A:C&T

Description: Matroid theory arose as a combinatorial generalisation of both linear algebra and graph theory. During the 20th century it has grown into a fascinating structural theory, unifying vast areas of combinatorics and establishing strong connections to many other mathematical disciplines, from optimisation theory to commutative algebra.
We will introduce the basics of matroid theory, with a particular focus on its geometric and polyhedral aspects, especially related to tropical linear spaces. In fact, this course can serve as a stepping stone towards Jan Draismas BeNeFri Spring lecture on Tropical geometry at the University of Bern. No particular prerequisites are needed except basic linear algebra.

Click on the [words in brackets] below in order to visualize or download the file.

Part 1- Definitions and examples

Warm-up : [Video], [Worksheet].

In-person lecture : Thursday, October 1, 8:45- 13:00, Uni Fribourg PER12 [click for a map]

Lecture Notes: [Lecture Notes part 1]

Work-out:

Video: diagrams for matroids of small rank. [Zipped video, big file] [Zipped video, smaller file]
Worksheet: [Worksheet 1]

Literature:

As a general reference for structural matroid theory I recommend
James Oxley, Matroid theory. Second edition. Oxford Graduate Texts in Mathematics, 21. Oxford University Press, Oxford, 2011. xiv+684 pp.
Download here: [Zipped folder] the beginning, including the pages needed in the worksheet.

Lecture 2 - Matroid polytopes and flats

Warm-up:

1) Generalities about polytopes. [Slides], [Video (~29 min.)]
2) Two constructions: vertex cones and products. [Slides], [Video (~15 min.)]

In-person lecture: Friday, October 9., 12:15 - ca. 16:15 PER 08 Room 2.52

Lecture Notes: [Lecture Notes part 2]
Video: review of proof of Theorem 1.10. [Zipped video]

Work-out:

Worksheet: [Worksheet 2]

Literature:

For the polytope warm-up see, e.g., the first chapter of
G.M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, 152. Springer-Verlag, New York, 1995. x+370 pp.
For matroid polytopes, the Lecture Notes are based on (parts of):
I. Gelʹfand, R. Goresky, M. MacPherson, V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. in Math. 63 (1987), no. 3, 301–316.
E. M. Feichtner, B. Sturmfels, Matroid polytopes, nested sets and Bergman fans. Port. Math. (N.S.) 62 (2005), no. 4, 437–468.
Download here [Zipped folder] the relevant part of Ziegler's book and the two research papers.

Zoom Q&A

Thursday, October 29., 15:30.

Lecture 3 - Bergman fans

Warm-up : Download the [Assignment], that will include watching the [Video] (here the [Slides] )

In-person lecture: Thursday, November 12., 8:15 - 12:00, online (TBD) .

Lecture Notes: [Lecture Notes part 3]
"Slides": [Slides for Lecture 3]

Work-out: Review Lecture Notes 3 (including all proofs) and as an exercise try to draw the Bergman complex of the uniform matroid of rank 2 on 5 elements.

Literature: aside from material mentioned above,

F. Ardila, C. Klivans, The Bergman complex of a matroid and phylogenetic trees. Journal of Combinatorial Theory, Series B, no. 96 (2006), 38-49.

Download here [Zipped folder] this paper.

Lecture 4 - The tropical connection, geometric lattices.

Warm-up : Review the previous lecture and work out the Bergman complex of the uniform matroid of rank 3 on 6 elements.

Live lecture: Friday, November 20., 12:30 - ca. 16:30 (Online)

Lecture Notes: [Lecture Notes part 4]
Blackboard: [Blackboard part 4]

Work-out:

Worksheet: [Worksheet 4]
Literature: aside from material mentioned above,
M. De Longueville, A course in topological combinatorics. Universitext. Springer, New York, 2013. xii+238 pp.
Download here [Zipped folder] the relevant selected chapters.

Zoom Q&A

Thursday, December 10, 15:30.

Zoom Q&A

Thursday, January 14, Afternoon (TBD).

Format: Lecture with some exercises. For those who need credit points, an oral examination.

Contact: SNSF-Prof. Emanuele Delucchi, emanuele.delucchi "at" unifr.ch