Polyhedral Methods in Algebraic Geometry

Christian Haase

class times: Tue 10:00-11:30, 12:30-14:00


Coming up:    Prüfungskolloquium Monday, August 26, room A3.024

You pick a topic (see below for examples) which you will present during our conference. The presentation will be followed by a discussion/questions which are not restricted to the specific topic. I will then write down a grade. This grade can be improved by active participation in the discussion during the other talks.

Possible topics: proofs, proof strategies or techniques left out in class; worked example of a result mentioned in class; a result or an application related to but not treated in class, ...
-- Gröbner walk algorithm
-- Boolean rings and VLSI design
-- path signature varieties
-- toric phylogenetic models (group based models)
-- proof of Kapranov's Theorem
-- Bernstein's Theorem via homotopy continuation
-- algorithmic complexity of mixed volume and mixed cells
-- polyhedral intersection theory (toric or tropical)
-- matroids, Grassmannians and Bergman fans
-- space of trees and M_0,n
-- toric patches in geometric modeling


Time Speaker Topic
10h00 Hannah Gröbner walk
10h40 Yumeng toric phylogenetic models
11h20 Marie signed tropical numbers
12h00   -- Lunch --
13h40 Karin toric Cartier divisors as polytopes
14h20 Xiangying hyperplane arrangements, Bergman fan
15h00 Kemal Grassmannians and the space of trees
15h40   -- Pause --
16h00 Lena Okounkov bodies -- general and toric
16h40 Niklas toric intersection theory
17h30   -- Kaltgetränke at Alter Krug --




0. Overview

I. Algebraic geometry primer
I.1 Ideals, varieties, coordinate rings
I.2 Gröbner basics


I.2 Gröbner basics, Hilbert function, dimension, degree

II. Polyhedral primer
II.1 Polyhedra and polytopes

04/23 II.2 Point lattices

III. Toric Varieties
III.1 Projective varieties
III.2 Toric ideals
III.3 Torus action

04/30 III.4 generation of toric ideals (Markov bases)

III.5 regular subdivisions and toric Gröbner bases


III.5 Gröbner fan, state polytope, secondary polytope

05/21 III.6.1 Polytopes with regular unimodular triangulation
III.6.2 Hierarchical statistical models, Conditional independence
III.6.3 Group based phylogenetic tree models

III.6.4 Path signature varieties

IV. Tropical geometry
IV.1 Tropical arithmetic


Problems for the algebraic geometry semester


IV.1 Tropical arithmetic
IV.2 Valued fields & Kapranov's Theorem


IV.3 Tropical varieties -- multiplicities and balancing


IV.4 Bernstein's Theorem and Seminar


IV.5 Grassmannians, matroids, trees

III.7 Moment map of toric varieties

07/09 Another interesting seminar @HU involving toric geometry. Please attend!



This course has (at least) two target audiences: on the one hand, this course is for students of algebraic geometry who are curious to see their field interact with polyhedral geometry, and to even see it applied to statistics, economics and optimization. On the other hand, this course is for students of discrete geometry, optimization or combinatorics, from numerical analysis, stochastics or other fields who want to get a glimpse of what algebraic geometry is about, and see examples of applications to their favorite objects.

tropical Bernstein's Theorem

The goal is to provide background and to get students up to speed so that they will be able to participate actively in a project group of the anticipated Thematic Einstein Semester Varieties, Polyhedra, Computation during the winter term 2019/20.