Denis Osin
Vanderbilt University, TN·Mathematics & Statistics
A group, in mathematics, is a set of symmetries — ways of transforming something that can be undone and combined. Geometric group theory studies groups by picturing them as collections of transformations of a space with a notion of distance.
- Fields
Mathematics & Statistics
- Topics
geometric group theory
hyperbolic groups
quasi-isometric rigidity
- Grants
- 1 known grant, through May 2027
What they work on
- Study rigidity properties of acylindrically hyperbolic groups by addressing auxiliary questions and conjectures connecting analytic and geometric properties, motivated by open problems on quasi-isometric and measure equivalence rigidity.
- Generalize previous results for relatively hyperbolic groups to groups acting cocompactly on simply connected hyperbolic simplicial complexes, including mapping class groups, many Artin groups and small cancellation quotients of graphs of groups.
- Continue joint work on automorphisms of von Neumann algebras and reduced C*-algebras associated with countable groups.
- Organize week-long conferences joining experts and young researchers across group theory, functional analysis, geometry and dynamical systems, with accompanying activities for undergraduate and graduate students.
More about the research
This turns out to be especially powerful when the space is negatively curved, or hyperbolic, which roughly means that triangles in it are very thin and paths spread apart quickly. Such geometry forces strong constraints on the group, and this project proposes to push those constraints further and to connect them to operator algebras, which are algebras of operators attached to a group and used to encode its symmetries analytically. Making progress draws on group theory, functional analysis, geometry and dynamical systems together.
Why it matters
This is foundational mathematics: it sharpens what geometric information about a group of symmetries determines its algebraic and analytic structure, linking geometry, group theory and operator algebras. Results could inform open rigidity problems that organize much of the field, and the planned conferences and student activities build expertise across these connected areas.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Srivatsav Kunnawalkam Elayavalli
University of Maryland, MD·Mathematics & Statistics
Apply Voiculescu's free entropy theory, a measure of size for noncommuting variables, to determine the structure of free products and ultrapowers of finite von Neumann algebras.
Tao Mei
Texas A&M University-College Station, TX·Mathematics & Statistics
Determine the Lp-approximation properties of group von Neumann algebras, which are operator algebras built from discrete groups, that is, how well these infinite-dimensional objects can be approximated by simpler pieces.
Efren Ruiz
University of Hawaii at Manoa, HI·Mathematics & Statistics
Develop new invariants for subshifts of finite type, symbol-sequence dynamical systems, by studying the graph algebras attached to them and how these algebras interact.
The school
Vanderbilt University: 22 labs on PI Pages, $1.3B a year on research, 494 research doctorates a year.
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