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Entry requirements:
Nikolay Bogachev
PhD
Supervisor’s research interests:
I study geometric actions of groups on Riemannian manifolds and the corresponding quotient manifolds and orbifolds. Sometimes, if a group H acts on a metric space X properly discontinuously, then the quotient X/H is an orbifold or manifold with some nice geometric and combinatorial properties. Various examples of such actions are provided by the theory of hyperbolic reflection groups developed by Vinberg in 1967. A natural fundamental domain of a discrete group generated by reflections with respect to hyperplanes is a Coxeter polytope, which can be described in the sense of Coxeter diagrams/graphs. The modern research of discrete groups combines algebraic, geometric, topological, combinatorial, dynamical, and number theoretical approaches. Sometimes, computer experiments are very helpful.
Research highlights:
Supervisor’s specific requirements:
Main publications:
- Master’s degree / equivalent in a related field
- B2 level of English
- Good track record of publications related to the topic of the intended research
- Strong research proposal 1,500 - 3,500 words
Nikolay Bogachev
PhD
Supervisor’s research interests:
I study geometric actions of groups on Riemannian manifolds and the corresponding quotient manifolds and orbifolds. Sometimes, if a group H acts on a metric space X properly discontinuously, then the quotient X/H is an orbifold or manifold with some nice geometric and combinatorial properties. Various examples of such actions are provided by the theory of hyperbolic reflection groups developed by Vinberg in 1967. A natural fundamental domain of a discrete group generated by reflections with respect to hyperplanes is a Coxeter polytope, which can be described in the sense of Coxeter diagrams/graphs. The modern research of discrete groups combines algebraic, geometric, topological, combinatorial, dynamical, and number theoretical approaches. Sometimes, computer experiments are very helpful.
Research highlights:
- In the framework of my research I collaborate with mathematicians from Switzerland, USA, Italy, Brazil, Germany, France, and Russia.
- My work was awarded by the Simons Foundation Prize for PhD students (2017, 2018), and also supported by grants of RSF, RFBR, Basis.
Supervisor’s specific requirements:
- Algebra: groups, rings, modules, number fields.
- Linear algebra: vector spaces, linear maps and operators, bilinear and quadratic forms.
- Topology: topology of R^n, topological spaces and manifolds.
- Geometry: convex polyhedra, smooth manifolds, Riemannian manifolds.
- Python knowledge would be a benefit.
Main publications:
- N. Bogachev, From geometry to arithmeticity of compact hyperbolic Coxeter polytopes, 2020, arXiv:2003.11944.
- N. Bogachev, A. Kolpakov, On faces of quasiarithmetic Coxeter polytopes, 2020, arXiv:2002.11445, to appear in Int.Math.Res.Notices.
- N. Bogachev, Classification of (1,2)-reflective anisotropic hyperbolic lattices of rank 4, Izvestiya Math, 2019, vol. 83:1, pp. 1-19.
- N. Bogachev, A. Perepechko, Vinberg’s algorithm for integral hyperbolic lattices, Math. Notes, 2019.
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