Prof. Dr. Johannes Nicaise


Technische Universität München
Zentrum Mathematik - M11
Boltzmannstr. 3
85748 Garching bei München

Büro:         MI 02.12.039
Telefon:     +49 89 289-17060


John von Neumann Lecture: Motivic integration, tropical geometry and rationality of algebraic varieties

Times and Venues

Monday, 14h00 - 16h00, Seminarraum MI 00.07.014
Thursday, 10h00 - 12h00, Seminarraum MI 03.10.011


The aim of this course is to introduce students to rationality problems in algebraic geometry and to some new techniques that grew from the theory of motivic integration. The course will be structured around the following key topics:

Semi-algebraic geometry and motivic integration

Besides its algebraic structure, the field K of complex Puiseux series also carries a valuation, the t-adic order of a Puiseux expansion. Semi-algebraic geometry over K studies the sets and maps one can define by using polynomial equations and inequalities between t-adic orders of polynomials. This type of geometry captures important aspects of the limit behaviour of complex varieties in one-parameter families, defined by algebraic equations that depend on t. A powerful tool to construct invariants of semi-algebraic sets over K is the theory of motivic integration (in the form developed by Hrushovski and Kazhdan), which allows us to extend geometric and cohomological invariants from complex varieties to semi-algebraic sets.

Tropical geometry

An important part of semi-algebraic geometry over K is tropical geometry, which studies the geometry of semi-algebraic sets in terms of their images under so-called tropicalization maps (defined by taking t-adic orders of coordinate functions). Tropical geometry has become a very active field of research in the last 30 years and connects algebraic geometry to many other branches of pure and applied mathematics. We will see how tropical geometry helps to define and compute the invariants appearing in motivic integration, and to construct semi-algebraic sets of geometric interest.

Applications to rationality problems

One of the most basic questions one can ask about an algebraic variety is whether it is rational, that is, birational to a projective space. This question turns out to be extremely hard and has a rich history that goes back more than a century. In the last ten years, there have been major breakthroughs in the theory thanks to the introduction of new degeneration techniques. We will see which role motivic integration has played in these breakthroughs, and how tropical geometry provides a very convenient computational tool to answer the rationality question for various types of algebraic varieties.

By the end of the course, students should be able to read contemporary research papers on these topics. The main references are

J. Nicaise and J.C. Ottem, “A refinement of the motivic volume, and specialization of birational types” – In: G. Farkas et al. (eds.), Rationality of varieties, volume 342 of Progress in Mathematics, Birkhaüser, pages 291-322 (2021), arXiv:2004.08161

J. Nicaise and J. C. Ottem, “Tropical degenerations and stable rationality” – To appear in Duke Mathematical Journal (2022), arXiv:1911.06138