A classic result in real algebraic geometry, due to Oleinik-Petrovsky,
Thom, and Milnor, bounds the topological complexity
(the sum of the Betti numbers) of basic
semi-algebraic sets. These bounds are used in many applications -- for
instance, for proving tight lower bounds on the complexity of several
problems in computer science. Over the past few years there have been
several improvements and generalizations of this bound due to Basu,
Basu-Pollack-Roy, and Pollack-Roy. In particular, it is now possible
to
prove better bounds on the individual Betti numbers of semi-algebraic
sets
rather than their sum. The talk will be a survey of these results.
R E F R E S H M E N T S
4:00 To 4:25 PM
Kenneth W. Anderson
Memorial Reading Room