# Algebraic Topology - Summer School

## Gulbenkian Foundation, LISBON — JULY 24 to 28, 2017

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

## Shapes of spaces

by Bjørn Dundas (University of Bergen)

Abstract: Algebraic topology is propelled by questions about the properties of spaces. These questions even arise in modelling: for instance, "is there a space such that...?" or "what are the properties of the space of solutions of...?"
The most striking results are those where deep geometric questions are linked to properties that do not change when the space is deformed which in turn can be determined by purely algebraic means.
As an "easy" example in low dimension, we may consider the problem of containing plasma by means of a magnetic field. It turns out that this is impossible unless the plasma is shaped as a doughnut! Why? (By the way: what is the shape of the space wherein we live? Are there ways of finding out?)
The aim is to provide basic tools. Some of these are effectively calculable which is valuable for applications. Others reshape our idea of what mathematics is able to talk about. Algebraic topology is the business of transforming problems about spaces into algebra (which we can calculate with), but ALSO translating problems in algebra into spaces (which are more flexible, and where a solution may be found after a deformation).

## Topology from the early days

by Daniel Dugger (University of Oregon)

Abstract: In mathematics, the study of continuous transformations between shapes is called topology. Historically, the subject probably first arose by people realizing that you can oftentimes "deform" a problem without changing any of the key features. The simplest example is in Euclidean geometry: if two lines in a plane intersect at a point, then moving the lines slightly doesn't change this fact...unless you move them too much! These lectures will discuss several examples of this kind of "deformation problem", in situations ranging from geometry to differential equations. Finding ways to measure qualities that stay the same under deformations leads to the subject of algebraic topology. We'll talk about projective spaces, Grassmannians, intersection theory, and the index theorem.

Useful tools: linear algebra.

## The topology behind the existence of an eversion of the sphere

by Pascal Lambrechts (Université Catholique de Louvain)

Abstract: A very famous result of Stephen Smale claims that it is possible to turn the sphere inside-out and one can find many movies on the internet describing such "eversions of the sphere". Interestingly enough, the classical proof of Smale's theorem is not an explicit construction of the eversion but a proof of existence which relies on the fact that a certain space is in a single piece (topologists say that the space is "connected"). We will explain the key ingredients of the proof of Smale's theorem as well as related results. On the way, we will study fundamental invariants like the "topological degree" which is related to counting the number of times a circle can wrap around itself but has far reaching generalisations and is ubiquituous in Mathematics.

# Schedule

 Monday, July 24 Tuesday, July 25 Wednesday, July 26 Thursday, July 27 Friday, July 28 9:30-10:00 opening 10:00-11:00 B. DUNDAS B. DUNDAS P. LAMBRECHTS D. DUGGER B. DUNDAS 11:00-11:30 coffee break coffee break coffee break coffee break coffee break 11:30-12:30 D. DUGGER D. DUGGER B. DUNDAS P. LAMBRECHTS D. DUGGER 12:30-14:00 lunch break lunch break B. DUNDAS (until 13.30) lunch break lunch break 14:00-15:00 P. LAMBRECHTS D. DUGGER P. LAMBRECHTS P. LAMBRECHTS 15:00-15:30 coffee break coffee break coffee break coffee break 15:30-17:00 PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION PROBLEM SESSION

Friday, July 28th at 6pm, 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 8pm at Restaurante Pano de Boca at Teatro Aberto, right next to the Gulbenkian Foundation.

# Mini-conference

On Saturday, July 29th, 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.

## Course assistants

Pedro Boavida () and Gustavo Granja ()

## 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.