Real Algebraic Geometry

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Next lectures
13.12.2024, 15:00 (S214)
20.12.2024, 15:00 (S214)
10.01.2025, 15:00 (S214)
17.01.2025, 15:00 (S214)
24.01.2025, 15:00 (S214)
31.01.2025, 15:00 (S214)
07.03.2025, 15:00 (S214)

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The course aims to introduce the study of real algebraic sets and semialgebraic sets from a topological and differential point of view. After reviewing the basic techniques in real algebra, we discuss the Tarski-Seidenberg theorem, showing that the projection of a semialgebraic set is semialgebraic. We then prove a semialgebraic version of Sard's theorem of critical values, and Hardt's semialgebraic triviality.

We apply the previous results to the study of real algebraic sets, bounding the number of their connected components (Harnack's theorem for curves), and more generally their Betti numbers (Thom-Milnor bound). We conclude by discussing the existence of algebraic models for compact real manifolds, i.e. the Nash-Tognoli theorem.

The text that will be used as a reference, and a more detailed program of the course, can be found at lorenzobaldi.github.io/teaching/

Keywords
Real Algebraic Geometry; Semialgebraic Geometry; Differential and Topological properties of Real Algebraic Sets; Algebraic Models of Manifolds

Prerequisites
Basic knowledge in topology and differential geoemtry. Beckground in basic algebraic geometry and commutative algebra is helpful but not necessary.

Language
English

Remarks and notes
During the course, exercise sheets on related topics not discussed in the lectures, or detailing proofs not completely detailed, will be provided.