Corso di Real Geometry

Docente L. Pernazza

Anno accademico 2022/2023


The course is devoted to introducing the main themes and tools of real algebraic and analytic geometry: Hilbert's 17th problem, Artin--Lang theory, semialgebraic and semianalytic sets, real algebra, relations between the algebraic and analytic frameworks. A good knowledge of basic topology and algebra (mainly ring theory) is is the only requirement.

Lezioni
Pavia, Dipartimento di Matematica
Aula: Sala Riunioni piano C (C12-C13)

Sarà (probabilmente) possibile seguire le lezioni anche via Zoom.

Registrazioni

Calendario (aggiornato al 27/3/2023)

Orario: martedì e giovedì 11-13 (se con * 10:30-13)

Diario delle lezioni


Argomenti che verranno (probabilmente) trattati

Real geometry: real algebraic varieties, semialgebraic sets.
Real algebra: ordered fields, real ideals, real radical of an ideal, real closure of a field.
Hilbert's 17th problem on sums of squares.
Real spectrum, fans, order spaces; real Nullstellensatz and Positivstellensatz.
Basic and principal semialgebraic sets, separation of semialgebraic sets and Br\"ocker's theorem.
Real analytic varieties, extension of orderings and Artin's approximation theorem.
Nash functions, Pfaffian functions, o-minimal structures and other real rings of functions.
Global real analytic geometry.

Libri consigliati
F. Acquistapace, F. Broglia, J.F. Fernando, Topics in Global Real Analytic Geometry, Springer Monographs in Mathematics, Springer, Cham, 2022
C. Andradas, L. Bröcker, J.M. Ruiz, Constructible Sets in Real Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 33, Springer, Berlin Heidelberg, 1996
C. Andradas, J.M. Ruiz, Algebraic and analytic geometry of fans, Memoirs of the AMS, 0553 (115), American Mathematical Society, Providence 1995
J. Bochnak, M. Coste, M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin 1998
M.A. Marshall, Spaces of Orderings and Abstract Real Spectra, Lecture Notes in Mathematics 1636, Springer, Berlin Heidelberg New York, 1996
A. Prestel, C.N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer Monographs in Mathematics, Springer, Berlin Heidelberg, 2001