# Abstracts for Conference on "Algebraic Topology, in memory of Hans-Joachim Baues", October 17 - 21, 2022

Alternatively have a look at the program.

## The Hill-Lawson spectral sequence and the telescope conjecture

In a 2021 paper on operadic tensor products, Hill and Lawson introduced a new spectral sequence converging to the stable homotopy groups of spheres. It may be useful for studying the telescope conjecture.

## Seiberg-Witten Floer Theory and Twisted Parametrised Spectra

Seiberg-Witten theory has played a central role in the study of smooth low-dimensional manifolds since their introduction in the 90s. Parallel to this, Cohen, Jones, and Segal asked the question of whether various types of Floer homology could be upgraded to the homotopy level by constructing (stable) homotopy types encoding Floer data. In 2003, Manolescu constructed Seiberg-Witten Floer spectra for rational homology 3-spheres, and in particular used these to settle the triangulation conjecture once and for all.

## Cosupport in tensor-triangulated geometry

We will give an introduction to the theory of cosupport in tensor-triangulated geometry, dual to the Balmer–Favi notion of support. This gives a method of classifying the colocalizing coideals of a tensor-triangulated category in terms of subsets of its spectrum of prime ideals. We will explain how this recovers and unifies a number of classification theorems in the literature, and give some new examples. This is joint work with Tobias Barthel, Natalia Castellana, and Beren Sanders

## Traces and categorification

The trace of a linear operator is simple to define, yet it is a surprisingly interesting and mysterious operation. From characters of representations, through fixed-point formulas, to various geometric transfer maps it appears all over mathematics. The theory of oo-categories and higher algebra allows one to organize many of these occurrences of the trace within a formal unified calculus. However, this calculus itself is more intricate and elaborate than one might expect.

## Recollections of Hans-Joachim Baues

This is not a talk on the work of Hans-Joachim Baues. According to MathSciNet Baues has 110 publications, among them 9 books. I feel I cannot do justice to his complete work. Instead this will be a talk in which I discuss some of his work which had overlaps with my research interests and report about our mathematical interaction.

## The universal Hochschild shadow: from bicategories to $(\infty,2)$- categories

(joint with Nima Rasekh)

## Duality for infinity-operads

We describe a bar-cobar duality between presheaves and copresheaves on a category of trees, and show that it restricts to such a duality for infinity-operads and infinity-cooperads. For ordinary operads, it specializes to the classical cases studied by Getzler-Jones and Ginzburg-Kapranov. (The lecture is based on joint work with Eric Hoffbeck.)

## Embedding n-categories into ($\infty,n$)-categories

While weak versions of n-categories are much more widespread in nature, strict version allows for modelling specific shapes of higher categories. For n=2, we have a complete understanding of their relationship. After introducing the terms and setting up the stage, I will report about this and - time permitting - about work in progress towards analogous results for n>2. This is joint work with Martina Rovelli.

## Contributed talk: From the secondary Steenrod algebra to $M\xi$

## Contributed talk: On Merge Trees and Discrete Morse Functions on Paths and Trees

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