Using toric geometry

03/11/2024
In the spring semester of 2024 I taught a course on toric varieties at the Max Planck Institute for Mathematics in the Sciences, Leipzig. 


Registration:

https://www.mis.mpg.de/events/event/using-toric-geometry

Short description: 

Toric varieties form a well understood class of algebraic varieties. They often arise as the image of a monomial map. These lectures offer an introduction to basic toric geometry and its applications in mathematics and other sciences. Covered topics include toric ideals, semigroup algebras, Cox rings, discriminants and algebraic moment maps. We will also discuss standard constructions of toric varieties from cones, polytopes and fans, and the role of toric varieties in convex optimization, algebraic statistics, and equation solving. 

Prerequisites:

Basic algebraic geometry: affine/projective varieties and their coordinate rings.

Schedule: 

Online attendance is possible for all lectures. Below, (Online) indicates that I will give the lecture remotely, using a tablet. All lectures will be given or broadcasted in room G3 10 at MPI MiS. 

  1. Wednesday April 3, 10-12AM. Monomial maps and toric ideals. Exercises: Thursday April 4, 10AM.
  2. Wednesday April 10, 10-12AM. Cones and affine toric varieties. Exercises: Thursday April 11, 10AM.
  3. Friday April 19, 10-12AM (Online). Polytopes and projective toric varieties. Exercises: Wednesday April 24, 10AM.
  4. Tuesday April 30, 2-4PM. Toric boundaries. Exercises: Monday May 6, 10AM. 
  5. Tuesday May 7, 10-12AM. Positive toric geometry. Exercises: Wednesday May 8, 9.30AM in A3 01.
  6. Friday May 10, 10-12AM. Toric geometry of linear programming. Exercises: Friday May 10, 1.30PM. 
  7. Friday May 17, 1.30-3.30PM. Fans and gluing toric varieties. Exercises: Tuesday May 21, 10AM. 
  8. Wednesday June 5, 10-12AM. Maps between toric varieties. Exercises: Friday June 7, 10.30AM.  
  9. Tuesday June 11, 9.30-11.30AM. Divisors on toric varieties. Exercises: Thursday June 13, 10AM.
  10. Wednesday June 19, 2h30-4h30PM. Divisors on toric varieties, part 2. Exercises: Thursday June 20, 10h30AM.
  11. Wednesday June 26, 10-12AM. The Cox ring of a toric variety. Exercises: Thursday June 27, 10h30AM. 

Follow-up lectures on statistics, solving equations and discriminants will be given during summer 2024.

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