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Semi-Algebraic Geometry

This course is supposed to provide understanding of basic concepts of semi-algebraic geometry, at a level suitable for master's students. It will be given by Maria Michalska, from the University of Lodz, visiting Aarhus under the Erasmus programme.

It will consist of four parts:

Preliminaries

  1. Definition of semi-algebraic sets
  2. The Tarski-Seidenberg theorem and quantifier elimination
  3. Geometric interpretation: stability of the class of semi-algebraic sets under projection and basic topological operations
  4. (if time allows) Tarski's Transfer Principle and real closed fields

Classical Theorems I

  1. Sturm's method of counting real roots
  2. Adapted cell decompositon and dimension of semi-algebraic sets
  3. (if time allows) Hermite's method

Classical Theorems II

  1. Finiteness theorems
  2. The ?ojasiewicz inequality for semi-algebraic functions
  3. (if time allows) The Bröcker-Scheiderer Theorem (without proof)

A subjective guide to recent developments

  1. Positivstellensätze
  2. Schmudgen's Positivstellensatz and semi-definite programs
  3. Open problems

Lecture dates

There will be four lectures:

  • Tuesday, 25 February, 2014, 12-14 in Koll. B3
  • Friday, 28 February, 2014, 14-16 in room D.01
  • Tuesday, 4 March, 2014, 12-14 in Koll. B3
  • Friday, 7 March, 2014, 12-14 in Koll. B3