# Spring School on Algebraic Topology (2015)

We have had another Spring School on Algebraic Topology in March, 2015.
In the survey at the end of the second spring school, $$\infty$$-categories and homotopy groups of spheres ranked first. We chose $$\infty$$-categories'' as the main theme of the third meeting. The homotopy groups of spheres are interesting, but it is not easy to make a good plan for a lecture series on this topic.
The following is a report.
• Date: From March 3 to 6 in 2015
• Venue: Department of Mathematical Sciences, Shinshu University
• Contents: Students' seminar and lectures on $$\infty$$-categories
• Students' seminar: During the second meeting, we had late-night students' seminars on a rather difficult paper, which was really hard. We decided to have students' seminar before lecture series, with the aim of helping understanding lectures. It has the following two objectives:
1. To understand the relationship between the structures of higher categories and homotopy. Hopefully to get convinced that homotopy theory is the right language for $$\infty$$-categories.
2. To get familiar with the language of simplicial sets and model categories used in lectures.
Seven students volunteered to give talks on these topics. The contents of their talks were suggested by this document.
• Lectures: From the afternoon of March 4th to the morning of March 6th by the following lecturers.
• Isamu Iwanari (Tohoku University)
• Takuo Matsuoka
• Schedule:
• March 3rd
• 13:00-14:00 Students' seminar (1) Homotopy theory of simplicial sets (Kenji Tobishima, Saitama University)
• 14:20-15:20 Students' seminar (2) Nerve functor (Shunsuke Kano, Chuo University)
• 15:40-16:40 Students' seminar (3) Simplicial sets and topological spaces (Naoya Suzuki, Nagoya University)
• 17:00-18:00 Students' seminar (4) Model categories (Genki Sato, The University of Tokyo)
• March 4th
• 9:00-10:00 Students' seminar (5) Cofibrantly generated model structures (Shun Wakatsuki, The University of Tokyo)
• 10:20-11:20 Students' seminar (6) The model structure on the category of simplicial sets due to Quillen (Tomomi Fujita, Tokyo Institute of Technology)
• 11:40-12:40 Students' seminar (7) The model structure on the category of simplicial sets due to Joyal (Jun Yoshida, The University of Tokyo)
• (Lunch)
• 14:00-15:30 Lecture (1) From categories and $$\infty$$-groupoids to $$\infty$$-categories (Takuo Matsuoka).
• 16:00-17:30 Lecture (2) Elementary concepts in $$\infty$$-categories (Isamu Iwanari).
• 19:00- Banquet
• March 5th
• 9:00-10:30 Lecture (3) Cartesian fibrations and marked simplicial sets (Takuo Matsuoka)
• 11:00-12:30 Lecture (4) Monoial structure and algebra (Takuo Matsuoka)
• (Lunch)
• 14:00-15:30 Lecture (5) Stable $$\infty$$-categories (Isamu Iwanari)
• 16:00-17:30 Lecture (6) Stable $$\infty$$-categories, continued (Isamu Iwanari).
• March 6th
• 9:00-10:30 Lecture (7) Advanced applications (Isamu Iwanari).
• 11:00-12:30 Lecture (8) Extended TQFT (Takuo Matsuoka).
• Lecture notes: Ye Liu, a graduate student at Hokkaido University, kindly offered me a note he took. It can be downloaded from here.
• Banquet: We had late-night students' seminar instead of banquet in previous meeting. It was fun but many people also wanted to have a banquet. We decided to have a banquet. We had much more than we expected in the banquet, surprising 74 people.
• Participants:
• Students 66, others 34, total 100 (!!).
• Affiliations of students: Hokkaido University 1, Tohoku University 1, The University of Tokyo (math) 24, The University of Tokyo (computer science) 1, Tokyo Institute of Technology 6, Tokyo Metropolitan University 1, Tokyo University of Science 2, Waseda University 1, Chuo University 1¿Í, Shibaura Institute of Technology 1, Saitama University 3, Shinshu University 6, Nagoya University 9, Kyoto University 2, Osaka University 3, Shimane University 1, Kochi University 1, Kyushu University 2: total 66.
• Undergraduates 27 (freshman 1, sophomore 0, junior 13, senior 13), graduates 39 (M1 9, M2 6, D1 11, D2 7, D3 5, other 1) total 66.
We had never expected to have a hundered people for this advanced topic. It is also amazing that the bulk of participants consists of juniors and seniors. The number of universities increased from 15 to 18, although many students are from the University of Tokyo.
This meeting is supported by two research grants from Shinshu University: PLAN the FIRST'' strategic research grant (Kazuyuki Inoue) and Dean's research grant from the Faculty of Science, Shinshu University (Dai Tamaki).