# Differential Geometry - Summer School

## Gulbenkian Foundation, LISBON — JULY 23 to 27, 2018

sketch for the panel Começar by José de Almada Negreiros - image reproduced with kind permission of the Gulbenkian Foundation

# Program

This school comprises three lecture courses (in English) aimed at 1st- or 2nd-year undergraduate students of mathematics, complemented by problem sessions.

# Courses

## Curves and Surfaces

by André Neves (University of Chicago)

Abstract: I will cover the basic theory of curves and surfaces in space. The basic idea is to associate some computable geometric quantities to a surface (like mean curvature or Gaussian curvature) and then infer information about the shape of the surface from those quantities. The foundational results to be covered are Gauss's Egregium Theorem and Gauss-Bonnet Theorem.

Prerequisites: Multivariable calculus, namely Implicit (or Inverse) Function Theorem, Jacobian of maps, area formula, and Stokes Theorem

## Hyperbolic Geometry - from graphs to geometry

by Danny Calegari (University of Chicago)

Abstract: We start with a combinatorial problem: given a planar graph G, when can we find a collection of round circles in the plane whose interiors are disjoint, and with one circle for every vertex of G, such that two circles are tangent if and only if the corresponding vertices are joined by an edge of G? This will be the starting point that leads to the Riemann mapping theorem, the construction of 3-dimensional hyperbolic polyhedra, and a beautiful family of connections between combinatorics, topology, geometry, and complex analysis.

## An introduction to Lie groups, symmetries, and symplectic geometry

by Robert Bryant (Duke University)

Abstract: The notion of groups of motions and symmetries of objects is a fundamental one in mathematics, and it is nowhere more important than dealing with continuous motions and and problems in physics and geometry. I will start by discussing familiar objects, such as the group of rotations in space and higher dimensions, and move on to other groups, mainly groups of linear transformations, so that we can stay as concrete as possible. After developing the theory of continuous groups and seeing how these objects can be studied using multivariable calculus and linear algebra, we will then apply these concepts to study some mechanical systems and the surprising connection between symmetries and conservation laws that is so fundamental in modern mathematics and physics.

Prerequisites: Linear algebra and multivariable calculus, mainly. Knowing something about calculus done with differential forms (for example, see these notes ) would be helpful, but it's not absolutely essential. [However, differential forms are an important tool in modern geometry and calculus, so it pays to learn about them early in any case.]

 Monday, July 23 Tuesday, July 24 Wednesday, July 25 Thursday, July 26 Friday, July 27 9:45-10:00 opening 10:00-11:00 A. NEVES A. NEVES D. CALEGARI R. BRYANT A. NEVES 11:00-11:30 coffee break coffee break coffee break coffee break coffee break 11:30-12:30 R. BRYANT R. BRYANT A. NEVES D. CALEGARI R. BRYANT 12:30-14:00 lunch break lunch break A. NEVES (until 13.30) lunch break lunch break 14:00-15:00 D. CALEGARI R. BRYANT D. CALEGARI D. CALEGARI 15:00-15:15 break break break break 15:15-16:00 PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION 16:00-16:15 coffee break coffee break coffee break coffee break 16:15-17:00 PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION

Friday, July 27th at 5.30pm, after the problem session, there will be a meeting in the lecture room where the participants will be able to talk to the lecturers about academic life in a relaxed setting.

On Friday we will go out to dinner at 7.30pm to Restaurante Pano de Boca at Teatro Aberto, right next to the Gulbenkian Foundation.

# Mini-conference

On Saturday, July 28th, there will be a student mini-conference in the same venue as the school where some of the participants in the undergraduate research program will present their projects. All school participants are invited to attend this mini-conference. Here is the schedule of the mini-conference. Here are the slides of Prof. Neves' presentation, and the slides of the undergraduate fellows' presentations.

## Course assistants

Miguel Martins dos Santos () and Miguel Moreira ()

## Venue

The school will take place at the headquarters of the Gulbenkian foundation with lectures and problem sessions in Sala 1. There is a map under Local Information.