# Spring School on Algebraic Topology (2014)

We have had another Spring School on Algebraic Topology in March, 2014.
The main topic of the second spring school is $$K$$-theory of topological spaces''. In the survey at the end of the previous spring school, $$K$$-theory was the most popular topics. There are, however, many kinds of $$K$$-theories, such as $$K$$-theory of topological spaces, algebraic $$K$$-theory, and $$K$$-theory for operator algebras. Because of the time constraint, we decided to focus on $$K$$-theory of topological spaces. $$K$$-theory of topological spaces itself is a huge subject, including the original construction by Atiyah and Hirzebruch, Atiyah's Real $$K$$-theory, equivariant $$K$$-theory, Morava $$K$$-theory, twisted $$K$$-theory, differential $$K$$-theory, and so on. We chose some of topics among them.
We also had a students' seminar based on a paper by Hohnhold, Stolz, and Teichner. Each of six students chose one of eight constructions of topological $$K$$-theory in the paper by Hohnhold-Stolz-Teichner and explained to other students.
The following is a report.
• Date: From March 6 to 8 in 2014
• Venue: Department of Mathematical Sciences, Shinshu University
• Talks: K-theory of topological spaces
• Lecturers:
• Kiyonori Gomi (Shinshu University)
• Takeshi Torii (Okayama University)
• Dai Tamaki (Shinshu University)
• Students' seminar:
Hohnhold, Henning; Stolz, Stephan; Teichner, Peter. From minimal geodesics to supersymmetric field theories. A celebration of the mathematical legacy of Raoul Bott, 207--274, CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, 2010. MR2648897
This paper compares eight spaces which represent $$K$$-theory. It is showen that all these spaces are homotopy equivalent. Each speaker chose one of constructions and explained an idea of the construction in about half an hour. This paper can be downloaded from here.
• We are pleased that we could support local expenses of all students.
• Talks:
• March 6th
• 13:30-14:50 Overview (Tamaki)
• 15:30-17:00 Basics of $$K$$-theory of topological spaces (Gomi)
• (Supper)
• 19:30-21:40? Students' seminar
• March 7th
• 9:00-10:30 $$K$$-theory as a cohomology theory (Torii)
• 11:00-12:30 Sample computations of $$K$$-theory (Gomi)
• 12:30-12:45 Group photo
• (Lunch)
• 13:30-15:00 Cohomology theories related to $$K$$-theory
• 15:30-17:00 Variants of $$K$$-theory (Gomi)
• (Supper)
• 19:30-22:00? Students' seminar
• March 8th
• 9:00-10:30 Cohomology theories with higher periodicities (Torii)
• 11:00-13:30 Problem session
• Lecture notes: The following scanned copy of lecture notes was taken by Toshiki Aoki, who was a junior at Shinshu University.
• Participants:
• Students: 53, others 23, total 76.
• Affiliations of students: Hokkaido University 2, Tohoku University 1, The University of Tokyo 18, Tokyo Institute of Technology 4, Waseda University 1, Seikei University 1, Tokyo University of Science 1, Gakushuin University 1, Shinshu University 9, Nagoya University 7, Kyoto University 3, Osaka University 2, Shimane University 1, Kochi University 1, Kyushu University 1: total 53.
• Undergraduates 22 (sophomore 2, juniour 11, seninor 9), graduates 31 (M1 9, M2 7, D1 6, D2 3, D3 6), total 53.
The number of students increased from 47 to 53, compared to the previous meeting. The number of juniors increased a lot. The number of universities also increased from 13 to 15.
This meeting was partially supported by the grant Plan the FIRST'' from Shinshu University.