Lecturers
 Bojan Mohar (Simon Fraser University & University of Ljubljana)
 Gelasio Salazar (Univ. Autónoma San Luis Potosí)
Description
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.
Contents
Bojan Mohar: Graphs on surfaces
The course will outline fundamental results about graphs embedded in surfaces. It will briefly touch on obstructions (minimal nonembeddable 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.
Download the notes for this course
Gelasio Salazar: Crossing numbers and related topics in combinatorial topology and geometry
The crossing number is an intriguing entity. Unlike most graphtheoretical 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 earlyish 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.
Download the notes for this course
Organization
There will be two lectures every morning, each 1h 45min long. In the afternoons, attendees and lecturers will work in open problems. There will be a seminar talk every afternoon.
– Lectures and seminar talks will take place in the Auditorium.
– Working and interacting will take place in rooms Pol1 and Pol2.
Program
Monday, May 7

Registration

Welcome

Crossing numbers and related topics in combinatorial topology and geometry
Gelasio Salazar

Coffee break

Graphs on surfaces
Bojan Mohar

Lunch time

Seminar
Triangulations and the BrunnMinkowski inequality in the plane
Oriol Serra
Universitat Politècnica de Catalunya 
Working & interacting
Tuesday, May 8

Graphs on surfaces
Bojan Mohar

Group photo

Coffee break

Crossing numbers and related topics in combinatorial topology and geometry
Gelasio Salazar

Lunch time

Seminar
Bounded Degree Vertex Deletion, Complexity and Applications to Graph Drawing
Fabian Klute
TU Vienna 
Seminar
Further Consequences of the Colorful Helly's Theorem Hypothesis
Leonardo Ignacio Martínez Sandoval
Ben Gurion University of the Negev 
Working & interacting
Wednesday, May 9

Crossing numbers and related topics in combinatorial topology and geometry
Gelasio Salazar

Coffee break

Graphs on surfaces
Bojan Mohar

Lunch time

Seminar
On the Density of Triangles with Periodic Billiard Paths
Ramona Charlton 
Working & interacting
Thursday, May 10

Graphs on surfaces
Bojan Mohar

Coffee break

Crossing numbers and related topics in combinatorial topology and geometry
Gelasio Salazar

Lunch time

Seminar
Square peg conjecture, can geometry help topology?
Sergey Avvakumov
Institute of Science and Technology Austria 
Seminar
Equiangular polygon contact representations
Hendrik Schrezenmaier
Technische Universität Berlin 
Working & interacting
Friday, May 11

Crossing numbers and related topics in combinatorial topology and geometry
Gelasio Salazar

Coffee break

Graphs on surfaces
Bojan Mohar

Lunch time

Seminar
Minimal Geometric Graph Representations of Order Types
Irene Parada
Graz University of Technology 
Working & interacting