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Events
Let’s get \(\mathbb{R}\)eal

MPI für Mathematik in den Naturwissenschaften Leipzig

G3 10 (Lecture hall)

conference

07/06/2023 08/06/2023

Let’s get \(\mathbb{R}\)eal

This 2-days event is dedicate to real algebraic geometry, and related areas. We will learn what a real closed field is, we will play with semialgebraic sets, and explore many applications, including convex geometry and optimization. There will be lectures and exercise sessions, as well as time for questions and discussions.

Program

09:00 - 10:30	Chiara Meroni (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) 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. Exercises Exercises Slides
10:30 - 11:00	Break
11:00 - 12:00	Eric Boniface (University of California, Berkeley, Berkeley, USA) Exercises
12:00 - 13:30	Lunch break
13:30 - 15:00	Chiara Meroni (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) 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. Exercises Exercises Slides
15:00 - 15:30	Break
15:30 - 16:30	Laura Casabella (MPI MiS Leipzig, Leipzig, Germany) Exercises
16:30 - 17:00	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. Slides
18:00 - 00:00	Dinner

09:00 - 10:30	Chiara Meroni (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) 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. Exercises Exercises Slides
10:30 - 11:00	Break
11:00 - 12:00	Viktoriia Borovik (Osnabrück University, OSNABRÜCK, Germany) Exercises
12:00 - 13:30	Lunch break
13:30 - 15:00	Chiara Meroni (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) 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. Exercises Exercises Slides
15:00 - 15:30	Break
15:30 - 16:30	Kamillo Ferry (Technische Universität Berlin, Berlin, Germany) Exercises
16:30 - 17:00	Clemens Brüser (TU Dresden, Dresden, Germany) 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. Slides

Let’s get \(\mathbb{R}\)eal

Program

Participants

Zaineb Bel-Afia

Barbara Betti

Eric Boniface

Viktoriia Borovik

Clemens Brüser

Laura Casabella

Joan Claramunt Caros

Sarah Eggleston

Kamillo Ferry

Claudia Fevola

Hannah Friedman

Leonie Kayser

Jianrong Li

Sommer Linus

Saiei-Jaeyeong Matsubara-Heo

Chiara Meroni

Dmitrii Pavlov

Raul Penaguiao

Elizabeth Pratt

Silke Prohl

Jesse Selover

Elima Shehu

Raluca Vlad

Maximilian Wiesmann

Maksym Zubkov

Scientific Organizers

Chiara Meroni

Administrative Contact

Mirke Olschewski