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.

Loop spaces

Homotopy associative algebras

Differential graded categories and \(A_{\infty}\)categories

The homology of fibered spaces via twisted tensor product

Schedule:

March 6h

13:0018:30 Students' seminar:

Naruki Masuda (The University of Tokyo): Loop spaces

Shun Wakatsuki (The University of Tokyo): Homotopy associative algebras

Shunsuke Kanou (Tokyo Institute of Technology): Differential graded categories and \(A_{\infty}\)categories

March 7th

10:0011:30 Students' seminar:

Kenji Tobishima (Shinshu University): The homology of fiber spaces via twisted tensor product

(Lunch)

14:0015:30 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

16:0017:30 Lecture (2)
Operads

Contents:
various equivalent definitions of an operad, algebras over an operad, free operad, cooperads, nonsymmetric operads.

19:00 Banquet
(Yamazato)

March 8th

10:0011:30 Lecture (3)
Homological algebra for operads

Contents:
homological algebra for operads and cooperad, operadic twisting morphisms, operadic bar and cobar construction.

(Lunch)

14:0015:30 Lecture (4)
Koszul duality theory for operads

Contents:
Koszul duality, the examples of As and ComLie, rewriting method.

16:0017:30 Lecture (5)
Homotopical algebra

Contents:
homotopy algebras (definitions and infinitymorphisms), homotopy transfer theorem, rectification.

March 9th

10:0011:30 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, nonstudents 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.
