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Laboratory for Computer Geometry and Quantum Topology

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As of 15.02.2021

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General information

Name of the project: Quantum topology

Strategy for Scientific and Technological Development Priority Level: а

Goals and objectives

Research directions: Quantum and low-dimensional topology

Project objective: Developing topological quantum theory field theory as a universal language for describing quantum phenomena in mathematics as well as for formalizing and solving problems in related areas of studies

The practical value of the study

  • Dijkgraaf-Witten invariants of three-dimensional manifolds have been computed. 
  • We have computed numerical invariants for an infinite series of three-dimensional manifolds measuring complexity of these manifolds.
  • Our research have classified nodes in thickened torus and in thickened Klein bottle. These results completely prove that quantum topology methods can be efficiently used for solving problems of three-dimensional topology. Results of research are also necessary to create a flexible topological approach to solving problems of quantum physics, in particular, of string theory.

Implemented results of research:

  • The Laboratory has created and registered the «Atlas of three-dimensional manifolds» that is a database of three-dimensional manifolds and their characteristics (complexity, values of manifolds, geometry type, graphic structures, generally accepted name).
  • We have created and registered the «Single-vortex spines of three-dimensional manifolds» computer program for plotting two-dimensional polyhedron of a special type (single-vortex spies) determining 3-manifolds.
  • We have created and registered the «HOMFLY-calculator» computer program for computing values of the classical HOMFLY polynominal of rational nodes.

Education and career development:

  • We have created 14 new education programs: «Additional chapters of topology (part I)», «Classifications of 3-manifolds», «Classification of Seifert manifolds and computing their quantum invariants», «Low-dimensional topology», «Quantum topology», «Computational topology», «Graph theory», «Homologies theory», «Complexity theory of geometrical objects», «Node theory», «Algebraic topology», «Graph-based algorithms», «Quantum and polynomial node invariants», «Single-vertex theory».
  • 2 doctoral dissertations, 12 candidate dissertations have been defended.
  • We have conducted 8 scientific conference in our area of studies.
  • 7 members of our team have enrolled into doctoral or postgraduate school.

Organizational and structural changes:

In 2017 the Research Laboratory or Quantum Topology was merged with the Laboratory of Computer Geometry to form the Laboratory of Computer Geometry and Quantum Topology. The new laboratory conducts research in quantum topology and computer geometry with financial support from the Russian Science Foundation and the Russian Foundation for Basic Research.

Collaborations: Indiana University (USA), University of Bologna (Italy), Georgia Institute of Technology (USA): scientific research and publications

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Vesnin A.Yu., Turaev V.G., Fominykh E.A.
Complexity of virtual 3-manifolds. Sbornik: Mathematics. Vol. 207. № 11: 1493–1511 (2016).
Matveev S.V., Nabeeva L.R
Tabulating knots in the thickened Klein bottle. Siberian mathematical journal. Том 57. Вып. 3. C. 542–548 (2016).
Vesnin A.Yu., Turaev V.G., Fominykh E.A.
Three-dimensional manifolds with poor spines. Proceedings of the Steklov institute of mathematics. Том 288. Вып. 1. C. 29–38 (2015).
Matveev S.V., Turaev V.G.
Dijkgraaf-Witten Invariants over Z(2) for 3-Manifolds. Doklady mathematics. Том 91. Вып. 1. С. 9–11 (2015).
Simon K.; Matveev S., Tarkaev V., Turaev V.
Dijkgraaf-Witten Z(2)-invariants for Seifert manifolds. Journal of knot theory and its ramifications. Том 26. Вып. 1: 1750006 (2017).
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