UiO Tropical Geometry Learning Seminar

This is the webpage for an introductory seminar in tropical geometry. Our ambition is to cover the fundamentals of tropical geometry, a selection of main results and achievements, and some of the most recent developments. The seminar is intended for students, and will include a few exercise sessions. For the more advanced topics of the later sessions, we will invite more experienced participants.

Practical information:

• Meeting times: Fridays at 14:15, 2 hours max
• Room: Zoom for the moment.
• Contact:
  • Edvard Aksnes (edvardak at math dot uio dot no),
  • Simen Westbye Moe (swm418 at ic dot ac dot uk),
  • Cédric Le Texier (cedricle at math dot uio dot no).
  • Kris Shaw (krisshaw at math dot uio dot no).

Talks

• Session 0 (external): Tropical homology and Hodge theory

  Date: August 23-27 2021

  Our seminar series will be kicked off by attending the Tropical homology and Hodge theory seminar which is being held in Leuven. The ambition for our seminar series will be to understand some of the topics discussed here.

• Session 1: Tropical algebra, curves and hypersurfaces

  Date: Tuesday, August 31, 2021, Time: 14.15, Speaker: Edvard Aksnes

  We will introduce tropical geometry through tropical algebra, define what a tropical polynomial. Then we will look at tropical curves and tropical hypersurfaces. The format is: 45 min talk + 15 min break + 45 min exercise session. Slides

• Session 2: Tropicalization

  Date: Tuesday, September 7, 2021, Time: 14.15, Speaker: Simen Westbye Moe

  This talk will be about the process of tropicalization, where we will build up to Kapranov's theorem. The format is: 45 min talk + 15 min break + 45 min exercise session. Slides

• Session 3: Tropical varieties and realisability

  Date: Friday, September 17, 2021, Time: 14.15, Speaker: Kris Shaw, Room: 1120

  We will define tropical varieties in (\mathbb{R}^n) of arbitrary codimension and consider the question of when these objects arise as tropicalisations of algebraic varieties defined over the field of Puiseux series and other valued fields. A main example will be Vigeland’s discovery of the excess number of lines on tropical cubic surfaces and how tropical intersection theory can be used as a tool to rule out their realisability. Another main example of non-realisable tropical varieties comes from matroid theory. This will be the topic of the next week. The format is: 45 min talk + 15 min break + 45 min exercise session. Slides

• Session 4: Matroids and their Bergman fans Date: Friday, September 24, 2021, Time: 14.15, Speaker: Eline Mannino, Room: 1120

  This talk will be about matroid theory and objects related to it, like the Bergman fan. Then we will look at how this is related to tropical varieties. The format is: 45 min talk + 15 min break + 45 min exercise session. Slides

• Session 5: Tropical linear spaces and Grassmannians

  Date: Friday, October 1, 2021, Time: 14.15, Speaker: Felipe Rincón, Room: 1120

  I will talk about parameter spaces for tropical linear spaces, such as Dressians and tropical Grassmannians. I will also explain how tropical linear spaces locally look like Bergman fans of matroids.

• Session 6: Oriented Matroids

  Date: Friday, October 8, 2021, Time: 14.15, Speaker: Chi Ho Yuen, Room: 1120

  I will introduce the notion of oriented matroids from different perspectives: combinatorial, geometric/topological, and algebraic. I will also describe how tropical geometry enters the picture.

• Session 7: An introduction to Logarithmic Geometry from the viewpoint of Algebraic Geometry

  Date: Friday, October 22, 2021, Time: 14.15, Speaker: Nikolai Thode Opdan, Room: 1120

  Logarithmic geometry was introduced in the 1980's as a way to deal with essentially two concepts arising in algebraic geometry: Compactifications and degenerations. Later it was realised that logarithmic geometry has a beautiful theory on its own, as an enlargement of algebraic geometry from rings to monoids.

  In this talk I will present the motivation as to why you would consider logarithmic geometry with a special emphasis on problems arising from algebraic geometry. I will then present the basic theory, and relate it to examples from (among others) toric- and tropical geometry.

• Session 8: Tropical Hyperfield and Modifications

  Date: Friday, October 29, 2021, Time: 14.15, Speaker: Cédric Le Texier, Room: 1120

  We want to consider two problems concerning tropical varieties: defining those as "zero sets" (which is not possible over the tropical semifield), and study non-transverse intersections of tropical varieties. We solve the first problem by introducing the tropical hyperfield, and the second problem using graphs of function over the tropical hyperfield, also called tropical modifications. Slides