Christian Steinert

Newton Okounkov Theory

4h graduate lecture by Prof. Dr. P. Littelmann. In German. Summer 2018.


Introduction

Let \(g(x,y) = \sum_{i,j}a_{i,j}x_iy_j\) be a polynomial in two variables \(x\) and \(y\). Then the Newton polygon \(New(g)\) is defined as the convex hull of \(\lbrace(i,j) \,|\, a_{i,j} \neq 0\rbrace\) in \(\mathbb R^2\). One of the surprising results about the Newton polygon is Bernstein's Theorem, that connects the number of solutions of \(g(x,y) = h(x,y) = 0\) in \((\mathbb C^*)^2\) for two polynomials \(g(x,y)\) und \(h(x,y)\) with the surface area of \(New(g)\), \(New(h)\) and the the Minkowski sum of \(New(g)\) and \(New(h)\). One goal of Newton-Okounkov Theory is to find such connections in a more general setting. One gets polytopes associated to various objects (graduated algebras, varieties, ....) and tries to conclude properties of the graduated algebra, the geometry of the variety and so on from the geometry of the polytope.

Requirements: Lineare Algebra I & II, Algebra, additional knowledge about commutative algebra and algebraic geometry can be useful.


Current Information


Lectures

The lectures will be given on Mondays and Wednesdays from 10:00 to 11:30 in Stefan-Cohn-Vossen-Raum (room 313).


Exercises

Exercise sessions will take place every Wednesday at 08:30 in S3 (room 314).
Exercises have to be handed in every Monday after the lecture. 50% of the points and active participation during exercise sessions are requirements for the final exam.


Exam

The exam will be held on Monday 09.07.2018 during the lecture.

Revision of the exam will be possible on Wednesday 11.07.2018, if the department is open, otherwise on Monday 16.07.2018.


References