# NCMW - Global Homotopy Theory (2020)

## Speakers and Syllabus

Homotopy theory has turned out to be one of the most prominent topics in Algebraic Topology starting from the 1950s. The celebrated computations of Serre and Toda of homotopy groups of spheres led to the systematic calculation of cobordism groups by Thom, Milnor, Wall, Stong and others. From the 1970s, it was clear that the stable homotopy calculations are very important and bear a lot of applications to problems in differential topology.

The homotopy theory for G-spaces (called equivariant homotopy theory), that is spaces with an action of a specific group G, has also been of great interest. One of the starting points of the subject was the spherical space form problem which was finally solved by Madsen, Wall and Thomas.The analogous statement for a product of spheres is one of the major unsolved problems in this subject called the rank conjecture. The techniques in equivariant homotopy theory also give a nice proof of the Atiyah-Segal completion theorem, among many other such results. These techniques are further carried forward in the relatively new topic in this area : Global homotopy theory which was introduced by Stefan Schwede.

The lectures in the workshop will be on classical topics on equivariant homotopy theory and stable homotopy theory. These will be accompanied by lectures by Stefan Schwede on global homotopy theory. Part of the background material was part of an Advanced Instructional School on homotopy theory that took place at ISI Kolkata in May, 2019. A number of participants in the AIS have expressed interest for such a

programme.

**Target Audience:**

The School is meant for PhD students working in Geometry and Topology. The topics may also interest students working in algebra and algebraic geometry especially those who are interested in homotopy theoretic techniques in these subjects. The participants must be familiar with Algebraic topology (homology theory and cohomology theories, and homotopy theory). It would be helpful if the participants have attended the AIS on homotopy theory in May, 2019 at ISI, Kolkata.

**Names and affiliations of speakers.**

(1) Stefan Schwede, Mathematisches Institut Universit ̈at Bonn

(2) Rekha Santhanam (RS), IIT Bombay

(3) Debasis Sen(DS) IIT Kanpur

(4) Somnath Basu (SB1), IISER Kolkata

(5) Samik Basu (SB2), ISI Kolkata

## Time Table

Time Table |
||||||

10:00-11:00 | 11:30-12:30 | 12:30-2:00 | 2:00-3:00 | 3:15-4:15 | 4:30 - 5:30 | |

10/02 | SB1 Lecture 1 | SB1 Lecture 2 | L | DS Lecture 1 | SB2 Lecture 1 | Tutorial SS, SB1, DS |

11/02 | DS Lecture 2 | RS Lecture 1 | U | SB2 Lecture 2 | SS Lecture 1 | Tutorial SS, SB2, RS |

12/02 | RS Lecture 2 | SB2 Lecture 3 | N | SS Lecture 2 | SB1 Lecture 3 | Tutorial SS, SB2, RS |

13/02 | RS Lecture 3 | DS Lecture 3 | C | SS Lecture 3 | SB1 Lecture 4 | Tutorial SS, SB1, DS |

14/02 | SS Lecture 4 | SB2 Lecture 4 | H | SS Lecture 5 | DS Lecture 4 | Tutorial SS, SB2, RS |

(SS)Stefan Schwede : Global Homotopy Theory

(SB1) Somnath Basu: Unstable equivariant homotopy theory

(DS ) Debasis Sen : Stable homotopy theory and Equivariant Stable homotopy theory

(SB2) Samik Basu : Spanier Whitehead Duality, Computation of π0 G (S 0) and EQuivariant Eilenberg Maclane spectra

(RS) Rekha Santhanam : Γ-spaces, Equivariant orthogonal spectra and Geometric fixed points.