Valid Until: 2025-01-15 23:59
Abstract: A theorem by Brehm and Kühnel (1987) says that a d-dimensional combinatorial manifold K (without boundary) with n vertices is PL homeomorphic to the sphere, provided that n is less than 3d/2+3. Moreover, if n is equal to 3d/2+3, then K is PL homeomorphic to either the sphere or a manifold like a projective plane, which exist in dimensions 2, 4, 8, and 16 only. There exists a 6-vertex triangulation of the real projective plane (the quotient of the boundary of regular icosahedron by the antipodal involution), a 9-vertex triangulation of the complex projective plane (Kühnel, 1983) and 15-vertex triangulations of the quaternionic projective plane (Brehm and Kühnel, 1992). Recently the speaker has constructed first examples of 27-vertex triangulations of manifolds like the octonionic projective plane and a lot of new 15-vertex triangulations of the quaternionic projective plane. I will speak about these results and also about symmetry groups of these traingulations.
Bio: Alexander Gaifullin is the Correspondent Member of the Russian Academy of Sciences, a Principal Researcher of Steklov Mathematical Institute of RAS, a Professor of Lomonosov Moscow State University (Faculty of Mechanics and Mathematics), and a Professor of Skolkovo Institute of Science and Technology. He was an invited speaker at the European Congress of Mathematics in 2012 and a plenary speaker at the European Congress of Mathematics in 2016. Alexander Gaifullin graduated from Moscow State University in 2005, defended his PhD thesis in 2008 (supervisor: Prof. Buchstaber), and doctoral thesis in 2010.