|Taras Panov||Professor, Moscow State University, Russia|
Bio: Higher geometry and topology chair, Professor. Research interests: Algebraic and differential topology, cobordism theories, toric topology. Honors: I. I. Shuvalov Prize, 1st degree, Moscow State University (2013), Moscow Mathematical Society award (2004).
Right-angled polytopes, hyperbolic manifolds and torus actions.
A combinatorial 3-dimensional polytope P can be realized in Lobachevsky 3-space with right dihedral angles if and only if it is simple, flag and does not have 4-belts of facets. This criterion was proved in the works of A.Pogorelov and E.Andreev of the 1960s. We refer to combinatorial 3 polytopes admitting a right-angled realisation in Lobachevsky 3-space as Pogorelov polytopes. The Pogorelov class contains all fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. There are two families of smooth manifolds associated with Pogorelov polytopes. The first family consists of 3-dimensional small covers (in the sense of M.Davis and T.Januszkiewicz) of Pogorelov polytopes P, also known as hyperbolic3-manifolds of Loebell type. These are aspherical 3-manifolds whose fundamental groups are certain extensions of abelian 2-groups by hyperbolic right-angled reflection groups in the facets of P. The second family consists of 6-dimensional quasi toric manifolds over Pogorelov polytopes. These are simply connected 6-manifolds with a 3-dimensional torus action and orbit space P. Our main result is that both families are cohomologically rigid, i.e. two manifolds M and M' from either family are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that a cohomology ring isomorphism implies an equivalence of characteristic pairs; in particular, the corresponding polytopes P and P' are combinatorially equivalent. This leads to a positive solution of a problem of A.Vesnin (1991) on hyperbolic Loebell manifolds, and implies their full classification. Our results are intertwined with classical subjects of geometry and topology such as combinatorics of 3-polytopes, the Four Colour Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology.
This is a joint work with V. Buchstaber, N. Erokhovets, M. Masuda and S.Park.
|Yi Liu||Professor, Beijing International Center for Mathematical Research|
Bio: Yi Liu is a professor at Beijing International Center for Mathematical Research (BICMR) in Peking University. His research interest lies primarily in 3-manifold topology and hyperbolic geometry. He received his Ph.D. degree in 2012 in University of California at Berkeley. In 2017, he received the Qiushi Outstanding Young Scholar Award. He has been a principal investigator of the NSFC Outstanding Young Scholar since 2019. Below are some selected research works of Yi Liu: (1) proving J. Simon’s conjecture about knot groups (joint with I. Agol, 2012); (2) resolving fundamental properties of the L2 Alexander torsion for 3-manifolds, (2017); (3) proving C. T. McMullen’s conjecture about virtual homological spectral radii of surface automorphisms (2020).
Finite covers of 3-manifolds.
In this talk, I will discuss some developments in 3-manifold topology of this century regarding finite covering spaces. These developments led to the resolution of Thurston’s virtual Haken conjecture and other related conjectures around 2012. Since then, people have been seeking for new applications of those techniques and their combination with other branches of mathematics.
The meeting will be held in the form of a webinar on the Zoom platform.
Pre-registration for the event is not required.
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Meeting ID : 626 6992 6224
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