Type of course (Veranstaltungsart)
Tropical geometry can be essentially intended as the geometry over the semiring of real numbers with addition replaced by minimum and multiplication replaced by addition.
The corresponding zero sets of polynomials obtained in this context are objects coming from polyhedral and convex geometry, and they can be seen as ''combinatorial shadows'' of their more classical algebro-geometric counterparts.
This young area of geometry has proven to be helpful both in complex and non-archimedean geometry and in combinatorics, to approach problems related to the Hodge conjecture, to define forms and currents on Berkovich spaces and to deal with matroidal objects, just to name a few.
The goal of these lectures is to introduce the main actors of tropical geometry and to show some applications of the theory.
We will start the course by giving some motivating topics from where the subject has found inspiration and source. These include the search for the shortest path in a weighted directed graph, the study of plane tropical curves, enumerative problems and amoebas.
We will then move to the two central results of the course, namely the so-called Fundamental Theorem of Tropical Geometry and the Structure Theorem, describing the shape of tropical varieties as balanced polyhedral complexes.
In doing this, we will need to touch topics like vauations, Newton polytopes, initial forms and Gröbner bases for ideals.
In the remaining of our course, we will explore some applications of the theory: stable intersection of tropical varieties with its application to root counting, matroids and their Hodge theory.
We conclude with an overview of a very recent paper by Adiprasito, Huh and Katz, which proves a conjecture on the log-concavity of the coefficients of the characteristic polynomial of a matroid.
- Diane Maclagan, and Bernd Sturmfels, Introduction to Tropical Geometry, volume 161 in the series ''Graduate Studies in Mathematics'' of the AMS (2015)
- Géométrie tropicale, papers from the Mathematical Days X-UPS held at the École Polytechnique, Palaiseau, May 14-15, 2008
- Karim Adiprasito, June Huh, and Eric Katz, Hodge theory for combinatorial geometries, Ann. of Math. (2) 188 2, pages 381-452 (2018)
Recommended previous knowledge
Linear Algebra and some elementary course in Algebra.
A previous exposure to Algebraic Geometry is certainly helpful, but it is not required, as the needed definitions and results will be recalled during the semester.
Tue, Thu 8-10
Course work (Studienleistungen)
- Preliminary registration for the organisation of exercise classes: at the end of the previous
semester via EXA or LSF (see announcement by the department)
- Registration for course work/examination/ECTS: FlexNow
- Successful participation in the exercise classes: 50% of the points in the exercises
- Oral exam: Duration: 25 minutes, Date: to schedule with the lecturer, re-exam: Date:
BV, MV, MArGeo