WARTHOG 2019

Expected background and suggested reading

Tropical geometry draws on several different areas of mathematics. While we don't expect most participants to be familiar with all of these, it will help to have met some of the new ideas before the workshop.

Do not worry if you do not have all the prerequisites yet; it should not be difficult to read some of the references below before the workshop begins. Below we give suggested readings. We will recall the needed facts and give examples — but we cannot expect anyone who's never seen any of these ideas to be able to follow most of the lectures.

We will be setting up a Googlegroup for this workshop, and we hope this will be a useful forum to ask questions or find online study groups. The core background will be topics from affine and projective algebraic geometry, tropical geometry, and matroids.

1. Affine/projective algebraic geometry.
   • Definitions of affine and projective varieties,
   • The Nullstellensatz
   • The Zariski topology
   • Familiarity with the Grassmannian and its Plücker embedding will also be helpful.
   Specific references:
   • Cox, Little, O'Shea, Ideals, Varieties and Algorithms (Chapters 1 and 4 - Chapter 2 is also useful),
   • Gathmann, Algebraic Geometry Course notes
   • Hartshorne, Algebraic Geometry (chapter I, section 1,2,3),
   • Hassett, Introduction to Algebraic Geometry (particularly chapters 3, 6, 9, 10),
   • Perrin, Algebraic Geometry: An Introduction (chapters 1,2),
   You should be comfortable going back and forth between ideals and varieties, and with how intersections and unions of closed/open sets are reflected in terms of ideals and localizations.
2. Tropical geometry. We recommend starting with an elementary survey paper, like the first parts of one of the following:
   • Brugallé, Shaw A bit of tropical geometry, https://arxiv.org/abs/1311.2360
   • Maclagan, Introduction to tropical algebraic geometry, https://arxiv.org/abs/1207.1925
   • Maclagan, Tropical Geometry Lecture notes from the LMS undergraduate summer school in 2017.
   • Johannes Rau, A first expedition to tropical geometry, https://www.math.uni-tuebingen.de/user/jora/downloads/FirstExpedition.pdf
   • Richter-Gebert, Sturmfels, Theobald, First steps in tropical geometry, https://arxiv.org/abs/math/0306366
   The basic pieces of background needed are:
   • The definition of the tropical semiring, and tropical polynomials.
   • The definition of a valuation.
   • The tropicalization of hypersurfaces.
   You should feel comfortable with drawing tropical curves in the plane, and solving tropical cubics in one variable.
3. Matroids.
   • First definitions, at least in the realizable (representable) case: Bases, circuits, independent sets, flats, duality. Cryptomorphism.
   • Examples - vector configurations, graphical matroids
   References:
   • Katz, Matroid theory for algebraic geometers, https://arxiv.org/abs/1409.3503 (sections 1-4)
   • Oxley's survey, What is a matroid? https://www.math.lsu.edu/~oxley/survey4.pdf (sections 1-3)
   • Oxley's book, Matroid Theory (Chapter 1, sections 1-7)
   You should be able to compute the bases, circuits, and independent sets for the matroid given by a vector configuration such as: {(1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,2,3) }.

Focus on the topics above before moving on to those below.

The following topics will be advantageous to know, but not strictly necessary:

1. Algebraic geometry:
   • The basics of scheme theory (affine schemes, structure sheafs, Proj)
   • Toric varieties (such as Chapters 1 and 2 of Cox, Little, Schenk, Toric Varieties ).
2. Tropical geometry:
   • Fundamental and Structure theorem for tropical varieties
   • Bergman fan of a matroid, tropical linear spaces
3. Matroids:
   • Valuated matroids
   • Bergman fans

Further pre-reading

• Payne, Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry https://arxiv.org/abs/1309.4403 (first three sections)
• Baker, An introduction to Berkovich analytic spaces and non-archimedean potential theory on curves http://swc.math.arizona.edu/aws/2007/BakerNotesMarch21.pdf, (sections 1 and 2)

Additional references

The content of the lectures will be based on the following papers (but you will not be expected to read them in advance):
Equations of tropical varieties by Giansiracusa and Giansiracusa.
Tropical schemes, tropical cycles, and valuated matroids by Maclagan and Rincón.
Tropical ideals by Maclagan and Rincón.
Scheme theoretic tropicalization by Lorscheid.
The universal tropicalization and the Berkovich analytification by Giansiracusa and Giansiracusa.
A moduli stack of tropical curves by Cavalieri, Chan, Ulirsch, and Wise.