Summer Graduate School

The goal of our school is to make connections between different flavors of geometry: euclidean geometry, hyperbolic geometry, translation surfaces and graph theory. We will focus on growth, asymptotics and extremal problems.

Problems related to growth, asymptotics and extremality play a prominent role in all the types of geometry mentioned above. Examples of questions are:

Euclidean geometry: What is the densest Euclidean sphere packing in a given dimension? What is the maximal possible kissing number of a sphere packing? How do these maxima
grow as the dimension tends to infinity?

Hyperbolic geometry: What are the maximal possible systole (the length of the shortest closed geodesic) and Laplacian eigenvalue on a hyperbolic surface of a given genus? How
does the number of geodesics of bounded length grow as this length tends to infinity? How do these quantities behave for a typical surface?

Translation surfaces: How does the number of sadle connections of bounded length grow as the length tends to infinity? What is the number of horizontal cylinders on a random
translation surface of large genus?

Graph theory: What are the maximal possible first eigenvalue of the adjacency matrix and girth (length of the shortest cycle) of a regular graph? How do these maxima grow as the size of the graph tends to infinity? What are they for a typical graph?

Even if the communities working in these different fields are somewhat separate, there are already multiple examples of ideas succesfully being transferred from one field to another. Recent examples are applications of ideas from graph theory in the study of spectral gaps on hyperbolic surfaces (by Hide–Magee, Anantharman–Monk, Hide–Macera–Thomas and others), Ramsey theoretic ideas in the study of sphere packings (by Campos–Jenssen–Michelen–Sahasrabudhe) and the connections between geodesic counting on hyperbolic surfaces and Masur–Veech volumes and Siegel–Veech constants on translation surfaces (by Delecroix–Goujard–Zograf–Zorich). The point of the school is to gather experts from these communities in order to present the latest developments and the methods that are behind them.

School Structure

There will be a total of 10 mini-courses, each consisting of 3 hours of lectures and 1 hour of excercises.  In addition, there will be an office hour at the end of each day for students to be able to ask questions.

Prerequisites

The techniques used in the subjects of our proposed school vary quite wildly, ranging from differential geometry and analysis to ideas from number theory, probability theory and combinatorics. One of the main goals of the school is to expose the students working on one particular geometry to the ideas used in the others. As such, we won’t assume any particular prerequisites, besides a general interest in the subject and the mathematical background expected from a graduate student. 

Application Procedure

SLMath is only able to support a limited number of students to attend this school.  Therefore, it is likely that only one student per institution will be funded by SLMath.

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Venue

The summer school will take place at Centre de recherches mathématiques (CRM), Montréal, Canada.