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Young Topologists’ Meeting 2015

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6–10 July 2015

EPFL, Ecublens, Switzerland

EPFL, Ecublens, Switzerland

The Young Topologists Meeting is an annual conference for graduate students, recent PhDs and generally young researchers in topology. It is organised in turns by the EPFL and the University of Copenhagen. The 2015 edition hosted at EPFL is already the ninth in this series.

You can find the conference booklet here: Booklet

And the conference’s group picture here: Group Picture

### Invited Lecturers

Apart from the presentations given by the participants, there were minicourses consisting of several lectures. The 2015 invited lecturers were Gunnar Carlsson of Stanford University and Emily Riehl of Harvard University.

### Gunnar Carlsson’s Minicourse

#### Videos:

Lecture 1, Lecture 2, Lecture 3, Lecture 4

#### Title: Methods of applied topology

#### Abstract:

There is a lot of discussion around the topic of “Big Data”, which means the study of large and complex data sets. From a mathematical point of view, these data sets typically have the structure of finite metric spaces, sometimes in numerous ways. The problem of analyzing these data sets is a very important one in the sciences, engineering, and commerce. Any kind of organizing principle for them would be a major contribution to the solution of these problems. It turns out that topology is a very useful such principle, both via versions of complex constructions as well as homology. This course will discuss these ideas, with numerous examples, and will as provide instruction on the use of persistent homology software.

### Emily Riehl’s Minicourse

#### Videos:

Lecture 1, Lecture 2, Lecture 3, Lecture 4

#### Title: Infinity category theory from scratch

#### Abstract:

We use the terms ∞-*categories* and ∞-*functors *to mean the objects and morphisms in an ∞-*cosmos*: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of “fibrant objects.” Quasi-categories, Segal categories, complete Segal spaces, iterated complete Segal spaces, and fibered versions of each of these are all all ∞-categories in this sense. In joint work with Dominic Verity, we show that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the classical concept.

In the first lecture, we define an ∞-cosmos and introduce its *homotopy 2-category*, the strict 2-category mentioned above. We illustrate the use of formal category theory to develop the basic theory of equivalences of and adjunctions between ∞-categories. In the second lecture, we study limits and colimits of diagrams taking values in an ∞-category and relate these concepts to adjunctions between ∞-categories. In the third lecture, we define comma ∞-categories, which satisfy a particular weak 2-dimensional universal property in the homotopy 2-category. We illustrate the use of comma ∞-categories to encode the universal properties of (co)limits and adjointness. Because comma ∞-categories are preserved by all functors of ∞-cosmoi and reflected by certain weak equivalences of ∞-cosmoi, these characterizations form the foundations for “model independence’” results. In the fourth lecture, we introduce (co)cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. We then describe the calculus of *modules*, between ∞-categories — comma ∞-categories being the prototypical example — and use this framework to state and prove the Yoneda lemma and develop the theory of pointwise Kan extensions along ∞-functors.

### Schedule

The lectures and talks were held at the EPFL from Monday morning until Friday at noon, so that participants were able to catch a flight in the afternoon. A precise schedule is found in the conference booklet or the following PDF-file.

### Slides

Gabriel Angelini-Knoll

Cristina Anghel

Scott Balchin

Eduard Balzin

Hugo Bello

Kaj Börjeson

Michael Catanzaro

Celeste Damiani

Ivan Dungan

Philip Egger

Boštjan Gabrovšek

Oihana Garaialde

Sara Kalisnik

Zhen Lin Low

Cary Malkiewich

Nina Otter

Sune Reeh

Massimiliano Ungheretti

Thomas Wasserman

Thomas Weighill

David White

Felix Wierstra

Sarah Yeakel

### Registration

YTM 2015 is over. Owing to the generous donations by the Swiss Academy of Sciences, the Swiss Mathematical Society, the Conférence Universitaire de Suisse Occidentale and the École Polytechnique Fédérale de Lausanne, and the Stiftung zur Förderung der mathematischen Wissenschaften in der Schweiz, we were able to provide financial support to cover housing expenses. In addition, we were able to obtain a grant from the US National Science Foundation, which covered all travel and housing expenses of US-students.

### List of participants

We had over 180 registered participants. You can find the list list of participants here.

### Travel Information

There are three big airports in Switzerland – Basel, Geneva and Zurich – of which Geneva is the one closest to Lausanne. The travelling time by train to Lausanne is approximately 45 minutes from Geneva and 2.5 hours from the two other airports. You might also be able to get decent flights to Bern-Belp, which is a smaller airport 2 hours from Lausanne.

From within Europe there is of course always the option of travelling by train. Taking the TGV, it takes approximately 4 hours to get from Paris to Lausanne. You can find more information on the (international) train connections and Swiss public transportation in general on sbb.ch.

### Housing

We provided rooms in several student residences owned by the FMEL. They were available to our participants from Wednesday, July 1 until Wednesday, July 15, allowing for extended stays.

Due to the expected number of participants, additional rooms were booked at the Lausanne GuestHouse, which lies within walking distance from the train station.

### Organisers

Joseph Hirsh (principal investigator NSF), Rachel Jeitziner, Martina Rovelli, Kay Werndli and Dimitri Zaganidis.