GAUS Workshop:
Motives and Higher Categories

Practical Information

Time: 20.10.2025 - 22.10.2025. For more information, see the schedule below.

Place: Johannes Gutenberg-University of Mainz, Institut für Mathematik, Staudingerweg 9, Mainz. The workshop takes place in the Hilbertraum (05-432) in the 5th floor of the Math building. 

How to get there: Bus 53, 54, 56, 58 from Hauptbahnhof to Friedrich-von-Pfeiffer-Weg, then a 10-minute walkor bus 75 from Hauptbahnhof to Duesbergweg, then a 3-minute walk.
If you do not have the Deutschlandticket, the best ticket is probably the "Sammelkarte Erw." it includes 5 rides and costs 13€. You can either purchase it in the "MainzerMobilität"-app or at a machine in front of the main station. See also here for more information.

Lunch: There are several lunch options. Zentralmensa (3-minute walk; pay with app), Baron (15-minute walk, or take the bus, pay with cash/card), La Oliva (25-minute walk, or take the bus, pay with cash/card)

Dinner: There will be a workshop dinner at Stadtbalkon on Monday, 19:00. 

Talks

William Balderrama: Unstable Synthetic Deformations
I will talk about joint work with Piotr Pstragowski on constructing synthetic deformations of unstable homotopy theories. In the motivating example, we construct a category of E-synthetic spaces for a sufficiently nice homology theory E, together with an unstable "cofiber of tau" formalism exhibiting E-synthetic spaces as a categorification of the unstable E-Adams spectral sequence. I'll sketch some of what goes into this.

Julius Frank: Hermitian Mackey Functors via Bispans
The algebraic structure on the homotopy groups of a ring spectrum with anti-involution has somewhat unwieldy descriptions. We discuss a cleaner definition in terms of bispans. On the way, we will modify bispans, which encode the combinatorics of commutative (semi)rings, to instead encode associative (semi)rings. Using work of Cnossen, Haugseng, Lenz and Linskens, this can be shown to work in an ∞-categorical context.

Niklas Kipp: Cohomologically smoothness via Chern classes
Strongly inspired by work of Zavialov, on one hand, and Annala-Hoyois-Iwasa and Tang on the other, we will learn that smooth morphisms in a six-function formalism, which admits a theory of first Chern classes, are cohomologically smooth. If we, in addition, assume the existence of a Deformation to the normal Bundle for sections of smooth morphisms together with a Jacobian criterion for the latter, we can also compute the dualizing sheaf in terms of the Tate twist without any further assumptions on the six-functor formalism. This strategy of proving cohomological smoothness thus applies to a wide range of six-functor formalisms. 

Ahina Nandy: Some remarks on metalinear algebraic cobordism
Metalinear or $\text{SL}^c$-orientation is of fundamental interest in $\mathds{A}^1$-homotopy theory. I would like to talk about metalinear cobordism $\text{MSL}^c$, that is the universal cohomology theory with respect to this particular orientation. One can think of this theory as an algebraic analogue of Stong's Complex-spin bordism. I would also report on attempts to compute some of its homotopy groups. This is joint work with Egor Zolotarev. 

Lucas Piessevaux: Equivariant synthetic spectra via motivic homotopy theory
The category of (even, MU-based) synthetic spectra has a deep connection to cellular motivic homotopy theory over the complex numbers. New advances in equivariant and global homotopy theory allow us to further push this connection into the equivariant direction. I will report on joint work with Keita Allen in which we prove that an appropriately cellular equivariant motivic stable homotopy category over the complex numbers gives rise to a suitable model of equivariant synthetic spectra at finite cyclic groups.

Florian Riedel: Higher algebra in positive characteristic
I will discuss some peculiar aspects of non-connective spectral algebraic geometry. In ordinary commutative algebra, we can always map out to more reasonable rings in a controlled manner and this idea was recently successfully carried over into higher algebra in the rational or T(n)-local categories in the form of the Chromatic Nullstellensatz. Analyzing the power operations in locally graded, F_p-linear categories, we learn two things: Firstly, that, at the prime 2, things work as expected, and as an application deduce that the rank of a finite free F_2-linear cell complex is always well-defined. Secondly, that, for p>2, the Bockstein operations completely obstruct even the most basic desiderata. 

Victor Saunier: Non-commutative motives of Poincaré categories as localizations
In recent work, Ramzi-Sosnilo-Winges have shown that the category of finitary localizing non-commutative motives of stable categories is a Dwyer-Kan localisation of the category of stable categories. There are two main ingredients to their proof: the existence of a factorization of every functor through a fully-faithful one followed by a universal K-equivalence, as well as categorifications of the suspension and the loop functors for localizing invariants. We explain how to adapt this proof, and notably its main two ingredients, to the realm of Poincare categories and Poincare-localizing invariants, such as Karoubi Grothendick-Witt and L-theory with the universal decorations. We will also shortly discuss the case of additive invariants and highlight the difference with the stable case.

Robert Szafarczyk: Animated delta-hat rings
In this talk we will introduce the concept of (animated) delta-hat rings, which is a variant on delta rings. We will study their categorical properties and stability under certain constructions. In particular, this will allow us to define delta-hat schemes. We will then relate delta-hat rings to the Tate-valued frobenius, which will allow us to obstruct certain schemes from lifting to the sphere spectrum.

Hari Sudarsan: Motivic Vorst's Conjecture
The classical conjecture of Vorst in algebraic $K$-theory predicts that the $K$-groups of a ring detect whether it is regular. Since it was proposed in 1979 this conjecture has been proven in characteristic 0 by Cortinas-Haesemeyer-Weibel in 2007. A weak version in finite characteristic was proven by Kerz-Strunk-Tamme in 2018 but the conjecture remains open in general. In this talk I will use the new motivic cohomology for qcqs schemes proposed by Elmanto-Morrow in 2023 to formulate a motivic analogue of Vorst's conjecture and discuss the proof of the conjecture in various cases.

Swann Tubach: Nilpotence of eta in étale motivic homotopy
In classical homotopy theory, the Hopf map is nilpotent in spectra. By contrast, this property fails in the stable motivic homotopy category SH(k) of Morel and Voevodsky: as shown by Morel, the algebraic Hopf map is never nilpotent, a consequence of his identification of the endomorphisms of the unit with the Grothendieck–Witt ring.  We prove that in the category of étale motivic spectra over any scheme, the algebraic Hopf map η is nilpotent, with nilpotence index four, precisely as in the classical stable homotopy category. In particular, every étale motivic spectrum is automatically η-complete. As an application, we deduce that the functor which assigns to an étale motivic spectrum its motive is conservative on connective objects. This is joint work with Klaus Mattis.

Anneloes Viergever:  Computing Quadratic Donaldson-Thomas invariants
(Zero-dimensional) Donaldson-Thomas-invariants "count" ideal sheaves of a given length which have zero-dimensional support on a smooth projective complex threefold. In case the threefold is toric, Maulik, Nekrasov, Okounkov and Pandharipande have proven a formula for the generating series of these Donaldson-Thomas invariants in terms of the MacMahon function. We discuss a conjectural quadratically enriched analogue of this result for smooth projective real threefolds satisfying an orientation condition, using a quadratic version of Donaldson-Thomas invariants taking values in Witt rings, which are constructed using work of Levine. We provide evidence for the conjecture coming from computations for $\mathbb{P}^3$ and $(\mathbb{P}1)3$. This talk is based on my thesis and on joint work with Marc Levine.

Sebastian Wolf: Categories of proétale motives
The proétale topology on schemes, introduced by Bhatt and Scholze, provides a framework for capturing the topological behavior of coefficient rings such as $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$ in étale cohomology. In this talk, I will define motivic analogues of the categories of proétale sheaves on a scheme and outline some of their basic properties. Finally, I will explain how a suitable version of the rigidity theorem continues to hold in this setting, once these categories are solidified in an appropriate sense. This is based on joint work in progress with Raphaël Ruimy and Swann Tubach.

Schedule

Contact

Organizers: Tom Bachmann, Julie Bannwart, Anton Engelmann, Klaus Mattis

Contact: Julie Bannwart, EMail: julie[DOT]bannwart[AT]uni-mainz[DOT]de