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Projects / Programmes source: ARIS

Maps on algebras

Research activity

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 
Keywords
Geometry of matrices, linear preservers, non-linear preservers, symmetries on bounded observables, noncommutative polynomials, central simple algebras, functional identities, structure of groups
Evaluation (rules)
source: COBISS
Researchers (23)
no. Code Name and surname Research area Role Period No. of publicationsNo. of publications
1.  19551  PhD Dominik Benkovič  Natural sciences and mathematics  Researcher  2010 - 2013  206 
2.  08721  PhD Matej Brešar  Natural sciences and mathematics  Researcher  2010 - 2013  824 
3.  15127  PhD Jakob Cimprič  Natural sciences and mathematics  Researcher  2010 - 2013  85 
4.  20267  PhD Karin Cvetko Vah  Natural sciences and mathematics  Researcher  2010 - 2013  118 
5.  05478  PhD Mirko Dobovišek  Natural sciences and mathematics  Researcher  2012 - 2013  147 
6.  18750  PhD Gregor Dolinar  Natural sciences and mathematics  Researcher  2010 - 2011  211 
7.  16331  PhD David Dolžan  Natural sciences and mathematics  Researcher  2010 - 2013  129 
8.  19550  PhD Daniel Eremita  Natural sciences and mathematics  Researcher  2010 - 2013  129 
9.  29707  PhD Mateja Grašič  Natural sciences and mathematics  Researcher  2012 - 2013  39 
10.  29584  PhD Marko Kandić  Natural sciences and mathematics  Researcher  2012 - 2013  62 
11.  22353  PhD Igor Klep  Natural sciences and mathematics  Researcher  2010 - 2012  306 
12.  23467  PhD Marjetka Knez  Natural sciences and mathematics  Researcher  2012 - 2013  189 
13.  12190  PhD Damjana Kokol Bukovšek  Natural sciences and mathematics  Researcher  2012 - 2013  148 
14.  07082  PhD Gorazd Lešnjak  Natural sciences and mathematics  Researcher  2010 - 2011  153 
15.  23340  PhD Janko Marovt  Natural sciences and mathematics  Researcher  2010 - 2013  249 
16.  20268  PhD Primož Moravec  Natural sciences and mathematics  Researcher  2010 - 2013  203 
17.  22723  PhD Polona Oblak  Natural sciences and mathematics  Researcher  2010 - 2013  129 
18.  24328  PhD Aljoša Peperko  Natural sciences and mathematics  Researcher  2012 - 2013  189 
19.  07680  PhD Tatjana Petek  Natural sciences and mathematics  Researcher  2010 - 2013  130 
20.  08728  PhD Pavle Saksida  Natural sciences and mathematics  Researcher  2012 - 2013  90 
21.  05953  PhD Peter Šemrl  Natural sciences and mathematics  Head  2010 - 2013  492 
22.  33617  PhD Špela Špenko  Natural sciences and mathematics  Junior researcher  2011 - 2013  36 
23.  19886  PhD Emil Žagar  Natural sciences and mathematics  Researcher  2012 - 2013  186 
Organisations (2)
no. Code Research organisation City Registration number No. of publicationsNo. of publications
1.  0101  Institute of Mathematics, Physics and Mechanics  Ljubljana  5055598000  19,615 
2.  2547  University of Maribor, Faculty of natural sciences and mathematics  Maribor  5089638051  17,650 
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
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