# Young Topologists Meeting 2017

The Young Topologists Meeting is an annual conference for graduate students, recent PhDs and other junior researchers in topology. Until now it has been organized in turns by the
EPFL, Switzerland, and the
University of Copenhagen, Denmark, but will be organized in 2017 by the departments of mathematics at
Stockholm University and the
Royal Institute of Technology in collaboration with Stockholm Mathematics Centre in Stockholm, Sweden. The Young Topologists Meeting 2017 will take place July 3-7 at the main campus of KTH Royal Institute of Technology in Stockholm.

The meeting is organized around mini courses by invited lecturers as well as presentations given by the participants. This year's mini courses will be given by
Benson Farb (University of Chicago) and
Brooke Shipley (University of Illinois at Chicago).

### Benson Farb's Minicourse

### Title: Topology interacting with number theory, representation theory and algebraic geometry

**Abstract: **

In these 4 lectures I hope to exhibit algebraic topology not as an isolated topic, but as a powerful viewpoint that is part of a rich and beautiful circle of ideas. In particular I will describe some connections to the classical theory of algebraic functions, to representation theory, to algebraic geometry and to number theory.

**Lecture 1: Braids, polynomials and Hilbert's (still open) 13th problem from a topological point of view**

There are still many fundamental open questions about polynomials, including Hilbert's 13th problem (widely believed (incorrectly) to be solved in 1957). In this talk I will explain the powerful point-of-view on such problems provided by topology. The topology of the space P(d) of monic, degree d, square-free polynomials is a central object here; it still has many mysteries to reveal. I will explain the close connections of P(d) with braid groups,algebraic functions, hyperplane complements, discriminants, and the cohomology of all of these objects.

**Lecture 2: Representation stability and FI-modules**

Many spaces or groups X_n have nontrivial automorphism groups G_n. In such cases, X_n almost never satisfies homological stability. I will explain the theory of representation stability and FI-modules, which gives a precise sense in which H_i(X_n) does stabilize, as a G_n-representation. This is a very active area of current research, with ideas coming in from (and out to) combinatorics, topology, arithmetic groups, geometric group theory, number theory, representation theory, and more.

**Lecture 3: Point counting for topologists**

In this lecture I will give a whirlwind tour of the amazing dictionary (proposed by Weil, built by Grothendieck, Artin, Deligne and many others) between the complex points of a variety and its points over a finite field. I will explain some very new entries in this dictionary, coming from homological stability and representation stability. I will also describe some open problems and new (and old) examples in topology that arise from this viewpoint.

**Lecture 4: How to make predictions in topology using number theory**

In this talk I will tell the story of how Melanie Wood, Jesse Wolfson and I were led to discover some amazing coincidences in topology purely by analogy with some classical analytic number theory. These coincidences are given in terms of a new topological concept: the ``homological density'' of one space in another. We have no explanation as to why these topological predictions end up being true. I will also explain why the following question is not completely crazy: why is the Riemann zeta function evaluated at n+1 like the 2-fold loop space of projective n-space?

### Brooke Shipley's Minicourse

### Title: The equivalence of differential graded modules and HZ-module spectra, applications, and generalizations

**Abstract:**

This sequence of lectures will explore the connections between the differential graded world and the spectral world. There will first be a brief introduction to model categories, stable homotopy, and symmetric spectra. Then we will discuss the equivalence between the homotopy theories of HZ-module (respectively, algebra) spectra and of differential graded modules (respectively, algebras or DGAs) where HZ here is the Eilenberg-Mac Lane spectrum associated with ordinary homology. We will then use this comparison to develop algebraic models of rational (equivariant) stable homotopy theories and to define topological equivalences of DGAs. In both of these applications we will discuss current on-going work; in the latter case, this uses a variant of Goerss-Hopkins obstruction theory to compute topological equivalences. As time permits we will also discuss extensions of the original comparison to the commutative (or E-infinity) case and to co-modules and co-algebras.

### Giving a talk

We strongly encourage the participants to give a talk. The talk does not have to contain original research results, expository talks are also welcome. If you are interested in giving a talk, please indicate this on the registration form along with a title and abstract. The talks should be 30 minutes in total, 25 minutes of presentation and 5 minutes for questions.

### Support

We will be able to cover accommodation for approximately 100 participants. In addition, we are applying for NSF funding to cover travel expenses for participants from US institutions. We are very thankful to Sarah Yeakel for being the principal investigator for the NSF application.

### Registration

The deadline for registration is closed. If you have registered, we will send an e-mail to you with more information. If you have applied for accommodation, an e-mail will be sent out giving information whether we were able to cover your accommodation or not.

### Accessibility

Participants requiring disability accommodations for the conference should contact the organizers as soon as possible.

### Venue

The conference will take place at the main campus of KTH Royal Institute of Technology. The lecture halls are D1 and D2, and lie in "D-Huset", Lindstedtsvägen 17. Follow this link for a map which shows you how to get to D1.

### Contact

If you have any questions, please contact us at youngtopologistsmeeting@gmail.com

### Organizers

- Eric Ahlqvist (ericahl@kth.se)
- Hadrien Espic (hadrien@math.su.se)
- Kaj Börjeson (kaj@math.su.se)
- Magnus Carlson (macarlso@kth.se)
- Matthias Grey (mgrey@math.su.se)
- Oliver Gäfvert (oliverg@kth.se)
- Petra Poklukar (petrap@math.su.se)
- Bashar Saleh (bashar@math.su.se)
- Ben Ward (bward@math.su.se)
- Stephanie Ziegenhagen (szie@kth.se)

### Sponsors

We gratefully acknowledge financial support from:

- Marcus Wallenberg Foundation for International Scientific Collaboration
- Stockholm Mathematics Centre
- Swedish Mathematical Society
- Vetenskapsrådet
- SVeFUM
- The Foundation Compositio Mathematica