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 :
Real fields and real varieties

This lecture will introduce the concept of a real closed field. We will discuss its properties and give some examples. Then, we will discuss real algebraic varieties and the real version of Hilbert’s Nullstellensatz.

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 the Tarski-Seidenberg principle and its consequences, such as 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.

Raluca Vlad :
Viro’s patchworking technique

We look at real plane algebraic curves. As a topological space, such a curve is a disjoint union of ovals and, if the curve has odd degree, a pseudoline. We discuss Viro’s patchworking technique, which is a combinatorial process that generates plane real algebraic curves with prescribed topology. We then use this patchworking to generate maximal curves, which are curves having the maximum possible number of connected components.