# TopICS: Topology Intercity Seminar

## Utrecht-Nijmegen

TopICS is the **Top**ology **I**nter**c**ity **S**eminar joint between Utrecht University and Radboud University, Nijmegen.

If you wish to receive seminar announcements and/or participate in the seminar please subscribe to the **mailing list****.**

The schedule for the 2020 edition of the seminar can be found **here**, and the schedule for earlier versions is available **here**.

# Autumn 2022 Schedule

**Thursday 6 October UTRECHT BBG069 @14:00-16:00**

Tobias Lenz (UU)

**Title:** G-global homotopy theory.

**Abstract:** In this talk I will introduce G-global homotopy theory as a synthesis of classical G-equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. I will then give an overview of several applications of the G-global theory to the study of purely equivariant or global questions, in particular regarding the corresponding notions of "coherently commutative monoids." Part of this is joint work in progress with Bastiaan Cnossen and Sil Linskens as well as with Michael Stahlhauer.

**Thursday 20 October UTRECHT BBG069 @15:00-16:00**

Max Blans (UU)

**Title:** Multiplicative structures on Moore spectra.

**Abstract:** The construction of coherent multiplicative structures on quotients in higher algebra is delicate and often impossible. It has for instance been known since the 70s that the mod p Moore spectra S/p do not admit an E_1 multiplication. Rather surprisingly, Robert Burklund showed in a recent preprint that a wide class of quotients do admit coherent multiplications. To name a few examples, he proves that S/8 admits an E_1-ring structure and S/p^{n+1} admits an E_n-ring structure for p an odd prime. The proof of these results makes use of an obstruction theory carried out in the category of synthetic spectra. I will explain this proof in my talk.

**Thursday 10 Novemeber UTRECHT BBG075 @15:00-17:00**

Lukas Brantner ( CNRS (Orsay) & University of Oxford)

**Title:** On deformations and lifts of Calabi-Yau varieties in characteristic p

**Abstract:** A smooth projective variety Z is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how Calabi-Yau varieties in characteristic p deform. More precisely, we show that if Z has degenerating Hodge–de Rham spectral sequence and torsion-free crystalline cohomology, then its mixed chracteristic deformations are unobstructed; this is an analogue of the classical BTT theorem in characteristic zero. If Z is ordinary, we show that it moreover admits a canonical lift to characteristic zero; this extends classical Serre-Tate theory. Our work generalises results of Achinger–Zdanowicz, Bogomolov-Tian-Todorov, Deligne–Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre–Tate, and Ward.

**Thursday 24 November UTRECHT MIN 2.06 @14:00-16:00**

Fabian Hebestreit (Münster)

**Title:** The stable cohomology of symplectic groups of the integers

**Abstract:** I want to report on a calculation of the so-called stable part of the cohomology of symplectic groups over the integers, in particular at the prime 2. The approach is via the group completion theorem, which relates this stable part to sympletic K-groups of the integers. The latter has recently seen advances in the case of general number rings and I will explain how these can be brought to bear. This is joint with M.Land and T.Nikolaus.

**Tuesday 6 December NIJMEGEN HG00.308 @15:30-17:30**

**Title:** A recognition principle for iterated suspensions as coalgebras over the little cubes operad

**Abstract:** In this talk I will discuss and prove a recognition principle for iterated suspensions as coalgebras over the little disks operad. This is based on joint work with Oisín Flynn-Connolly and José Moreno-Fernández.

**Luc Illusie (Paris): Lectures on the de Rham Complex**

**Thursday 15 December UTRECHT, KGB Pangea @ 16:00-17:00**

**Title: **A brief historical survey.

*This talk will be followed by drinks in the HFG library.*

**Friday**** 1****6**** December UTRECHT, KGB Pangea @ 1****1****:00-1****2****:00**

**Title:** De Rham complexes in mixed characteristic I.

**Friday 16 December UTRECHT, KGB Pangea @ 1****4****:00-1****5****:00**

**Title:** De Rham complexes in mixed characteristic II.

# Spring 2022 Schedule

**Thursday 3 March UTRECHT BBG.023 @17:15-18:15**

**Title:** Condensed Mathematics Reading Seminar

**Thursday 17 March UTRECHT BBG.017 @17:00**

**Title:** Condensed Mathematics Reading Seminar

***Monday 28 March* NIJMEGEN HG02.032 @*10:30***

**Title:** Condensed Mathematics Reading Seminar

**Thursday 14 April UTRECHT BBG.023 @17:00**

**Title:** Condensed Mathematics Reading Seminar

**Thursday 28 April UTRECHT BBG.023 @17:00**

**Title:** Condensed Mathematics Reading Seminar

**Title:** Homotopy coherent nerve for (∞,n)-categories

**Abstract:** In joint work with Nima Rasekh and Martina Rovelli, we are developing a new approach to limits in an (∞,2)-category, by defining them as terminal objects in the corresponding double (∞,1)-category of cones. To verify that this gives the correct universal property, we need to compare our definition to the established definition of limits in an (∞,2)-category seen as a category enriched in (∞,1)-categories. The difficulty of this comparison arises in the fact that there is no direct Quillen equivalence between the (∞,2)-categorical models of categories enriched in complete Segal spaces and 2-fold complete Segal spaces.

As a first step towards the comparison, we construct a direct Quillen equivalence between the above mentioned models. This construction is not specific to the case n=2 and we in fact obtain a direct Quillen equivalence between categories enriched in complete Segal Theta-n-spaces and complete Segal objects in Theta-n-spaces, which both model (∞,n+1)-categories. In particular, this construction generalizes the homotopy coherent nerve from Kan-enriched categories to quasi-categories.

**Thursday 12 May UTRECHT BBG.023 @17:00**

**Title:** Condensed Mathematics Reading Seminar

**Title:** Global homotopy theory via partially lax limits

**Abstract:** Global homotopy theory is the study of equivariant objects which exist uniformly and compatibly for all compact Lie groups in a certain family, and which exhibit extra functoriality. In this talk I will present new infty-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limit to formalize the idea that a global object is a collection of G-objects, one for each compact Lie group G, which are compatible with the restriction-inflation functors. This is joint work with Sil Linskens and Denis Nardin.

**Thursday 2 June NIJMEGEN HG03.054 @ 15:30**

**15:45-16:45 **

**Title:** Condensed Mathematics Reading Seminar

**17:00-18:00 **

**Title:** Global equivariant Thom spectra.

**Abstract:** The aim of this talk is to explain a systematic formalism to construct and manipulate Thom spectra in global equivariant homotopy theory. The upshot is a colimit preserving symmetric monoidal global Thom spectrum functor from the infinity-category of global spaces over BOP to the infinity-category of global spectra. Here BOP is a particular globally-equivariant refinement of the space Z x BO, which simultaneously represents equivariant K-theory for all compact Lie groups.

I plan to give two applications of the formalism. Firstly, a specific and much studied morphism mU--> MU between two prominent equivariant forms of the complex bordism spectrum is a localization, in the infinity-category of commutative global ring spectra, at the ‘inverse Thom classes’. Secondly, by joint work with Gepner and Nikolaus, the infinity-category of global spectra can be describe as a pushout of parameterized symmetric monoidal infinity categories along the global Thom spectrum functor.

**Thursday 23 June NIJMEGEN HG03.085 @ 11:00-13:00**

**Title:** Condensed Mathematics Reading Seminar

**Title:** A descent principle for compact support extensions of functors.

**Abstract:** A characteristic property of compact support cohomology is the long exact sequene which connects the compact support cohomology groups of a space, an open subspace and its complement. Given an arbitrary invariant of, say, algebraic varieties, taking values in a stable infinity category C, one can wonder when it makes sense to define a "compact support" version of this invariant, such that this long exact sequence exists by construction. In this talk, I give an answer in terms of an equivalence of categories of C-valued sheaves on certain sites of algebraic varieties. I will discuss some applications of this result, and, if time permits, speculate about some related things that I haven't proven yet.

# Autumn 2021 Schedule

**9 November UTRECHT HFG 6.11 @ 15:30-18:00**

**Title:** A spherical HKR Theorem

**Abstract:** The classical Hochschild-Kostant-Rosenberg theorem identifies Hochschild homology of a commutative ring which is smooth over the base field with its de Rham complex. In this talk, we provide a generalisation to topological Hochschild homology of commutative ring spectra, replacing the de Rham complex by an "η-deformed de Rham complex" which incorporates the E_∞ structure. (Joint with Thomas Nikolaus).

**Title:** Infinity-operads as analytic monads

**Abstract:** Joyal proved that symmetric sequences in sets (or “species”) can be identified with certain endofunctors of Set, namely the “analytic" functors. Under this identification, the composition product on symmetric sequences corresponds to composition of endofunctors, and this allows us to identify operads in Set with certain “analytic” monads. Moreover, the monad corresponding to an operad O is precisely the monad for free O-algebras in Set. In this talk I will explain how to obtain an analogous identification for infinity-operads: assigning to an infinity-operad O (in Lurie’s sense) the monad for free O-algebras in spaces identifies infinity-operads with analytic monads. This builds on previous work with Gepner and Kock where we developed the theory of analytic monads in the infinity-categorical setting.

**23 November NIJMEGEN HG01.028 @ 15:30-18:00 **

**Title:** Log Hochschild homology via the log diagonal

**Abstract:** Log geometry is a variant of algebraic geometry in which mildly singular varieties can be treated as if they were smooth. Rognes has extended the definition of Hochschild homology to allow for log rings - the affine schemes of log geometry - as input.

As the Hochschild--Kostant--Rosenberg theorem identifies the Hochschild homology of smooth rings with its de Rham complex, it is natural to ask whether Hochschild homology of log smooth log rings are related to the log de Rham complex. I will give a reformulation of Rognes' definition that will allow us to tackle this problem in much the same way as for ordinary Hochschild homology.

Parts of the talk will be based on joint work with Binda--Park--Østvær.

**Title:** The straightening theorem

**Abstract:** Lurie’s straightening theorem is one of the cornerstones of ∞-category theory. It provides an equivalence between functors from an ∞-category C valued in Cat_∞, the ∞-category of small ∞-categories, and particular kinds of ∞-categories fibered over C called cocartesian fibrations. This equivalence gives an efficient way of writing down Cat_∞-valued functors via these fibered ∞-categories, which are otherwise hard to write down directly because of the coherence issues one then has to deal with. In this talk we will see a handful of applications of this straightening construction, and we will give an outline of a new proof of the straightening theorem (this is joint work with Fabian Hebestreit and Gijs Heuts).