# Spring School on Algebraic Topology (2016)

We have had the fourth Spring School on Algebraic Topology in March, 2016.
The main theme of the fourth meeting was operads.
The following is a report of the school.
• Date: From March 6 to 9 in 2016
• Venue: Lecture room 1, Faculty of Science, Shinshu University
• Contents: Student's seminar and lectures on operads
• Lecture:
• Student's seminar: We had student's seminar, as we did last year. The following four topics were chosen from seven topics suggested by Prof. Vallette.
1. Loop spaces
2. Homotopy associative algebras
3. Differential graded categories and $$A_{\infty}$$-categories
4. The homology of fibered spaces via twisted tensor product
• Schedule:
• March 6h
• 13:00-18:30 Students' seminar:
1. Naruki Masuda (The University of Tokyo): Loop spaces
2. Shun Wakatsuki (The University of Tokyo): Homotopy associative algebras
3. Shunsuke Kanou (Tokyo Institute of Technology): Differential graded categories and $$A_{\infty}$$-categories
• March 7th
• 10:00-11:30 Students' seminar:
1. Kenji Tobishima (Shinshu University): The homology of fiber spaces via twisted tensor product
• (Lunch)
• 14:00-15:30 Lecture (1) Kosuzul duality for algebras
• Contents: homological algebra for associative algebras and coassociative coalgebras, twisting morphisms, bar and cobar construction, Koszul duality.
• Group photo
• 19:00- Banquet (Yamazato)
• March 8th
• 10:00-11:30 Lecture (3) Homological algebra for operads
• (Lunch)
• 14:00-15:30 Lecture (4) Koszul duality theory for operads
• Contents: Koszul duality, the examples of As and Com-Lie, rewriting method.
• 16:00-17:30 Lecture (5) Homotopical algebra
• Contents: homotopy algebras (definitions and infinity-morphisms), homotopy transfer theorem, rectification.
• March 9th
• 10:00-11:30 Lecture (6) Applications
• Contents: deformation theory (Kontsevich formality of Poisson manifolds), moduli spaces of genus 0 curves and little disks operads.
• Video and Photo: Norihiko Minami at Nagoya Institute of Technoology took videos and photos during the school. I plan to upload them soon.
• Participants:
• Students 45, non-students 44, total 89.
• Affiliations of students: Hokkaido University 1, Tohoku University 2, The University of Tokyo (math) 14, Kavli IPMU 1, Tokyo Institute of Technology 3, Tokyo University of Science 2, Waseda University 1, Keio University 1, Shibaura Institute of Technology 2, Shizuoka University 1, Shinshu University 4, Nagoya University 7, Ritsumeikan University 1, Kyoto University 1, Osaka University 2, Yamaguchi University 1, University of the Ryukyus 1: total 45.
• Undergraduates 11 (freshman 0, sophomore 1, junior 8, senior 2), graduates 32 (M1 7, M2 6, D1 3, D2 6, D3 10, unknown 2) total 45.
The number of participants is slightly smaller than the previous school, but 89 is still a big number.
• Impression: During student's seminar, they wanted to talk more than I expected. Each student talked for more than 90 minutes. Participants enjoyed the series of lectures by Prof. Vallette very much, which was well structured with clear goal. Before the school began, I wondered if participants can ask questions in English, since most Japanese are shy and afraid of speaking in English. It turned out that we had many questions and discussions, which made every lecture go overtime for half an hour.
This meeting is supported by Dean's research grant from Faculty of Science, Shinshu University and JSPS KAKENHI Grant Number 15K04870.