Division of Physical and Mathematical Sciences
Speaker: Prof. Basudeb Datta, Professor, Department of Mathematics
Title: Minimal triangulations of manifolds
Date & Time: 28th April, 2022 at 4.00 pm (on MS Teams)
Prof. Govindan Rangarajan, Director, IISC will preside
Please use the following link to join the meeting:
Abstract: A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, a topological space M is an n-dimensional manifold if each point of M has a neighborhood homeomorphic to an open subset of the n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, the torus, and also the Klein bottle. At any fixed moment, our universe is a 3-dimensional manifold. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept also has applications in computer graphics given the need to associate pictures with coordinates.
We know that surfaces can be expressed as union of triangular regions. A collection X of simplices is called a simplicial complex if (i) a sub-simplex of a simplex in X is also in X, and (ii) the intersection of any to members in X is also in X. Simplices of dimensions 0, 1, 2, and 3 in a simplicial complex are called the vertices, edges, triangles, and tetrahedra, respectively. If the union of all the simplices of a simplicial complex X is homeomorphic to a manifold M, then we say that X is a triangulation of the manifold M. Now, it is known that there are manifolds which do not have triangulations. Sometimes, it is useful and convenient to have a vertex-minimal triangulation of a given manifold. In this direction, it is natural to ask for the following: (i) Finding the minimal number of vertices required to triangulate a given manifold; (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional manifolds; (iii) Classifications of all the triangulations of a given manifold with same number of vertices. In this talk, I would like to discuss these problems and my works related to these problems.
About the Speaker: Professor Basudeb Datta obtained his MSc degree in Pure Mathematics from Calcutta University in 1982 and PhD in Mathematics from Indian Statistical Institute, Kolkata in 1988. He joined IISc as an Assistant Professor in 1992. He held visiting positions at TIFR, Mumbai; ICTP, Trieste, Italy; ISI, Bangalore; Ecole Polytechnique, Paris, France; Universite de Lille, France; Ruhr-Universitat Bochum, Germany; Universitat Leipzig, Germany; University of Queensland, Brisbane, Australia; MSRI, Berkeley, USA, over the course of his career. Prof. Datta received Sir C. V. Raman Young Scientist Award in 2001 from the Government of Karnataka. He is a fellow of the Indian Academy of Sciences and the National Academy of Sciences, India. His research interests lie broadly in Combinatorial Topology, Piecewise Linear Topology, and Convex & Discrete Geometry. Last few years he is working on Tight & stacked triangulations of manifolds, and Semi-regular tiling of surfaces.
ALL ARE WELCOME