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 ↗

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 $1\le r\le \lfloor \frac{10n-2}{n-1}\rfloor$.

I would like to report on recent advances on the theory of templicial objects of Lowen–Mertens; in particular: deformation properties; relation to A_∞-categories and an example in endofunctors.

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.

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.

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.

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.