- Bojan Mohar (Simon Fraser University & University of Ljubljana)
- Gelasio Salazar (Univ. Autónoma San Luis Potosí)
This course will cover hot topics in combinatorial and discrete geometry, focusing on a selection of techniques that have enabled some of the latest breakthroughs in the area. The activity will consist of two lectures every morning, each at most 2h long, followed by work and interacting in the afternoons.
Bojan Mohar: Graphs on surfaces
The course will outline fundamental results about graphs embedded in surfaces. It will briefly touch on obstructions (minimal non-embeddable graphs), separators and geometric representations (circle packing). Time permitting, some applications will be outlined concerning homotopy or homology classification of cycles, crossing numbers and Laplacian eigenvalues.
Gelasio Salazar: Crossing numbers and related topics in combinatorial topology and geometry
The crossing number is an intriguing entity. Unlike most graph-theoretical parameters, we do not even know its value for the usual suspects: complete graphs and complete bipartite graphs. The rich history of this parameter goes back to the renowned combinatorialist and number theorist Paul Turán, and to British artist Anthony Hill, a remarkable figure of the Constructivist Group. Determining the exact crossing number of familiar collections of graphs remains a stubbornly open problem, and perhaps for this reason most early research on crossing numbers focused on evaluating this parameter for specific families of graphs, or even for a single graph. Among the few notable early-ish exceptions we have the Crossing Lemma, whose proof(s), and applications we will cover in this course. In the last twenty years or so, the field has become a mainstream part of Topological Graph Theory, due to several important theorems of a structural character. Quite a few important questions on crossing numbers remain open to (elementary? intricate?) ideas from newcomers to the field. We will survey a biased selection of what is known about this parameter, from its early history back to Turán, Zarankiewicz, and Hill, up to the most recent developements, with a strong emphasis on open questions. A selected collection of related problems and results in combinatorial geometry and topology will also be covered. Our plan is to highlight how tools from algebraic, probabilistic, and pure combinatorics shed light on an eminently topological problem for which, at some basic level, the only topological tool available is the Jordan curve theorem.
There will be two lectures every morning, each 1h 45min long. In the afternoons, attendees and lecturers will work in open problems.
Monday, May 7
Tuesday, May 8
Wednesday, May 9
Thursday, May 10
Friday, May 11
Note: Afternoon seminars will be announced in-place.