Mini Symposium In Mathematics

Center of Excellence in Mathematics (CEMS)
Department of Mathematics, DSB Campus
Kumaun University, Nainital
29th April, 2013

Professor M S Raghunathan
(Head, National Center of Mathematics, IIT Bombay)
Title of Talk:
"Construction of certain compact hyperbolic 3-manifolds"
As is well known the group G of all holomorphic automorphisms of the unit (open) disc D2 leaves invariant thre Riemannian metric (dx2+dy2)/(1-r2)2 . This metric has constant Gaussian curvature. The group G is naturally isomorphic to PSL(2,R) (which is the same as the identity connected component SO_o(2,1) of the Lorentz group SO(2,1)). The uniformization theorem for Riemann surfaces tells us that every compact Riemann surface of genus > 2 has for its universal covering the unit disc so that every compact Riemann surface carries a metric of constant curvature. Algebraic geometry provides us a wealth of examples of Riemann surfaces of genus > g. It follows that there are lots of examples of discrete subgroups D of SO_o(2,1) such that SO_o(2,1)/D is compact. In dimension 3, we can consider the metric (dx2+dy2+dz2)/(1-r2)2 in the unit disc D3 in R3 ; this metric has constant sectional curvature. But the existence of compact Riemannian 3-manifolds is a somewhat delicate to prove. In this talk I will outline a method of construction of such manifolds via Number theory.
Professor S Kesavan
(IMSc, Chennai)
Title of Talk:
"Some problems in the calculus of variations based on elementary geometry"
Starting from a simple proof of the fact that the shortest distance between two points in the plane is the length of the line segment joining them, some problems in shape optimization will be discussed using very elementary geometrical ideas. This will be gradually developed to prove the two-dimensional classical isoperimetric inequality.
Professor Jugal Verma
(IIT Bombay)
Title of Talk:
"The Upper Bound Theorem for spheres"
Richard Stanley solved the Upper Bound Conjecture for triangulations of Spheres in 1975 using the theory of Hilbert functions and Cohen-Macaulay rings. We shall present an outline of his proof.
All are cordially invited.
Prof. H S Dhami, Pro Vice Chancellor
Coordinator, CEMS
Dr. Sanjay Kumar Pant
Asst. Coordinator, CEMS

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