## Thomas Poguntke Memorial WorkshopBarcelona (CRM)31 January — 1 February, 2019 |

Thomas Poguntke passed away on September 24, 2018, after a serious illness, just a few weeks after arriving in Barcelona. He was 28 years old and about to hand in his PhD thesis (University of Bonn) on higher Segal spaces, K-theory and Hall algebras, the main parts of which are available as two papers on the arXiv.

This workshop is dedicated to his memory and to his mathematics, which will live on.

- Tobias Dyckerhoff
- Imma Gálvez-Carrillo
- Gustavo Jasso
- Joachim Kock
- Mark Penney
- Claudia Scheimbauer
- Walker Stern
- Tashi Walde
- Matthew Young

**Preliminary list of participants:**
Jaume Aguadé,
Carles Broto,
Louis Carlier,
Guillermo Carrió,
Carles Casacuberta,
Natàlia Castellana,
Alex Cebrian,
Tobias Dyckerhoff,
Wilson Forero,
Imma Gálvez,
Javier Gutiérrez,
Matías del Hoyo,
Gustavo Jasso,
Joachim Kock,
Mikel Lluvia,
Jesper M. Møller,
Mark Penney,
Detlev Poguntke,
Sune Precht Reeh,
Ricard Riba,
Albert Ruiz,
Claudia Scheimbauer,
Jérôme Scherer,
Walker Stern,
Andy Tonks,
May Underhill-Proulx,
Tashi Walde,
Matthew Young.

CLICK ON THE 2-SIMPLEX TO SHOW OR HIDE THE ABSTRACT

— T H U R S D A Y — | |

Thursday 14:35—14:40 | Welcome |

Thursday 14:40—15:40 |
▼ Imma Gálvez-Carrillo: Decomposition spaces and symmetric functions
Abstract: I will show how to use decomposition spaces to give purely combinatorial constructions of various Hopf algebras of symmetric functions (Sym, QSym, NSym, FQSym, WQSym, etc.). The constructions are objective and do not make reference to infinite alphabets or inductive limits of polynomial rings. |

Thursday 15:40—16:20 | Coffee Break |

Thursday 16:20—17:20 |
▼ Mark Penney: Independence of parabolic induction from the perspective of higher Segal spaces
Abstract: In a soon-to-appear joint paper with Adam Gal and Lena Gal we provide a new, groupoid-based proof of the independence of parabolic induction on the choice of parabolic subgroup for finite groups of Lie type. In this talk I will discuss how this proof, at least in the type A case, naturally arises in the theory of double 2-Segal spaces. |

Thursday 17:30—18:30 |
▼ Joachim Kock: Operadic categories and 2-Segal spaces
Abstract: I will give a characterisation of Batanin and Markl's operadic categories in terms of the decalage comonad, and explain how discrete 2-Segal spaces are a special case of operadic categories. This leads to an interesting result about 2-Segal spaces. This is joint work with Richard Garner and Mark Weber. |

Thursday evening | (probably we go out to find some light tapas,
and not too late) |

— F R I D A Y — | |

Friday 9:30—10:30 |
▼ Tobias Dyckerhoff I: Higher Segal structures in algebraic K-theory
Abstract: The classical S-construction is a simplicial space which lies at the heart of Waldhausen's approach to algebraic K-theory. In this talk, we discuss a higher version of the S-construction as introduced and studied in Thomas' thesis work. In particular, we explain its relevance for algebraic K-theory and the theory of higher Segal spaces. |

Friday 10:30—11:00 | Coffee Break |

Friday 11:00—12:00 |
▼ Gustavo Jasso: Representation-theoretic aspects of the higher Waldhausen S-constructions
Abstract: In this talk I will present a representation-theoretic viewpoint on the higher Waldhausen S-constructions of Dyckerhoff and Pogunkte. More precisely, I will explain how they relate to Iyama's higher Auslander algebras of type as well as some interesting consequences of this observation. This is part of a joint work with Tobias Dyckerhoff and Tashi Walde. |

Friday 12:10—13:10 |
▼ Tashi Walde: Stable higher Segal structures and the Dold-Kan correspondence
Abstract: We classify higher Segal objects in abelian categories and stable infinity-categories in terms of horn-filling conditions and the Dold-Kan correspondence. We explain how we expect to categorify this story by studying 2-categorical horn-filling conditions and Segal objects in the (infinity,2)-category of stable infinity-categories via Dyckerhoff's categorified Dold-Kan correspondence. This talk is based on joint work in progress with Tobias Dyckerhoff and Gustavo Jasso. |

Friday 13:10—14:30 | Lunch Break |

Friday 14:30—15:30 |
▼ Tobias Dyckerhoff II: Equivariant motivic Hall algebras
Abstract: Hall algebras have come to play an important role in the study of counting invariants in the context of moduli problems equipped with stability conditions. In this talk, we discuss Thomas' construction of a new flavor of Hall algebra, the equivariant motivic Hall algebra, and explain how it can be used to obtain new Harder-Narasimhan type recursion formulas. |

Friday 15:40—16:40 |
▼ Walker Stern: 2-Segal objects and algebras in spans
Abstract: In this talk, we discuss relationships between algebraic conditions in infinity-categories of spans in C and the 2-Segal conditions in C. In particular, we describe equivalences relating associative algebras to 2-Segal simplicial objects and Calabi-Yau algebras to 2-Segal cyclic objects. These equivalences have consequences for the construction of open topological field theories. |

Friday 16:40—17:00 | Coffee Break |

Friday 17:00—18:00 |
▼ Matthew Young: Degenerate versions of Green's theorem for Hall modules
Abstract: An interesting class of modules over Hall algebras arises from
the ℛ |

Friday 18:10—19:10 |
▼ Claudia Scheimbauer: The Waldhausen construction and 2-Segal spaces
Abstract: Waldhausen's S-construction gives examples of unital 2-Segal objects, aka decomposition spaces. I will explain a generalization thereof, which leads to an equivalent description in terms of its input data, namely certain "stable" double (homotopical) categories. The adjoint is given by a "totalization", an idea that was also present in Thomas Poguntke's work. Moreover, this generalized S-construction recovers the previous well-known S-constructions for stable ∞-categories (Barwick, Blumberg-Gepner-Tabuada, and Dyckerhoff-Kapranov), proto-exact ∞-categories, and the relative Waldhausen construction for exact functors. |

Friday evening | We plan to go out for dinner |

Last updated: 2019-01-30 by Joachim Kock.