The theory of 2-Segal spaces, or decomposition spaces, has found a wide range of applications since its introduction over a decade ago. One of the early motivations, and a source of many interesting examples, was the recognition that applying Waldhausen's S-construction to an exact category produces a 2-Segal space. Since this construction plays a central role in algebraic K-theory, this result establishes 2-Segal spaces as closely linked to that theory. Subsequently, it was shown, in joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, that all 2-Segal spaces arise from applying the S-construction to more general structures known as augmented stable double Segal spaces. Indeed, they can be regarded as a kind of universal input for K-theory. Around the same time, Campbell and Zakharevich introduced CGW categories as another general input for algebraic K-theory, including examples that are not algebraic in nature. In a recent preprint with Shapiro and Zakharevich, we establish an explicit relationship between augmented stable double Segal spaces and CGW categories.

In this minicourse, we'll begin with the theory of 2-Segal spaces and how they arise from exact categories. We'll then discuss the equivalence with augmented stable double Segal spaces, as well as how CGW categories fit into the picture. Time permitting, we'll talk about other related ideas and work in progress, such as higher variants of the S-construction for higher Segal spaces and for cyclic 2-Segal spaces, a proposed version for real K-theory, and a description of 2-Segal spaces associated to CGW categories.

Julie Bergner's website ↗

The cyclic category was introduced in the work of Connes and Tsygan to capture the cyclic symmetry found on the standard model for Hochschild homology, the cyclic bar construction. In this original context, the key observation was that it encodes an extra differential leading to several variants of cyclic homology, a central invariant of noncommutative geometry. From the point of view of homotopy theory, the cyclic category models the classifying space of the circle thus giving rise to a combinatorial method to describe circle actions.

In this lecture series, we will discuss some new perspectives on the cyclic category (and its variations) connecting it to constructible/perverse sheaves, categorification, and little disks operads.

Tobias Dyckerhoff's website ↗

Artin–Mazur and Friedlander introduced the étale homotopy type as a homotopical refinement of the étale fundamental group of a scheme. Just as the étale fundamental group is a profinite group, the étale homotopy type is a pro-homotopy type. The first goal of this mini-course is to give a modern introduction to étale homotopy theory — whose foundations have been made much cleaner by higher category theory — as well as to explain a number of interesting applications and results.

The second goal is to explain some recent work on refining the étale homotopy type using condensed mathematics. In general, even the étale fundamental group of a scheme is insufficient to fully capture Q_ℓ-local systems, which are of great interest in number theory. In their work on the proétale topology, Bhatt and Scholze explained how to improve the situation by enlarging the étale fundamental group to their proétale fundamental group. However, the proétale fundamental group is a genuine topological group that does not generally arise as the limit of a pro-system, so standard techniques do not immediately allow one to refine it to a homotopy type. We will explain how condensed mathematics can be used to do this, as well as what is currently known about the condensed homotopy type of a scheme.

The talks will cover several different works, joint with Barwick–Glasman and Holzschuh–Lara–Mair–Martini–Wolf.

Peter Haine's website ↗

The little n-cubes operad classifying n-fold loop spaces is one of the most classical examples of a topological operad, and led to the notion of an E_n-operad as one being equivalent to the little n-cubes operad. In spite of the lack of a more conceptual and generic definition of this notion, there are many different examples of E_n-operads coming from more combinatorial data such as posets, graphs, and monoidal categories proposed and used in work of Fiedorowicz, Vogt, Jeff Smith, McClure, and Berger, to mention a few examples. The goal of these lectures is to explain these different models and the relations between them.

Ieke Moerdijk's page ↗

The notion of height does more than order the infinite set of elliptic curves: it turns their totality into a geometric space. After a brief motivation over ℚ, I will explain this principle over three rational function fields.

Over k(t), an elliptic curve can be viewed as an elliptic surface over ℙ¹_k, and its height is the degree of its classifying map to M_1,1. The resulting height moduli stack is of finite type and parametrizes elliptic curves of fixed Faltings height n, with a natural correspondence between its strata and their Kodaira fibre configurations.

Over 𝔽_q(t), motivic identities in the Grothendieck ring of stacks specialize to exact counts of elliptic curves of bounded height in every characteristic, including p=2 and p=3. Every lower-order term can be traced to a special stratum, automorphism locus, or minimality defect.

Over ℂ(t), fixing a Kodaira fibre stratum and applying the Shioda–Tate formula identifies Mordell–Weil rank jumps with additional Hodge classes. Special cycle modularity organizes their possible height pairings, while a separate period map argument produces analytically dense rank jumps throughout the expected transverse range of $1\le r\le \lfloor \frac{10n-2}{n-1}\rfloor$.

Power operations for commutative ring spectra are a classical theme in homotopy theory. In this talk, I will discuss their interaction with higher semiadditivity, both chromatically, where the chromatic filtration organizes stable homotopy-theoretic phenomena, and in categorical settings. I will explain how this perspective naturally gives rise to the notion of a β-ring, clarifies its relation to the more familiar notion of a λ-ring, and suggests connections with redshift phenomena. The talk is based largely on the work of my student Beckham Myers, together with ongoing joint work with him and Bastiaan Cnossen and Shachar Carmeli.

A symmetric (simplicial) set is a presheaf on the category of nonempty finite sets and all functions. Symmetric sets that satisfy the ordinary Segal condition are just the nerves of groupoids. This talk is about the interaction between two weakenings of the Segal condition that were introduced at about the same time 10–15 years ago. The first defines the class of partial groupoids, symmetric sets for which the Segal maps are injective. These were thought up by Chermak for use in p-local finite group theory. The second weakening defines the d-Segal spaces of Dyckerhoff–Kapranov and (for d = 2) Gálvez-Carrillo–Kock–Tonks.

The higher Segal conditions are normally formulated using triangulations of cyclic polytopes. By work of Tashi Walde, they can be viewed as exactness conditions with respect to certain easily understood classes of coverings of objects of the simplex category. Using this point of view, we develop tools for understanding the "degree of higher Segality" of a partial group. They are based on the discrete geometry of partial group actions and ultimately involve solving Helly type problems in abstract closure spaces. Most of our attention has been computing the degree invariant for interesting classes of partial groups (still not so easy), and I'll describe what sort of group theory shows up in this. Beyond computations, we wonder about some other things, which I'll try to highlight in the last part of the talk. This is joint work with Philip Hackney.

In this talk I will report about recent work with Arne Mertens, where we established that the templicial A_∞-nerve determines an equivalence of categories. After introducing quasi-categories in vector spaces, I will highlight properties of the templicial A_∞-nerve, sketch the construction of a quasi-inverse and prove the desired equivalence.

If X is a smooth projective variety over a field k, then Beilinson (1978) showed that the bounded derived category of X could be identified as the subcategory of the unbounded derived category of quasi-coherent sheaves on X satisfying natural finiteness conditions. I will discuss some generalizations of this result to certain types of algebraic stacks with infinite stabilizers.

Given a Hopf algebra on a finite dimensional vector space V, with an invertible antipode, its double is a quasi-triangular Hopf algebra on V⊗V^*. Classically, the construction of the double uses the compact closure of finite dimensional vector spaces to produce V^* and much of the constituent data. One way to relax this requirement is via the (partial) Int construction, which takes a (partially) traced symmetric monoidal category and produces a compact closed (para)category. Recently, Hasegawa showed the construction of the double extends to traced symmetric monoidal categories via the Int construction. In this talk, we shall extend the construction of the double to partially traced symmetric monoidal categories via explicating the minimal partial trace. As an example, we shall see how these constructions help explain Enriquez and Halbout's quantization of (coboundary) Lie bialgebras. This talk is based on joint work with Philip Hackney and Marcy Robertson.

I'll talk about a new invariant in contact geometry, due to myself and Lisa Traynor, taking values in stable homotopy types. We can prove that this invariant is stronger than classical ones. If time allows, I'd like to sketch the definition of a comultiplication on these invariants — we expect to have a "co-category" whose morphism spaces are these invariants, and the comultiplication of a single invariant is just the structure of endomorphisms. I'll give the talk assuming no background in contact geometry, but with an eye toward building conversations, in case some participants want to join a project constructing an appropriate model for an "A_∞ co-category enriched in spectra" to articulate the structures present in contact geometry.

Derived algebraic geometry is a natural variant of algebraic geometry, which uses the category of animated rings as the affine model. This introduces higher nilpotent thickenings and consequently lets us interpret some phenomena more geometrically, such as non-transverse intersections or the tangent complex. Another natural variant of algebraic geometry is supergeometry, which introduces fermionic nilpotent thickenings. Leveraging the geometric similarities we define the notion of virtual fundamental class in supergeometry. As an application we obtain the Θ-classes on the moduli of curves as the pushforward along a natural map of the virtual fundamental class of the moduli of SUSY-curves.

If time permits, we will also discuss how classical theorems in supergeometry follow by definition, when using definitions from derived algebraic geometry, and future directions.

The Kashiwara–Vergne problem connects Lie theory, low-dimensional topology, and quantum algebra through the structure of the Baker–Campbell–Hausdorff formula. Its genus-zero solutions admit several equivalent interpretations, including automorphisms of free Lie algebras, universal finite-type invariants, and formality isomorphisms for Goldman–Turaev Lie bialgebras.

In this talk, I will present an operadic construction of genus-zero KV solutions. Building on work of Alekseev–Enriquez–Torossian, we show that these solutions arise from isomorphisms between moperads — monoids in the category of right modules over a given operad — associated with braids and chord diagrams. This extends the classical description of Drinfeld associators as operadic isomorphisms and explains the passage from associators to KV solutions as a natural shift from operads to their modules.

This is joint work with Zsuzsanna Dancso, Iva Halacheva, Guillaume Laplante-Anfossi, and Marcy Robertson.

Let X be a simplicial object in an additive category. There is a natural action of the symmetric group on n+1 elements on X_n, and as n varies, these actions are compatible with the face and degeneracy maps in the sense that X extends to a presheaf on finite sets. The phenomenon underlying this observation is a (unique) functorial retract to the inclusion of the augmented simplex category inside the category of finite sets after ℤ-linearisation: put differently, any map between finite sets can be expressed as a ℤ-linear combination of monotone maps in a functorial way.

There are some interesting combinatorial aspects of this construction: it appears that the coefficients that turn up are always 0, 1, or −1. If f is a permutation, the coefficient of the identity function is equal to sgn(f), so that one can sensibly extend the definition of the sign of a permutation to define the sign of any function in a functorial manner. We do not have a conceptual interpretation of the other coefficients that could explain why only 0, 1 and −1 appear.

This retract is closely related to the Dold–Kan construction, which in the augmented setting (and after appropriate shifts) arises naturally as a monoidal equivalence, deriving from the functor classifying the commutative dg-algebra ℤ[e]/e². So, after linearisation, the universal associative algebra happens to be commutative, and Day convolution on simplicial objects in an additive category can be upgraded to a symmetric monoidal structure in a canonical manner. This point of view allows for a construction of our retract that does not invoke generators and relations, allowing us to replace the integers by the sphere spectrum and extend our symmetric group actions to simplicial objects in additive ∞-categories. This is joint work in progress with Lior Yanovski.

The K-theory of G-sets for a finite group G has been well-understood for some time. In contrast, if we relax our setting to consider A-sets for some finite monoid A, most of our intuition breaks down. In this talk, based on joint work in progress with Brandon Shapiro and Inna Zakharevich, I will try to explain exactly what goes wrong, and what we can do to fix it, and recover an analogous version of the classical computations for G-sets.

Let S be a noetherian and normal scheme and f : X → S a geometric fibration (e.g. a smooth and proper morphism) with geometrically connected fibres. Friedlander proved that, after completion away from char(S), the étale homotopy type of a geometric fibre of f coincides with the homotopy fibre of the induced map on étale homotopy types. In this talk, I will explain a more conceptual proof (strategy) of Friedlander's result that works for arbitrary qcqs schemes S. This is joint work in progress with Alexander Schmidt and Jakob Stix.

Maru explained to us how to compute the K-theory of sets with an action of a monoid even when the monoids and actions are quite terrible. In this talk, based on ongoing joint work with C. Weibel, I will describe a different approach that excludes the terrible monoids and actions. The resulting K-theory extends to monoid schemes (built out of abelian monoids like schemes are built out of commutative rings) and behaves quite similarly to K-theory of (ordinary) schemes. In particular, there are versions of the Fundamental Theorem of K-theory, projective bundle formulas, and blow-up formulas — although these are painful to prove.

How to understand ∞-categories like we understand homotopy theory. Joint work with Hadrian Heine.

Higher rank graphs (k-graphs) were introduced by Kumjian and Pask to construct a nice class of generalisations of Cuntz–Krieger algebras. They are a generalisation of directed graphs that correspond to discrete Conduché fibrations above the additive monoid ℕ^k (Brown–Yetter, 2017). In this talk, I will outline an abstract nerve theorem for k-graphs and indicate some potential applications. This is current work with David Pask.