Real algebraic geometry II, SS 2020
- Monday 11.45-13.15, Wednesday 11.45-13.15;
tutorial on Thursday 10.00-11.30.
Everything in remote teaching mode. For details see ILIAS.
- Start of class on Monday 20 April 2020
- Tutorials are held by Thorsten Mayer, start on 30 April 2020
- 9 ECTS credits
- Verwendbarkeit: Hauptmodul (Master Mathematik), Wahlmodul (Master
Mathematik, Master Lehramt Gymnasium)
- Die Modulprüfungen werden am Semesterende mündlich abgenommen
About this course:
We first study archimedean quadratic modules or preorderings and
prove the Positivstellensatz. It has plenty of applications to
sum-of-squares type representations of positive polynomials, and
we'll discuss some of those. After a few complements in commutative
algebra and algebraic geometry (power series rings, regular and
singular points of varieties) we next treat the archimedean
local-global principle and give applications to Nichtnegativstellensätze.
Further topics will include moment problem, stability questions and
applications of sums of squares to optimization, e.g. Lasserre
relaxation.
The course is a continuation of Real algebraic geometry I (WS 2019/20),
and requires familiarity with part I (ordered and real closed fields,
real spectrum, sums of squares, semialgebraic geometry)
Future outlook:
For WS 2020/21 I am planning a 2+1 course (specialization
module) building upon this one, probably on hyperbolic forms.
For SS 2021 I am planning a seminar for Master students
that will build upon RAG I and II.
Problem sheets
Recommended reading:
- M. Marshall:
Positive Polynomials and Sums of Squares.
Mathematical Surveys and Monographs 146, American Mathematical Society,
Providence, RI, 2008.
