Clemens Brüser :
Student presentation: Algorithms for counting real roots
In the 17th century, Rene Descarte first developed ideas to count the number of positive real roots of a univariate real polynomial based on its coefficients alone. These ideas have come a long way and today provide us with useful algorithms to answer the following questions:
- Given f in R[x], how many of its roots are real? How many are positive? How many lie in [0,1]?
- Given f,g in R[x], how many points a are there such that both f(a) = 0 and g(a) > 0?
We will discuss some of these algorithms and how they can be applied to answer some of these questions.

Chiara Meroni :
Abstract perspective on real numbers
This lecture will introduce the concept of a real closed field and the Tarski-Seidenberg principle. We'll discuss the properties of real closed fields and give examples, as well as explore the use of the Tarski-Seidenberg principle in real algebraic geometry.

Chiara Meroni :
Semialgebraic sets
In this lecture, we will explore semialgebraic sets, which are sets that can be defined by finitely many polynomial inequalities. We will discuss their nice properties, and in particular the cylindrical algebraic decomposition.

Chiara Meroni :
Positivity and convexity
In this third lecture, we will focus on positive polynomials and their applications in optimization. We will discuss their properties and their relationship to important convex semialgebraic cones.

Chiara Meroni :
Volumes of semialgebraic sets
The final lecture in this series will show you how to compute or approximate the volume of a semialgebraic set using semidefinite optimization. In this context, we will discuss recent developments based on Lasserre hierarchies.