Специални хомогенни пространства в геометрията, теорията на представянията, физиката и теорията на кодирането
Проект по договор КП-06-Н92/2
Базова организация:
Институт по Математика и Информатика при Българска Академия на Науките.
Източник на финансиране:
Фонд „Научни изследвания”, конкурс за финансиране на фундаментални научни изследвания – 2025 г.
Срок на изпълнение:
декември 2025 – декември 2028 г.
Special homogeneous spaces in geometry, representation theory, physics, and coding theory
Project under contract KP-06-Н92/2
Base organization:
Institute for Mathematics and Informatics, Bulgarian Academy of Sciences
Funding:
The Bulgarian National Science Fund, competition for funding of fundamental scientific research – 2025
Period:
December 2025 – December 2028
Колектив
- Доц. д-р Валдемар Василев Цанов – ИМИ-БАН, ръководител
- Проф. д-р Величка Василева Милушева – ИМИ-БАН
- Доц. д-р Михаил Школников – ИМИ-БАН
- Доц. д-р Хамед Пежан – ИМИ-БАН
- Гл.-ас. д-р Елица Иванова Христова – ИМИ-БАН
- Гл.-ас. д-р Мариам Бажалан – ИМИ-БАН
- Гл.-ас. д-р Виктория Герасимова Бенчева-Петрова – ИМИ-БАН и ВТУ „Св. св. Кирил и Методий“
Team
- Valdemar Vasilev Tsanov, PhD, assoc. prof. at IMI-BAS, project leader
- Velichka Vasileva Milousheva, PhD, prof. at IMI-BAS
- Mikhail Shkolnikov, PhD, assoc. prof. at IMI-BAS
- Hamed Pejhan, PhD, assoc. prof. at IMI-BAS
- Elitza Ivanova Hristova, PhD, chief assis. prof. at IMI-BAS
- Maryam Bajalan, PhD, chief assis. prof. at IMI-BAS
- Victoria Gerasimova Bencheva-Petrova, PhD, chief assis. prof. at IMI-BAS and VTU “St. St. Cyril and Methodius”
Research objectives
We plan research in several interconnected areas of mathematics and physics. The motivation for a common project is found in several geometric and algebraic constrictions, notably involving special transformation groups, which appear as models for key concepts in more than one context, offering opportunities for exchange of ideas and collaboration. The main directions of research are the following.
Flag varieties and Geometric Invariant Theory (GIT):
We aim at systematic descriptions of massless quantum fields in de Sitter space-time as representations of the conformal group U(2,2) and their restrictions to the subgroup SO(4,1). The key novelty lies in employing the conformal Clifford algebra cl(4,2) in this context. We intend to develop new tools relevant to higher-spin theory, holography, and quantum gravity models in positively curved spacetimes.
Quantum systems in de Sitter space-time:
The Hilbert-Mumford framework of GIT applied for reductive group actions on flag varieties allows to relate several representation theoretic problems, e.g. description of invariant rings, to problems about the geometry of projective orbits and momentum maps. We study the variations of momentum images of flag varieties and incorporate the use of compatible real forms and Heinzner-Schwarz stratifications, aiming at new structural descriptions of momentum images, Kempf-Ness sets, Littlewood-Richardson cones.
Tropical geometry
We study tropicalizations of special subvarieties (linear spaces, algebraic curves, subgroups, etc) of complex reductive Lie groups, extending initial work on SL(2,C). Key challenges concern the concepts of Ronkin function, Viro’s patchworking, and the descriptions of spherical and hyperbolic coamoebas of special surfaces, e.g. minimal or with constant mean curvature.
Representations of classical groups on associative algebras:
We study representations of GL(n) on special algebras with polynomial identities, with special attention to long commutator identities, in the framework of the non-commutative invariant theory of Domokos and Drensky. We aim to develop new constructions of irreducible GL(n)-submodules suited for computing structural entities such as the sequence of codimensions.
Coding theory:
We study skew polycyclic codes over Ore rings, aiming at new results on the classification, duality, Hamming isometric equivalence, minimum distances for such codes. The guiding idea is to characterize these codes as invariant subspaces under suitable groups of linear transformations.
