Speaker

:

Dr. Abhinav Kumar

Affiliation

:

Microsoft research

Subject Area

:

Mathematics

Venue

:

Lecture Hall - III, Dept of Mathematics

Time

:

4.00 pm

Date  

:

December 19, 2006 (Tuesday)

Title

:

Universally optimal distribution of points on spheres

Among the various characterizations of ``dense" sphere packings in Euclidean space,

 or ``dense" spherical codes, we may consider configurations of points which minimize 

potential energy for a suitable repulsive potential. For instance, we can try to minimize the
function $f(C,k) = \sum\limits_{x\neq y \in C} d(x,y)^{-k}$, for $C \subset S^{n-1} \subset 

\mathbb{R}^n$ a spherical code of fixed size $N$ (here $d(x,y)$ is the usual Euclidean 

distance between $x$ and $y$). We find a large class of spherical codes which are optimal for
every positive value of $k$ (and indeed, for a larger class of potentials), and show their 

uniqueness in many cases. Our techniques involve linear programming bounds and build on  

those of Kolushov and Yudin and of Andreev. We also conjecture that $A_2, E_8$ and the
Leech lattice are optimal for a large class of potentials, among all periodic configurations 

of the same density. This is joint work with Henry Cohn.