Welcome to the Real Algebraic Geometry Reading Seminars taking place on Thursdays 13:00-14:00 CET on Zoom. 

Registration:

The relevant materials will be shared with all the attendees in advance. We will email you the link/password on the day of the meeting.

Overview:

The main aim of this reading group is to get familiar with the basics of Real Algebraic Geometry. Our main motivation is to cover the theory up to the level so that we are equipped to understand its usage in algorithms and applied problems. Our plan is to cover major parts from the lecture notes by Markus Schweighofer and Tim Netzer. 

Tentative Schedule:

• March 03, 2022: An invitation to real algebraic geometry - Notes

• March 10, 2022: Oliver Clarke -- Ordering on Fields - Notes

• March 17, 2022: Oliver Clarke -- Real closed fields and semi-algebraic sets - Notes

• March 24, 2022: Sebastian Debus -- Projection theorem and sums of squares - Notes

• March 31, 2022: Francesca Zaffalon -- Hilbert's 17th Problem and Artin's Solution - Notes

• April 07, 2022: Oskar Henriksson -- Real Spectrum and prime cones - Notes

• April 14, 2022: Ayush Kumar Tewari -- Real radical ideals and constructible sets - Notes

• April 28, 2022: Ahmed Umer Ashraf  -- Real Stellensätze - Notes

• May 5, 2022: Job D. Rock -- Schmüdgen’s Positivstellensatz

• May 12, 2022: Sebastian Debus - "On invariant non negatives and sums of squares" - Slides

Abstract: In this talk we consider invariant non-negative polynomials and sums of squares. Our focus lies on symmetric and even symmetric polynomials. We present how one can complexity reduced represent the invariant sums of squares under the action of a reflection group. Then, we consider even symmetric and symmetric forms. We observe that we can identify invariant polynomials in a sufficiently large number of variables. Thus, we can consider their limit. We present test sets for the verification of non-negativity of certain even symmetric forms and answer the limit question of non-negativity versus sums of squares. The first part of the talk is based on joint work with Cordian Riener and the second part on joint work with Jose Acevedo, Greg Blekherman and Cordian Riener.  

Lecture Notes:

• Real Algebraic Geometry, Positivity and Convexity by Markus Schweighofer

• Real Algebra and Geometry by Tim Netzer

Books:

• Real Algebraic and Semi-algebraic Sets

• Algorithms in Real Algebraic Geometry

• Semidefinite Optimization and Convex Algebraic Geometry