# 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

- 09.07.2018: Revision of the exam will be possible on Wednesday 11.07.2018, if the department is open, otherwise on Monday 16.07.2018.
- 09.07.2018: The results of the exam are online.
- 26.06.2018: Please remember to hand in the Antrag auf schriftliche Prüfung im Aufbaumodul if needed.
- 25.06.2018: Registration for the exam via KLIPS should be open. Please contact me Mrs. Georg if you have any problems with the registration.
- 09.04.2018: The first exercise session (organization and questions) will take place on 18.04.2018 at 9:00 in S3 (room 314).
- 09.04.2018: Please send me an email if you want to participate in the exercises.
- 09.04.2018: The date of the exam is Monday 09.07.2018 10:00 - 12:00.

### Lectures

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

- Script.
*In German. Updated on 25.06.2018.*

### 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.

- Exercise Sheet 1. Submission on 23.04.2018.
- Exercise Sheet 2. Submission on 30.04.2018.
- Exercise Sheet 3. Submission on 07.05.2018.
- Exercise Sheet 4. Submission on 14.05.2018.
- Exercise Sheet 5. Submission on 28.05.2018.
- Exercise Sheet 6. Submission on 04.06.2018.
- Exercise Sheet 7. Submission on 11.06.2018.
- Exercise Sheet 8. Submission on 18.06.2018.
*New Version. Explanation added.* - Exercise Sheet 9. Submission on 25.06.2018.
*New Version. Confusing typos corrected.* - Exercise Sheet 10. Submission on 02.07.2018.
- Short Questions. No submission.
*Just preparation/repetition for the exam.*

### Exam

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

- Exam with solutions.
*In German.* - Results.

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

### References

**Boucksom, Sebastien.***Corps d’Okounkov (d'après Okounkov, Lazarsfeld-Mustata et Kaveh-Khovanskii),*Astérisque No. 361 (2014), Exp. No. 1059, vii, 1–41.**Kaveh, Kiumars; Khovanskii, Askold. G.***Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory,*Ann. of Math. (2) 176 (2012), no. 2, 925–978.**Kaveh, Kiumars; Manon, Christopher.***Khovanskii bases, higher rank valuations and tropical geometry,*arXiv:1610.00298.**Kiritchenko, Valentina; Smirnov, Evgeny; Timor, Vladlen.***Ideas of Newton-Okounkov bodies,*Snapshots of modern mathematics from Oberwolfach, DOI:10.14760/SNAP-2015-008-EN.