Jonathan Barmak in 2000 gives the proof of this theorem based on an interesting object - the cylinder of a poset map.

In the talk we described this proof as well as the analogous proof of the homological version of Quillen’s theorem. The speaker will also tell about the approximate homological version of Quillen’s theorem - the hypothetical generalization of this theorem to persistent homology. He will also indicate a way to prove this result. The approach is based on ideas of Barmak and the technique of spectral sequences. Similar ideas can be found in the work of Govc and Skraba, An Approximate Nerve Theorem, 2017, but we supposed that some of these ideas can be naturally generalized.

Quillen’s theorem can be interesting from the applied viewpoint, since it can be used to reduce the dimension of data, saving their homotopy type. To find a persistent version of this theorem seems a natural problem.

## 08.05 High-dimensional expanders

**Identifier: **820 9694 0694 **Password: **019388

**Speaker - **Konstantin Golubev, postdoc at ETH Zürich.

Expander graphs and, in particular, Ramanujan graphs, possess many extremal combinatorial properties (expansion, mixing, chromatic number). These properties are intimately related to spectral and geometrical properties of graphs. Many results in the theory of expander graphs had found applications in other areas of mathematics and computer science. During last years, there is an active research in attempt to generalize expander graphs to higher dimensions: to hypergraphs and simplicial complexes. In his lecture Konstantin will tell about several results in this area concerning colorings and cohomological expansion, as well as their applications in extremal combinatorics and computer sciences.

## 24.04 Canonical forms = persistence diagrams

**Identifier: **831 7726 5045 **Password: **011699

**Speaker **- Sergei Barannikov, Scoltech, Paris Diderot University

A filtered complex over the field F can be reduced, by means of linear transformations preserving the filtration, to a so called canonical form, that is to the canonically defined direct sum of filtered complexes of two types: one-dimensional filtered complexes with the trivial differential: d(e_{t_i})=0 and two-dimensional filtered complexes with trivial homology d(e_{s_j})=e_{r_j}. In the talk we will discuss the proof of this theorem, which was first published in the work of the speaker “Framed Morse and its invariants“ Adv. in Sov. Math, 21:93-115 in 1994. In this work, the invariants called the canonical form of a filtered complex, were applied to Morse complexes, describing the sublevel homology of functions.

Starting from the middle of 2000’s, these invariants are widely used in applied mathematics under the name «persistence diagrams» or «persistence barcodes». The aforementioned result is called the Persistence homology Main (or Structure, or Principal) Theorem in applied science.

## 17.04 Factorization of toric maps and the problem of a common subdivision of a triangle

**Conference ID:** 856 7022 5298 ** Password:** 012143

**Speaker -** Alexander Perepechko, research fellow, MIPT

In 1978 Tadao Oda proposed the strong factorization conjecture of toric varieties:

Every toric (i.e. equivariant) birational map between two complete smooth toric varieties X and Y decomposes in a sequence of toric blow-ups and blow-downs.

Combinatorially, a complete toric variety is described by a complete fan of rational polyhedral cones, and its blow-up - by the subdivision of this fan. It is known that any toric birational map is described by a sequence of such subdivisions and their inverses, the is called the weak factorization. Oda conjecture tells that these operations can be ordered: first we perform all subdivisions, then - all inverse operations. In particular, for the fans of varieties X and Y there exists a common subdivision describing a map between those varieties.

In 3-dimensional case this conjecture reduces to the existence of a common subdivision of any pair of triangle’s subdivisions. In 2009, Silva and Karu (arXiv:0911.4693) proposed an algorithm which conjecturally always finds a common subdivision. We describe the structure of triangle’s subdivisions and understand this algorithm.

Practically, this algorithm can be simplified, and the problem of the least common subdivision is computationally hard. Theoretically, common subdivision can serve as a secret kew, recovered from a pair of given subdivisions. The speaker proposes the converse problem: to find an efficient algorithm which tells, for a randomly generated secret, the pair of subdivisions such that the secret is their minimal common subdivision. Such algorithm would give a one-direction function, which may be of use in cryptography.

## 10.04 The random simplicial complexes and toric topology

**Conference ID:** 757 830 213 **Password:** 023367

**Speaker** - Djordje Baralic, Mathematical Institute of the Serbian Academy of Sciences and Arts

In the talk we are going to introduce several models of random simplicial complexes and review some of the most known results in stochastic topology. We will emphasize a model of the random complexes inspired by Erdos-Renyi model of the random graph. In this model, the random d-complex Y_d(n,p) is defined as a simplicial complex on the vertex set [n] which contains the complete (d-1)-skeleton of the simplex on n vertices and each possible d-dimensional face appears independently with probability p.

Toric topology studies toric spaces from various point of view. One of the crucial constructions from toric topology is a functorial assignment a topological space with an action of torus to a simplicial complex, known as the polyhedral product functor. The results coming from toric topology open new way for studying various topological and combinatorial features of the random complexes in the context of stochastic topology. We are going to explain some of these ideas and some of recent results in this direction. Particularly, we are going to establish the law of large numbers for the bigraded Betti numbers of Y_d (n, p).

The talk is based on a joint work with Vlada Limic.

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