Results in Real Algebraic Geometry Concerning Semi-Algebraic Sets

Series: 
Research Horizons Seminar
Wednesday, March 12, 2014 - 12:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
School of math
We will discuss a few introductory results in real algebraic geometry concerning semi-algebraic sets. A semi-algebraic subset of R^k is the set of solutions of a boolean combination of finitely many real polynomial equalities and inequalities. These sets arise naturally in many areas of mathematics as well as other scientific disciplines, such as discrete and computational geometry or the configuration spaces in robotic motion planning. After providing some basic definitions and examples, we will outline the proof of a fundamental result, the Oleinik-Petrovsky-Thom-Milnor bound of d(2d-1)^{k-1} on the sum of the Betti numbers of a real algebraic variety, as well as indicate the direction of recent and ongoing research generalizing this result.