Совместный семинар с BIMSA

27 октября 2025
Anton Ayzenberg
Homological obstructions to the existence of matrix diagonalization algorithms

In applied linear algebra many problems reduce to finding eigenvalues of a real symmetric matrix A of size n. This can be done quite efficiently with asymptotical algorithms, such as QR-algorithm. Essentially, the algorithm produces a sequence A0, A1, A2, ... of isospectral matrices such that A0=A, and lim Ai is diagonal. In practice, the initial matrix A is usually sparse, meaning that the number of non-zero entries is much less than n^2. In computations with really large matrices, RAM happens to be a bottleneck, so the meaningful theoretical question: is there a diagonalization algorithm all of whose intermediate steps Ai belong to the same sparsity class as A?  The answer depends on the sparsity class. For Hessenberg matrices, i.e. matrices of staircase shape, QR-algorithm does the job. Our main result: non-Hessenberg sparsity classes (matrices that cannot be reduced to staircase shape by permutation of rows and columns) do not admit a diagonalization algorithm.  To prove this, we investigated topology of the manifold M(Γ, λ) of all sparse matrices with the specified spectrum λ, for each sparsity class Γ. Some particular cases are well-known: real full flag manifold (for full matrices), Tomei manifold (for tridiagonal matrices), twins of real Hessenberg varieties (for Hessenberg matrices). We made a systematic study of all sparsity classes and proved the alternative: (1) If Γ is of Hessenberg class then the total Betti number β(M(Γ,λ)) equals n!. (2) For any non-Hessenberg Γ we have β(M(Γ,λ)) > n!. In this case Morse theory implies our main result. This alternative is proved by combination of classical results in structural graph theory, our recent results in toric topology, and a number of computational experiments with cohomology of sheaves over finite posets.  In my talk, I will explain the theories and theorems behind the proof, omitting the mathematically loaded details. The talk is based on my works with V. Buchstaber, V. Cherepanov, M. Masuda, G. Solomadin, and K. Sorokin.

5 мая 2025
Михаил Тужилин
Properties of real networks and centrality measures

One of the most important questions in the network science is which characteristics differentiate artificial networks from real ones based on real experimental data. Centrality measures or shortly centralities play important role in this question. There are two main invariants that distinguish real networks from random ones: degree centrality and local clustering coefficient. For real networks, degree centrality obeys a power law, unlike the distribution of random networks (the so-called scale-free property). For small-world networks, the threshold of the average clustering coefficient and the average shortest path length differ from random ones (the small-world property).
There are many mathematical models that simulate these two properties. For example, the Watts-Strogatz network was the first mathematical network that satisfied the small-world property. However, this network is not scale-free. The Barabasi-Albert network is a scale-free network, but the average clustering coefficient is not large enough. These problems were solved in the network proposed by Boccaletti, Hwang, and Latora, which is scale-free and has a large average clustering coefficient.
In the first part of our talk, we will present theorems on the relationships between various centralities and other network characteristics. More precisely, we will show the relationships between stress, betweenness, radiality, and other small-world characteristics. We will present simple network properties in terms of local clustering centrality, where the average clustering coefficient is greater than the global clustering coefficient and vice versa. We will also show the case for a geodesic network where there exists a relationship between the average clustering coefficient and the average shortest path.
In the second part of our talk, we will present a new invariant for real networks, called ksi-centrality. We will show that this ksi-centrality not only distinguishes random networks from real ones, but also prove that it is related to the local clustering coefficient, the algebraic connectivity of the network, and the Cheeger constant. Moreover, Watts-Strogatz, Barabasi-Albert and Boccaletti, Hwang, and Latora networks are generally classified as random or artificial networks by this centrality, but there is a narrow set of parameters for which Watts-Strogatz and Barabasi-Albert networks have the same properties as real networks by this centrality. In this case, Watts-Strogatz and Barabasi-Albert networks have a more tree-like structure, like real networks.

14 апреля 2025
Fengling Li
Knot data analysis using multiscale Gauss link integral

In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. Although knot theory and related subjects are a focus of study in mathematics, their success in practical applications is quite limited due to the lack of localization and quantization. We address these challenges by introducing knot data analysis (KDA), a paradigm that incorporates curve segmentation and multiscale analysis into the Gauss link integral. The resulting multiscale Gauss link integral (mGLI) recovers the global topological properties of knots and links at an appropriate scale and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science. This is a joint work with Li Shen, Hongsong Feng, Fengchun Lei, Jie Wu and Guo-Wei Wei.


3 марта 2025
Rongling Wu
Statistics at a crossroads: How it can revolutionize artificial intelligence

Artificial intelligence (AI) is profoundly impacting science and society by applying algorithms and machine learning to enable machines to perform humanlike tasks. Statistics as a branch of mathematics, lying at the core of AI and data science, is facing an unprecedented challenge with the surge of complex, heterogenous data across a variety of platforms. In a real sense, statistics is at a crossroads to leverage its central role in revolutionizing the foundational and fundamental framework of AI. In this talk, I will present several state-of-the-art statistical methods that have been widely used in AI across various fields. I will focus on how to develop statistically principled reasoning and theory to validate the application of AI and enhance its interpretability and sustainability. Our approach builds on statistical mechanics theory and methodology derived from interdisciplinary integration.

20 января 2025
Антон Казаков
Inverse problems related to electrical networks and the geometry of non-negative Grassmannians

An electrical network is just a graph equipped with positive edge weights denoting conductivities, which nodes are divided on two sets: a set of inner nodes and a set of boundary nodes. Applying voltages  to its boundary nodes, we obtain the unique harmonic extension on all vertices voltages , which might be found out by the Ohm's and Kirchhoff's laws. Studying different properties of these harmonic extensions has given rise to many combinatorial objects: electrical response matrices, effective resistances and partition functions of spanning groves. All of them have appeared in many theories from the statistical physics (see, for instance,  Potts models and its relation to Abelian sandpile models ) to some areas of chemistry.
In the focus of my talk will be the theory of the planar circular electrical networks, which closely relates to the geometry of non-negative Grassmannians. We will present the explicit construction of the embedding of electrical networks to the non-negative part of Grassmannian  by their effective resistance matrices. Using it, we will provide the sketch of the cluster solution of the network topology reconstruction problem, which has the application in phylogenetic network theory.

23 декабря 2024
Василий Горбунов
Geometry of Electric Networks and data analysis in phylogenetic

A classic problem in data analysis is studying the systems of subsets defined by either a similarity or a dissimilarity function which is either observed directly or derived from a data set X.For an electrical network there are two functions defined by the resistance matrix and the response matrix either of which defines the network completely. We argue that these functions should be viewed as a similarity and a dissimilarity function moreover as such they are related via the covariance mapping also known as the Farris transform or the Gromov product. We will explore the properties of electrical networks from this point of view.
It is known that the resistance matrix defines a metric on the nodes of the electrical networks. Moreover for a circular planar electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. The main results in the talk will be a complete description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier.
One important area of applications where the Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalisation of phylogenetic trees. Our results allow to use the powerful methods of reconstruction of the minimal graphs of electrical networks in phylogenetics.
This is a joint work with Anton Kazakov.