Projects / Programmes
| Code |
Science |
Field |
Subfield |
| 1.01.04 |
Natural sciences and mathematics |
Mathematics |
Algebra |
| Code |
Science |
Field |
| P120 |
Natural sciences and mathematics |
Number theory, field theory, algebraic geometry, algebra, group theory |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
Geometry of matrices, linear preservers, non-linear preservers, symmetries on bounded observables, noncommutative polynomials, central simple algebras, functional identities, structure of groups
Researchers (23)
Organisations (2)
Abstract
We will continue with research in the areas that have already been treated by the research groups whose principal investigators were Peter Šemrl and Matej Brešar. In this project two younger but already well-established colleagues Igor Klep and Primož Moravec will join our research group. As a consequence we expect that, on one hand, we will start working in some new directions, and on the other hand, their broad knowledge of algebra will help us prove new results in areas that we have worked on over the last few years. We will be mainly interested in maps acting on various algebras and their subspaces (matrix and operator algebras and spaces, and more general algebras and spaces). These maps will have certain algebraic properties and/or certain preserving properties. The main objective is to describe the general form of such maps.
The central problems we will be concerned with are the following:
1. Improvements of Hua's fundamental theorems of geometry of matrices. We intend to obtain optimal versions of these classical results. In the hermitian case we would like to extend Hua's theorem to the infinite-dimensional case in order to apply it in studying symmetries on bounded observables.
2. Finite-dimensional nonassociative algebras will be treated. In particular, we will be
interested in those algebras in which every non-scalar element generates an algebra isomorphic to the algebra of complex numbers.
3. We will continue with the study of linear preservers of invertibility.
4. Recently, the first results on general (non-linear) preservers on matrix and operator algebras have been obtained. We hope to get some new results in this direction.
5. We shall continue to investigate functional identities (FI). Our primary focus here will be on identifying new areas od applications of FI's.
At the same time we will join forces with our new younger colleagues to study the following two topics in noncommutative algebra:
6. Free positivity, i.e., the study of images of polynomials in noncommuting variables under well-behaved representations. In contrast to classical representation theory, we are interested in the image of a fixed element of the free algebra (a noncommutative polynomial) under all representations in a suitably chosen class. Of particular importance are various notions of positivity (e.g. positive semidefiniteness, positivity of the trace, etc.). Our focus will be mainly on finite dimensional *-representations of the free *-algebra with a foray into finite von Neumann algebras as needed for Connes' embedding conjecture (see below for more details). This forces the theory into two branches. The dimensionfree (in the sense that the we are considering evaluations at tuples of matrices of all sizes) setting is developed much better due to the works of Helton with coauthors and also Klep (together with coworkers, including Brešar). This branch of the theory has a certain operator-algebraic flavor. On the other hand, a mixture of central simple algebras, quadratic forms and valuation theory is the dimension dependent branch of the theory, initiated by Procesi and Schacher studying the Albert-Weil notion of positive involutions and orderings on central simple algebras. How this relates to dimension dependent positivity in free algebras has been studied by Klep and Unger and will be pursued further in this proposed project.
7. We will develop a general theory of functional identities in groups. This will be used in studying some important classes of group functions, such as cocycles in the cohomological theory of groups, Lie homomorphisms, and functional identities arising in Freiman's additive set theory and Quillen's conjecture.
Significance for science
Our research group publishes in international research journals of high quality. The easiest way to prove the relevance of our research to the development of our scientific field is to check the total citations of the leading researcher Peter Šemrl: -normalized number of citations (!993-2014): 3808 -normalized h-index: 27 -total citations according to MathSciNet: 1923 Matej Brešar, another member of the research group, has been cited even more. Few other facts proving the relevance of our research: -the leading researcher was elected to be the president of the International Linear Algebra Society -the leading researcher is Editor-in-Chief of Linear Algebra and Its Applications, the leading journal in the area of linear algebra -the leading researcher is a member of two more editorial board of mathematical journals with impact factor -Matej Brešar is a member of editorial boards of two journals with impact factor
Significance for the country
Mathematics is the common language of science and technology. All highly developed countries have excellent schools of mathematics. It is hard to believe that some country can be economically successful without having well-developed all sciences and in particular, mathematics. Our research group is included in international exchange of knowledge (the leading researcher has around 40 coauthors from abroad). This is very important when supervising young researchers. It also adds to the international promotion of our country.
Most important scientific results
Annual report
2010,
2011,
2012,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2010,
2011,
2012,
final report,
complete report on dLib.si
