The Van Est isomorphism, part II [IMPRS Seminar]
Contact: Christian Kaiser (kaiser@mpim-bonn.mpg.de)
tba
Contact: Pieter Moree (moree@mpim-bonn.mpg.de)
On Birch–Swinnerton-Dyer volumes for hypergeometric Calabi-Yau motives
Contact: Pieter Moree (moree@mpim-bonn.mpg.de)
We study the arithmetic of rank 4 weight 3 Calabi--Yau
motives in the framework of a broader program that approaches
regulators via differential equations that control them. We
introduce biextension periods for certain hypergeometric
CY motives and compare them numerically to the first derivative
of their $L$--functions at $s=2$, obtaining evidence in favor
of a B-SD-type conjecture.
The Van Est isomorphism, part I
Contact: Christian Kaiser (kaiser@mpim-bonn.mpg.de)
Symplectic Calabi-Yau 4-manifolds: some recent progress
Contact: Christian Kaiser (kaiser@mpim-bonn.mpg.de)
Detecting non-permutative elements of $K_1(\text{Var})$ using point counting
The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field $k$) modulo the relation that for a closed immersion $Y\to X$ there is a relation that $[X] = [Y] + [X \setminus Y]$. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field $k$ is finite for a variety $X$ we can consider the "almost-finite" set $X(\bar k)$, which represents the local zeta function of $X$. In this talk we will discuss how to detect interesting elements in $K_1(\text{Var})$ (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on $X$.
Topological cyclic homology and the Fargues-Fontaine curve
I will explain how the Fargues-Fontaine curve appears from localizing invariants, including K-theory and topological cyclic homology, and indicate how the twistor projective line should appear analogously.
Euler classes, pairings, and duality
Classical Morita equivalence defines an Euler class for modules that generalizes the Euler characteristic for spaces. Using this understanding of the Euler class, fundamental structure of the class becomes accessible. Among many choices, in this talk I'll focus on compatibility of the Euler class with a familiar pairing on Hochschild homology since it also allows the exploration for different forms of duality.
The family signature theorem
Hirzebruch's signature theorem relates the signature of the intersection form of a manifold with the integral over the manifold of a certain characteristic class, namely the L-class. This was extended to families of smooth manifolds (i.e. smooth fibre bundles) by Atiyah, using the family index theorem for the fibrewise signature operator. In this setting it relates the Chern character of a certain vector bundle constructed from the local system of intersection forms of the fibres, with the fibre integral of the L-class.
I will explain an elaboration of this result in two directions: to families more general than smooth fibre bundles (e.g. topological fibre bundles), and to an equation in symmetric L-theory rather than rational cohomology. I will then say something about how the L-theoretic version can be analysed, using recent advances in Grothendieck--Witt theory.
Towards conjectures of Rognes and Church--Farb--Putman
In this talk I will give an overview of two related projects. The first project concerns the high-degree rational cohomology of the special linear group of a number ring $R$. Church--Farb--Putman conjectured that, when $R$ is the integers, these cohomology groups vanish in a range close to the virtual cohomological dimension. The groups $SL_n(R)$ satisfy a twisted analogue of Poincaré duality called virtual Bieri--Eckmann duality, and their rational cohomology groups are governed by $SL_n(R)$-representations called the Steinberg modules. I will discuss a recent proof of the "codimension two" case of the Church--Farb--Putman conjecture using the topology of certain simplicial complexes related to the Steinberg modules. The second project concerns Rognes’ connectivity conjecture on a family of simplicial complexes (the "common basis complexes") with implications for algebraic K-theory. I will describe work-in-progress proving Rognes' conjecture for fields, and its connections to $SL_n(R)$ and the Steinberg modules. This talk includes past and ongoing work joint with Benjamin Brück, Alexander Kupers, Jeremy Miller, Peter Patzt, Andrew Putman, and Robin Sroka.
Yang-Baxter elements and a new proof of homological stability for the mapping class group of surfaces
A Yang-Baxter element in a monoidal category gives a weak form of braiding. We explain how such an element allows to define a semi-simplicial set whose connectivity rules homological stability for certain automorphism groups in the category, and how this can be used in the category of bimarked surfaces to give a quite direct proof of slope 2/3 stability of the homology of the mapping class groups of surfaces. This is joint work with Oscar Harr and Max Vistrup.
Syntomic Complexes
I will discuss the theory of syntomic complexes (due to Bhatt--Morrow--Scholze and Bhatt--Lurie) and their identification for regular schemes (joint with Bhatt), inspired by some work in topological Hochschild and cyclic homology.
Weights in homotopy theory
Weight decompositions were first introduced in the setting of rational homotopy as a tool to study p-universal spaces. The same notion may be adapted to many other algebraic contexts and, in general, positive and pure weight decompositions have strong homotopical consequences, often related to formality. A main source of weights is algebraic geometry, either via the theory of mixed Hodge structures on de Rham cohomology or the theory of Galois actions in étale cohomology. In this talk, I will review such weight structures defined at the cochain level, together with some main homotopical implications related to formality over the rationals and also over finite fields. This is mostly joint work with Geoffroy Horel and some work in progress with Bashar Saleh.
On the $K$-theory of $\mathbb{Z}/p^n$
I will report on joint work with Achim Krause and Thomas Nikolaus on an algorithm to compute the higher algebraic $K$-groups of rings such as $\mathbb{Z}/p^n$.
Artin reciprocity via spheres
I will explain a very strange proof of the Artin reciprocity law. At the heart of it is the construction of a sphere from every locally compact Q-vector space.
Reflection
This talk will introduce a notion of a stratification of a (stable presentable) category. A stratification of a scheme determines a stratification of its category of quasi-coherent sheaves. A stratification of a topological space determines a stratification of its category of linear sheaves. Such a stratification determines two reconstruction theorems, each reconstructing the category in terms of its strata and gluing data. An abelian group can be reconstructed in terms of its p-completions and its rationalization, as well as from its p-torsion and its corationalization. Concatenating these two reconstruction theorems results in a sort of duality — reflection — which is a categorification of Möbius inversion for posets. Examples of reflection are abundant, because stratifications are abundant. In algebra, reflection recovers the derived equivalences of quivers coming from BGP reflection functors. In topology, reflection is closely related to Verdier duality, which generalizes Poincaré duality. This talk will discuss all this, through many examples. This is a report on joint work with Aaron Mazel-Gee and Nick Rozenblyum.
Analytic Geometry
We will discuss some of our joint work with Dustin Clausen on Analytic Geometry, based on Condensed Mathematics.
Descent on Analytic Adic Spaces via Condensed Mathematics
| In this talk, I am going to explain the main results of my recent preprint (arXiv:2105.12591). The primary goal will be to prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective modules over $\mathcal{O}_X(X)$ form a stack with respect to the analytic topology on $X$. The proof relies on the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics. |
The genus filtration on the modular surface operad
| The moduli spaces of surfaces assemble into a modular $\infty$-operad, closely related to the $2$-dimensional bordism category. I will establish an obstruction theory for algebras over this surface modular operad where the obstruction for extending from genus $g-1$ to $g$ is controlled by the curve complex $C(\Sigma_g)$. For invertible algebras this yields a spectral sequence with $E^1$-page given by unstable homology of mapping class groups, which converges to the spectrum homology of $\mathit{MTSO}_2$. The required cancellations in this spectral sequence imply for instance that the top-dimensional homology group $H_{14}(\mathcal{M}_5)$ has at most rank $2$. |
The homology of configuration-section spaces
| Configuration-section spaces parametrise fields with singularities on a given manifold, and may be viewed as an enrichment of configuration spaces by non-local data. Hurwitz spaces are two-dimensional examples of these, parametrising branched coverings of surfaces, and the behaviour of their homology is important for questions in analytic number theory, as shown in a celebrated result of Ellenberg, Venkatesh and Westerland on the Cohen-Lenstra conjecture. I will talk about joint work with Ulrike Tillmann (some published and some in progress) on homological stability and the stable homology of configuration-section spaces. Time permitting, I will also explain how similar techniques may be applied to asymptotic monopole moduli spaces. |
| © MPI f. Mathematik, Bonn | Impressum & Datenschutz |
English
Deutsch