Abstracts

Aravind Asok: Motivic vs. algebraic vector bundles

Given a smooth algebraic variety $X$ over a field, a motivic vector bundle of rank $n$ on $X$ is a motivic homotopy class of maps from $X$ to the classifying space of the general linear group $GL_n$. There is always a comparison map from algebraic to motivic vector bundles. If $X$ is affine, then the comparison map is a bijection, but it fails to be bijective in general. I will describe joint work with Jean Fasel and Mike Hopkins analyzing this comparison map, and also links to the problem of determining when a topological complex vector bundle on a smooth complex variety admits an algebraic structure.

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Tom Bachmann: A $C_2$-equivariant generalization of Gabber's presentation lemma

Report on work in progress. Gabber's presentation lemma is a cornerstone of motivic homotopy theory over fields, where it can be used among other things to prove exactness of Gersten complexes and Morel's "stable connectivity" theorem. To the best of my knowledge, a generalization to the equivariant context so far has not been proposed - and indeed the Gersten complex even for $C_2$-equivariant K-theory is known not to be exact. I will explain a slightly modified Gabber presentation lemma which does seem to hold $C_2$-equivariantly. Encouragingly, this form implies stable connectivity but *not* exactness of the Gersten complex.

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Federico Binda: p-adic cohomology theories and motivic monodromy

In this talk, we will discuss some recent advances in the theory of motives in the context rigid analytic geometry. We offer a new construction of the Hyodo-Kato cohomology, without making any reference to log schemes or the log-de Rham Witt complex. As a consequence, we can construct Clemens-Schmidt-style complexes in the mixed characteristic setting, confirming an expectation of Flach and Morin. This is a joint work with Alberto Vezzani and Martin Gallauer.

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Robert Burklund: Algebraic K theory, redshift and the telescope conjecture

I will discuss joint work with Jeremy Hahn, Ishan Levy and Tomer Schlank wherein we show that the algebraic K-theory of the K(1)-local sphere is a counterexample to the height 2 telescope conjecture.

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Hélène Esnault: Survey on arithmetic properties of rigid local systems

Adapting to our broad public, I’ll try to give a small survey of the recent advances on arithmetic properties of rigid local systems, based on joint work in the recent years with Johan de Jong, Michael Groechenig, and Moritz Kerz. [A guiding thread is the Springer Lecture Notes based on the Eilenberg Lectures I gave at Columbia U. (NY) in the fall 2022, which comes out in October.]

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Markus Hausmann: On the moduli stack of equivariant formal groups

The thick subcategory theorem of Hopkins-Smith can be formulated as saying that the Balmer spectrum of finite spectra is homeomorphic to the space of points of the moduli stack of formal groups. In my talk I will explain that the analogous statement is true A-equivariantly for every abelian compact Lie group A. (joint with Lennart Meier).

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Moritz Kerz: Lefschetz pencils of semi-stable schemes

Lefschetz pencils of a projective variety are an important tool to study the arithmetic and topology the variety: from Lefschetz's 1924 proof of the Weak Lefschetz Theorem to Deligne's 1974 proof of the Weil conjecture. In my talk I will describe the geometry of Lefschetz pencils for semi-stable schemes over discrete valuation rings. (Joint with A. Beilinson, H. Esnault)

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Josefien Kuijper: The algebraic K-theory spectrum of varieties and compactly supported cohomology theories

In this talk I will discuss a new construction of the K-theory spectrum of varieties K(Var), using the framework of so-called squares categories. With this approach, a spectral analogue of Bittner's presentation can be formulated (but remains conjectural): is K(Var) equivalent to a spectrum built from just smooth and complete varieties? In the second half of the talk, I define a category of "compacly supported cohomology theories". Here squares show up again, and one can show that a compactly supported cohomology theory is determined by its image on smooth and complete varieties, which could remind one of Bittner's presentation. A connection between K(Var) and this notion of compactly supported cohomology theories exists, in the form of a derived motivic measure. This talk is partially based on joint work in progress with Jonathan Campbell, Mona Merling and Inna Zakharevich.

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Lennart Meier: Topological Jacobi Forms

Jacobi forms are an amalgam of elliptic functions and modular forms, with many natural examples. I will present joint work in progress with Tilman Bauer, in which we define topological Jacobi forms TJF. These are $E_\infty$-rings that behave to Jacobi forms like topological modular forms (TMF) to modular forms. I will explain how equivariant TMF is important in understanding TJF. If time remains, I will include some speculative remarks about their relationship to field theories.

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Thomas Nikolaus: Algebraic K-Theory in Geometric Topology

While algebraic K-theory has many applications and uses in modern algebra and arithmetic, its origins actually lie in geometric topology through Whithead's work on simple homotopy theory. We will review this in modern language, to elucidate the nature of simple homotopy types. This will also lead to parametrized versions of results of West and Chapman through the use of Efimov-K-Theory. The key is a K-theoretic model of assembly maps which is of independent nature and should have many more applications in the furture. This is joint work with A. Bartels and A. Efimov.

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Oliver Röndigs: A motivic version of the Adams conjecture

In joint work in progress with Alexey Ananyevskiy, Elden Elmanto, and Maria Yakerson, the following version of the Adams conjecture is obtained: Given a vector bundle $E$ over a smooth scheme $X$ over a field $F$ and an integer $k$ invertible in $F$, there exists a natural number $N$ such that the Thom spectrum of the $k^N$-fold direct sum of $E$ is equivalent to the Thom spectrum of the $k^N$-fold direct sum of the $k$-th Adams operation of $E$. This equivalence exists in the motivic stable homotopy category $SH(X)$.

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Julia Semikina: Cut and paste invariants of manifolds via algebraic K-theory

The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope $P$ in $\mathbb{R}^n$, one can cut $P$ into a finite number of smaller polytopes and reassemble these to form $Q$. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the $SK$-equivalence ("schneiden und kleben" is German for "cut and paste"). In this talk I will explain the construction that will allow us to speak about the "K-theory of manifolds" spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups $SK_n$. I will show how to relate the spectrum to the algebraic $K$-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further I will describe the connection of our spectrum with the cobordism category.

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