# 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.

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.