Maria Infusino - Teaching

Summer semester 2015

Master Fachseminar Real Algebraic Geometry
with Prof. Dr. Salma Kuhlmann and Dr. Charu Goel

2 hours, Thursday: 11.45 – 13.15, Room F426

The aim of this cycle of seminars is to develop in more details some topics that were introduced in the course "Real Algebraic Geometry I" from the WS 2014-15 and that could be a starting point for a master thesis in the research stream investigated in the Schwerpunkt Reelle Geometrie und Algebra. The talks could focus for instance on the cylindrical decomposition of semialgebraic sets, the Hilbert 1888 theorem for ternary quartics, the representation of positive polynomials as sum of squares (Positivstellensätze), some aspects of the moment problem (e.g. the proof of Haviland's theorem, the original operator theoretical proof of the Schmüdgen theorem of 1991). Each talk will provide a review of the results already known in literature about one of these topics and point out some related open questions.

Target group
The seminar is primarily aimed at master students, but is also suitable for students of Diplom and Lehramt.

References

J. Bochnak, M. Coste, M. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer Vol. 36, Berlin,1998
S. Kuhlmann, Notes on Real Algebraic Geometry (WS 2009-10) available here.
S. Kuhlmann, Notes on Positive Polynomials (SS 2010) available here.
M. Marshall, Positive Polynomials and Sum of Squares, Vol. 146, Mathematical Surveys and Monographs, AMS, 2008
R.K. Rajwade, Squares, LMS Lecture Notes Series 171, Cambridge University Press, 1993.

Preliminary meeting
A preliminary discussion (Vorbesprechung) for the assignment of the topics (Themenvergabe) will take place on Thursday 16th of April from 11.45 to 13.15 in Room F426.
For further information, see also the announcement of this cycle of seminars and the list of possible topics.

Schedule

April, 23 : Saturation of preorderings in $\mathbb{R}[\underline{X}]$, (Natalia Walder). Abstract
May, 7: Positive polynomials and sum of squares in formal power series rings, (Siqi Yao). Abstract
May, 21: Theorems of Pólya and Reznick, (Patrick Michalski). Abstract
May, 28: Proof of Hilbert’s 1888 Theorem for ternary quartics, (Charu Goel). Abstract
June, 11: Topological definition of dimension of semi-algebraic sets, (Patrick Michalski). Abstract
June, 18 : The C^p-m-reparameterization theorem for O-minimal expansions on $\bar{\mathbb{R}}$, (Simon Müller). Abstract
June, 25: The necessity part of Hilbert's 1888 Theorem: an explicit example of psd not sos ternary sextic, (Andrea Winkler). Abstract CANCELLED!
July, 2: Moment Problem: Haviland’s Theorem, (Maria Infusino). Abstract
July, 16: "Moment Problem: Schmüdgen Theorem (1991)", (Maria López Quijorna). Abstract

Weekly deadlines

Monday: submission of the final title and abstract by noon (to maria.infusino@uni-konstanz.de)
Thursday: seminar handout at the beginning of the talk (max 10 pages, about 10 copies)

IMPORTANT DEADLINE: One week after the day of your talk, you should submit the final version of the report (to maria.infusino@uni-konstanz.de)

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