Real Algebraic Geometry

Saturday, April 14, 2007 - 2:00pm - 3:30pm
EE/CS 3-180
Claus Scheiderer (Universität Konstanz)
While algebraic geometry is traditionally done over the complex
field, most problems from real life are modelled on real numbers and
ask for real, not for complex, solutions. Thus real algebraic
geometry --- the study of algebraic varieties defined over the real
numbers, and of their real points — is important. While most
standard techniques from general algebraic geometry remain important
in the real setting, there are some key concepts that are fundamental
to real algebraic geometry and have no counterpart in complex
algebraic geometry.

In its first part, this talk gives an informal introduction to a few
such key concepts, like real root counting, orderings of fields, or
semi-algebraic sets and Tarski-Seidenberg elimination. We also sketch
typical applications. In the second part, relations between
positivity of polynomials and sums of squares are discussed, as one
example for a currently active and expanding direction. Such
questions have already been among the historic roots of the field.
New techniques and ideas have much advanced the understanding in
recent years. Besides, these ideas are now successfully applied to
polynomial optimization.