# Operations in Highly Structured Homology Theories (16w5108)

Arriving in Banff, Alberta Sunday, May 22 and departing Friday May 27, 2016

## Organizers

(University of Illinois, Urbana-Champaign)

(University of Glasgow)

## Objectives

Our programme is primarily intended to bring together algebraic topologists actively developing or using techniques associated with highly structured ring spectra and their associated (co)homology theories. Given the wide range of rapidly developing activity across the subject, a specialist meeting of this type in the near future is desirable and likely to have a significant impact through cross-fertilization between different parts of the subject.

We aim to bring together a wide ranging group of mathematicians including a mixture of internationally leading experts on the relevant parts of algebraic topology and \emph{younger topologists} (late in doctoral training or beginning postdoctoral activity) who will benefit from exposure to state of the art talks and interaction with more established mathematicians. We will ensure that the talks include a kernel of high quality expository or overview talks aimed at introducing people to subject areas and encouraging future research activity. We will also encourage talks by younger people to highlight current activity by future subject leaders.

In particular the following broad topics will be covered.

• Homology and cohomology theories of structured ring spectra. Topological André -Quillen theories for $H_\infty$ and $E_\infty$ algebras and Hochschild theories. Generalizations to $E_n$-algebras, such as chiral homology.

• Operations. Classical operations associated with $H_\infty$ and $E_\infty$ ring spectra. Chromatic analogues, including operations for elliptic cohomology theories and Lubin-Tate theories. Analogues for $H_\infty$ and $E_\infty$ algebras; operations in TAQ and THH.

• Calculational aspects. Obstructions theories for operadic structures; stable homotopy calculations (classical and chromatic). Classification of $H_\infty$, $E_\infty$, and $E_n$ structures on chromatic spectra such as $BP$ and $BP\langle n\rangle$. Construction of structured orientations for such theories.

• Equivariant stable homotopy. Analogues and generalizations of $H_\infty$ and $E_\infty$ ring structures for $G$-spectra. Global equivariant spectra and ultracommutative ring structures.

• Motivic stable homotopy. Existence of $H_\infty$ and $E_\infty$ ring structures for motivic theories.

The size of the workshop with about 40 participants seems likely to enhance the chances of interactions between small groups of individuals and we expect that significant collaborative research will be carried out at the workshop or as a direct result of it.

The programme of the meeting would include a series of major invited talks or short series of talks surveying subject areas, more specialized talks on current research by individual participants selected from titles offered after acceptance of invitations to attend, and talks by early-career participants together with informal discussion sessions and problem sessions. We believe that it is particularly important not to overload the programme, as a full schedule leaves little time for less structured but often very productive interactions between individuals. Therefore we propose a maximum of 5 one hour or half-hour talks per day, with adequate breaks in between to allow for discussion and audience reaction.

We have already informally sounded out a number of people and there has been an enthusiastic reaction to the idea of a meeting of this type. We have no doubt that we would attract the participation of an impressive range of leading mathematicians and strong young people. The lists below give an indication of established mathematicians whom we would consider inviting, and inevitably this excludes very young people that we are not yet aware of but might well invite if the programme goes ahead.