Federico Binda-Universität Regensburg
Title: Towards a motivic homotopy theory without A1-invariance
Abstract: Motivic homotopy theory as conceived by Morel and Voevodsky is based on the crucial observation that the affine line A1 plays in algebraic geometry the role of the unit interval in algebraic topology. Inspired by the work of Kahn-Saito-Yamazaki, we constructed an unstable motivic homotopy category "with modulus", where the affine line is no longer contractible. In the talk, we will sketch this construction and we will explain why this category can be seen as a candidate environment for studying representability problems for non A1-invariant generalized cohomology theories.
Joana Cirici- Universität Münster
Title: Formality of symmetric monoidal functors and operads via mixed Hodge theory
Abstract: It has long been observed that Hodge theory is a powerful tool for proving formality results, both in the operadic setting or in the setting of rational homotopy theory. In this talk, I will explain a "purity implies formality" result in the abstract setting of symmetric monoidal functors. I will then develop some applications to: 1) the formality of certain operads defined in the category of complex schemes, and 2) the formality of morphisms of schemes in the sense of rational homotopy. This is joint work with Geoffroy Horel.
Benjamin Collas - Universität Bayreuth
Title: Moduli Stack of Curves and Motivic Homotopy
Abstract: The stratification of the moduli spaces of curves given by the Deligne-Mumford compactification presents an important arithmetic combinatoric, which is expressed both in the context of Geometric Etale Galois representation and of Mixed Tate Motives. This talk will focus on the stack stratification of the spaces -- such as given by the automorphism groups of objects. This structure being known to possess similar arithmetic properties, we will present how the motivic homotopy theory is especially adapted in capturing these properties at the motivic level, as well as in providing a natural bridge between the motivic and Galois étale theories.
John Greenlees - University of Sheffield
Title: The Balmer spectrum of the category of rational equivariant cohomology theories
Abstract: We we consider the tensor triangulated category of rational G-equivariant cohomology theories when G is a torus (or the toral part for a more general compact Lie group). One may classify thick tensor ideals, and the Balmer spectrum is precisely the space of subgroups under cotoral inclusion. The first key ingredients are the Localization Theorem and tom Dieck’s description of the rational Burnside ring. Since this space of subgroups is the basis for the algebraic model for the category, it suggests that the strategy that worked in that case might apply elsewhere.
Markus Hausmann - University of Copenhagen
Title: Symmetric products and subgroup lattices
Abstract:: Let G be a finite group. In this talk I will describe a relationship between symmetric products of G-representation spheres and the subgroup lattice of G. More concretely, it turns out that the rational homotopy groups of the symmetric products of the G-sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice, which in many cases allows an explicit computation of the former. If time permits, I will further discuss how this isomorphism can be refined to obtain an algebraic model for the global equivariant homotopy type of rational symmetric products.
Thomas Hudson - Universität Wuppertal
Title: Segre classes for algebraic cobordism and their application to
Abstract: Since the introduction by Levine and Morel of algebraic cobordism, several attempts have been made to lift to arbitrary oriented cohomology theories the classical results of Schubert calculus. In this talk I will explain how the use of generalised Segre classes allows one to extend (among others) the Kempf-Laksov formula, which expresses as a determinant in Chern classes the fundamental classes of the Schubert varieties of Grassmann bundles. This work is joint with Tomoo Matsumura.
Moritz Kerz - Universität Regensburg
Title: Algebraic K-theory and descent for blow-ups
Abstract: We study descent of algebraic K-theory along blow-ups and apply this to Weibel's conjecture on the vanishing of negative K-groups. This is joint work with F. Strunk and G. Tamme
Adeel Khan - Universität Regensburg
Title: Homotopy K-theory of E∞-ring spectra
Abstract: Weibel’s homotopy K-theory is a variant of algebraic K-theory which satisfies the property of homotopy invariance with respect to the affine line. One can construct an analogue of homotopy K-theory for E∞-ring spectra, which satisfies homotopy invariance with respect to the “brave new affine line”. It turns out that, for connective E∞-ring spectra, homotopy K-theory is insensitive to derived nilpotent thickenings; that is, the homotopy K-theory of R is equivalent to that of π0(R). We will sketch a proof of this fact using the motivic homotopy theory of brave new schemes.
Lennart Meier - Universität Bonn
Title: Real spectra and their duals
Abstract: Real bordism is a C2-equivariant spectrum constructed by Landweber and Araki and is in some sense analogous to algebraic cobordism. There are many C2-spectra built from it, like the BPR
Irakli Patchkoria - Universität Bonn
Title: Proper equivariant stable homotopy theory
Abstract:In this talk we will present a setup for doing equivariant stable homotopy theory with infinite discrete groups and finite isotropy. We develop a theory of proper G-spectra for a discrete group G and show that various equivariant cohomology theories on proper G-spaces are represented in this category of G-spectra. Our main examples include equivariant K-theory and equivariant stable cohomotopy. This is a joint work with Degrijse, Hausmann, Lück and Schwede. Next, we will also mention connections to geometric group theory and relate proper G-spectra to classical finiteness questions for groups. The latter is a joint work with Barcenas and Degrijse. At the end we will also indicate how proper Qp/Zp-spectra are related to cyclotomic spectra and topological cyclic homology.
Simon Pepin-Lehalleur - FU Berlin
Title: Constructible 1-motives
Abstract: Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise, we have at our disposal a mature theory of triangulated categories of mixed motivic sheaves with rational coefficients over general base schemes, with a "six operations" formalism and the expected relationship with algebraic cycles and algebraic K-theory. A parallel development has taken place in the study of Voevodsky's category of mixed motives over a perfect field, where the subcategory of 1-motives (i.e., generated by motives of curves) has been completely described by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to combine these two sets of ideas to study the triangulated category of 1-motivic sheaves over a base. Our main results are the definition of the motivic t-structure for 1-motivic sheaves, a precise relation with Deligne 1-motives, and the extraction of the "1-motivic part" of a general motivic sheaves via a "motivic Picard functor".
Steffen Sagave - Radboud University
Title: Logarithmic topological Hochschild homology
Abstract: Logarithmic ring spectra form a common generalization of the concept of a structured ring spectrum appearing in homotopy theory and the concept of an affine log scheme appearing in algebraic geometry. In this talk I will explain the definition of logarithmic ring spectra and the construction of their logarithmic topological Hochschild homology. Topological K-theory spectra give rise to logarithmic ring spectra that sit between the corresponding connective and periodic K-theory spectra, and I will present both structural and computational results about the logarithmic topological Hochschild homology of these logarithmic ring spectra. (This is report on joint work with John Rognes and Christian Schlichtkrull.)
Stefan Schwede - Universität Bonn
Title: Global homotopy theory
Abstract: This is an overview lecture on aspects of global homotopy theory.
Nikita Semenov- LMU München
Title: Cohomological invariants of algebraic groups and the Morava K-theory
Abstract: In the talk I will discuss an approach to cohomological invariants of algebraic groups over a field based on the Morava K-theories which are universal generalized oriented cohomology theories in the sense of Levine-Morel with respect to the Lubin-Tate formal group law. I will explain that the second Morava K-theory detects the triviality of the Rost invariant and I will describe a method to compute the Morava K-theory of some affine varieties.
Vesna Stojanoska - University of Illinois-Urbana-Champaign
Title: The Gross-Hopkins duals of higher real K-theory spectra
Abstract: The Hopkins-Mahowald higher real K-theory spectra are generalizations of real K-theory; they are ring spectra which give some insight into higher chromatic levels while also being computable. This will be a talk based on joint work with Barthel and Beaudry, in which we compute that higher real K-theory spectra at prime p and height p-1 are Gross-Hopkins self-dual with shift (p-1)2. We expect this will allow us to detect exotic invertible K(n)-local spectra.
Marcus Zibrowius - Universität Düsseldorf
Title: The symmetric complex over projective space
Abstract: Projective spaces were the first truly geometric objects over which symmetric vector bundles were studied. In 1980, Arason showed that essentially all symmetry is inherited from the ground field: up to the usual K-theoretic stabilization process and up to hyperbolic factors, any symmetric bundle over a projective space can be pulled back from a point. For complexes, Gille, Walter and later Nenashev showed by a variety of methods that the situation is markedly different: on each projective space, there is essentially one additional symmetric complex that is not visible over the point. Unfortunately, the elegant simplicity of the result is in stark contrast to the intricacies of the existing calculations, and, in particular, to the sharp distinctions between several cases each relies on. We will present a short historic survey and offer a similarly brief remedy for this conceptual deficiency.