The seminar takes place Thursdays, 8.30-10 in Arnimallee 3, Room 019.
The seminar will be advised by Giulia Codenotti (giulia.codenotti@fu-berlin.de) and by Ansgar Freyer (freyer@posteo.de). Please get in touch for any questions.
RULES
To get credit for this seminar, you must:
- set up a meeting with the instructors by getting in touch two weeks before your planned talk
- discuss in the meeting what will be presented in the talk, in particular, which examples, which proofs, and any questions you have about the topic or the presentation.
- give a clear exposition of your assigned topic; not everything needs to be proved, but everything should be justified, for example through a meaningful worked out example.
- write a 4 page report of your work by the end of the semester; this should be readable and include plenty of examples and explanations about the results presented, especially when omitting proofs!
- actively participate in other talks, by asking questions and giving feedback.
The actual talk should be 60 minutes long, plus 15 minutes for the (hopefully many) questions and discussions in between. The last 10-15 minutes will be spent discussing the execution of the talk, everyone is invited to give some positive feedback and some aspects that could be improved.
SCHEDULE
| 17.10 |
Presentation |
Giulia |
| 24.10 |
Introduction to lattices |
Kyle |
| 31.10 |
Exercise session on lattices and lattice polytopes |
Kyle |
| 07.11 |
Paper 0 (see list below) |
Erfan |
| 14.11 |
Paper 1 |
Ansgar |
| 21.11 |
Paper 3, part 1 |
Katarina |
| 28.11 |
Paper 3, part 2 |
Steffi
|
| 5.12 |
Paper 4, part 1 |
Anna |
| 12.12 |
Paper 4, part 2 |
Gregor |
| 19.12 |
Extra time for missing bits |
|
| 09.01 |
Paper 5 |
Negar |
| 16.01 |
Paper 6 |
Lea |
| 23.01 |
Paper 7 |
Mara |
| 30.01 |
Paper 7, second part |
Ansgar/Giulia |
| 06.02 |
Wrap up, further topics |
Giulia/Ansgar |
TOPICS
This seminar will focus on lattice polytopes--polytopes whose vertices have integer coordinates. These objects are central in various branches of mathematics: algebraic (toric) geometry and (integer) optimization, for example. We will investigate extremal problems about lattice polytopes: how large/wide/volumous can a lattice polytope with/-out some property be? Some property is for example "not containing points with integer coordinates" or "not projecting onto a polytope with the property above", etc. We will see that there is a conceptual difference between polytopes with "some interior lattice points" and those with no "interior lattice points", which will be a common theme in the questions that we discuss.
This seminar is accessible to anyone who has taken discrete geometry 1 or an equivalent course introducing polytopes. Having some knowledge of Ehrhart theory, covered in discrete geometry 2, can be helpful, but is not required. On the other hand, the seminar aims to quickly reach current research topics, and can be an excellent starting point for a bachelor or master thesis. The topics are chosen to fit into a research project that we recently started in collaboration with Cottbus, and will therefore be also streamed for virtual participants. Talks will however take place in person at FU and everyone is encouraged to come weekly.
We start with the basics, introducing the tools we need and then look at (a selection of) the following papers:
0) Lagarias& Ziegler: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice.
1) Averkov: A proof of Lov´asz’s theorem on maximal lattice-free sets (https://arxiv.org/pdf/1110.1014)
4) Averkov, Krümpelmann, Weltge. Notions of Maximality for Integral Lattice-Free Polyhedra:The Case of Dimension Three
7) Averkov,Kruempelmann &Nill: Largest integral simplices with one interior lattice point ( https://arxiv.org/pdf/1309.7967 )