• Nilpotent orbits and commuting matrices
The Client : Javna agencija za raziskovalno dejavnost RS
Project type: Bilateral projects
Project duration: 2019 - 2020
  • Description

Overview The concepts of matrix theory and linear algebra are crucial for understanding the theory behind data science, machine learning (especially deep learning), balancing chemical equations, design engineering, etc. Big data analysis has become the crucial part of todays industry and the applications of data analysis methods range from applications in environmental engineering and food production (e.g. they are used to figure out what type of manure is most suitable for a given type of land) to applications in space industry and financial markets (e.g. predicting price movement of a given stock or a portfolio). Amongst the many techniques developed for data analysis, machine learning is the most widely used. Methods of machine learning developed are at the crossroads of numerous areas of mathematics and computer science. At the heart of almost all methods lies linear algebra which is applied throughout science and engineering because it allows us to model natural phenomena and to compute them efficiently. The matrix multiplication is not a commutative operation and the study of pairs of commuting matrices has a long history. Pairs of commuting matrices are not only studied in different branches of mathematics (e.g. in linear algebra, the theory of Lie algebras, differential and partial differential equations, combinatorics, etc.), but also have many applications in computer science, physics, engineering, etc. Moreover, there is a natural action of a semisimple Lie algebra on a set of its nilpotent elements by conjugation. The orbits under this conjugation are called nilpotent orbits. Besides their intrinsic mathematical importance in geometry and representation theory, nilpotent orbits also have a significant bearing on theoretical physics, in particular, on the problem of studying asymptotically flat black hole solutions to extended supergravities. In that context are of particular relevance real semisimple Lie algebras which admit a Z_2-grading. Aims and scopes The description of the structure of the variety of commuting matrices turned out to be a difficult problem and is still very incomplete, despite its long history and an active recent research. The structure of the varieties of commuting pairs of matrices and of commuting pairs of nilpotent matrices is not yet well understood. It was proved by Motzkin and Taussky that the variety of pairs of commuting matrices was irreducible. Guralnick showed that this was no longer the case for the variety of triples of commuting matrices. Moreover, the variety of commuting pairs of nilpotent matrices is irreducible as well. Our motivation to study the problem is to contribute to better understanding of the structure of this variety and which might also help in understanding the irreducibility of the variety of triples of commuting matrices. The object of the study in this project is to understand the intersection of pairs of nilpotent orbits with the set of pairs of commuting matrices. The answer to the question could be considered as a generalization of the GerstenhaberHesselink Theorem on the partial order of nilpotent orbits. Recent developments in the areas of data analysis opened up completely new perspectives. The inspiration for the current research comes from the connections of the computational methods developed within the frame of representation theory to the machine learning algorithms. These connections include new algorithms developed to study important invariants, such as the commutativity and factorization of matrices. These invariants are very important in the related fields of data analysis and data fusion. In our research we will focus on the construction of new algorithms for computation of the centralizers of a given class of matrices over various rings. Since these matrices are not defined over a field, but rather over rings of polynomials, it is necessary to develop different approaches from the classical ones when dealing with this kind of mathematical objects. The starting point for our research will be the association of graphs to these sets of matrices and translate the problems involving matrices into the problems involving these discrete structures. Our approach to the above families of matrices involves combinatorial, homological, geometrical representation theoretical and computational methods. These methods represent a combination of classical methods, and new methods originating in machine learning. Justification of such a choice of methods lies in the fact that this is one of the ground-breaking approaches that has the potential to produce very important results in a short time span. The researchers from both countries have experiences in working on similar problems from the past, but each team considered these problems from a different perspective. The researchers from Slovenia have a lot of experiences with problems about commuting matrices and nilpotent orbits, as well as with creating optimal algorithms for computation with various classes of matrices, while researchers from Bosnia and Herzegovina have a very deep understanding of problems about representations of algebraic groups and graded algebras. Therefore we believe that the joint cooperation of both teams will enable us to solve some deeper problems that require expertise from more fields of mathematics.