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AG-Seminar Summer 2020

• Do 12-14 online on Zoom

No	Date	Title / Abstract	Speaker
1	07.05	Local Lefschetz-Verdier traces	Benedikt Preis
2	14.05.	Verdier-Lurie duality	Marco Volpe
3	21.05.	Holiday
4	28.05.	Relative Atiyah duality	Marco Volpe
5	04.06.	The tangent complex of K-theory	Benjamin Hennion, University Paris-Saclay
5	11.06.	Holiday
6	18.06.	Motivic rational homotopy types	Isamu Iwanari, Tohoku University
7	25.06.	Local structure of algebraic stacks and applications	David Rydh, Royal Institute of Technology (KTH)
8	02.07.	A cubical model of (∞,1)-categories	Christian Sattler, Chalmers University of Technology
9	09.07.	A cubical model for weak ω-categories	Yuki Maehara, Macquarie University
10	16.07.	Microlocal sheaf categories and the J-homomorphism (time change: 14:30-16:00)	Xin Jin, Boston College
11	23.07.	Dimensional reduction for (-1)–shifted cotangent stacks	Tasuki Kinjo, University of Tokyo
12	30.07.	Neeman Duality for stable ∞-categories (time change: 13:15-14:45)	Paul Bärnreuther
13	06.08.	Nisnevich Descent for Algebraic K-theory	Sebastian Wolf

AG-Seminar Winter 2019/20

• Do 12-14 MA009

No	Date	Title / Abstract	Speaker
1	17.10	Left exact infinity categories and injectivity of assembly maps	Ulrich Bunke
2	24.10	Cartesian modules over representations of small categories Abstract: We introduce the concept of cartesian module over a pseudofunctor R from a small category to the category of small preadditive categories. Already the case when R is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of a classical Representation Theorem of Makkai and Paré. As an application, we relate and extend some previous constructions of the pure derived category of a scheme.	Simone Virili (G. Raptis)
3	31.10	The intrinsic normal cone for Artin stacks Abstract: We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.	Dhyan Aranha and Piotr Pstrągowski (A. Khan)
4	7.11	Localization theorems in derived rigid analytic geometry In the seminal joint work of Thomason-Trobaugh, the authors proved an important localization theorem for perfect complexes on schemes. The latter can be thought as an excision statement for dg-categories of perfect complexes on schemes. In this talk, we propose an alternative proof which has the advantage that works in the (derived) rigid analytic setting. More precisely, let k denote a non-archimedean field of rank 1 valuation and, O_k its ring of integers. Given a (derived) O_k-adic scheme X and X^rig its rigidification, we prove that the canonical functor Perf(X) -> Perf(X^rig) is a Verdier localization of idempotent complete stable \infty-categories. If time permits, we will also mention some applications of our results. This is a joint work with Mauro Porta and Gabriele Vezzosi.	Jorge António (D.-C. Cisinski)
5	14.11	Group actions on moduli spaces For a complete complex variety X with a C^*-action we may subdivide the points of X according to where their C^*-orbit converges at infty; the result is a Bialynicki-Birula decomposition. I will describe a very far reaching functorial generalization of this decomposition and sketch an application to understanding geometry and topology of moduli spaces of projective subschemes. Parts of this is joint work with Lukasz Sienkiewicz.	Joachim Jelisiejew (M. Yakerson)
6	21.11	Regularity of spectral stacks, their algebraic K-theory, and weight structures Commutative rings are regular if and only if a homological criterion holds. This criterion can be formulated for connective E∞-ring spectra and also for connective spectral stacks. We study this property and show its peculiar behaviour. We do it by putting the whole situation in the context of weight structures and by constructing a weight structure on the category of perfect complexes on spectral stacks that are good quotients of an affine spectral scheme. This also allows us to prove the nil-invariance property of theories like fib(K → TC) and KH for spectral stacks. This is based on a joint work with Adeel Khan, and also on a work in progress joint with Tom Bachmann, Adeel Khan, and Charanya Ravi.	Vladimir Sosnilo (A. Khan)
7	28.11	Generalized bornological coarse spaces	Daniel Heiß
8	05.12	No seminar	Bavarian Geometry & Topology Meeting (Augsburg)
9	12.12	Tba	Edoardo Lanari/Andrea Gagna (HK Nguyen)
10	19.12	Derived modular functors Abstract: In my talk, I will present a new approach to a class of non-semisimple representation categories, more specifically non-semisimple modular tensor categories, via homotopy theory and low-dimensional topology. This will lead to so-called derived modular functors. It is already well-known that for a semisimple modular tensor category, the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus, we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).	Lukas Woike (U. Bunke)
11	09.1	Magnetic Laplacians Abstract: The goal of this talk is to develop some features of the spectral theory of 2D 2-periodic magnetic laplacians. The goal of the talk is to explain a simple example in the intersection of mathematical physics, spectral theory and coarse geometry. The main result will be the calculation of the K-theory classes of spectral projections of this operator as classes in the K-theory of the group C^{*}-algebra using the families index theorem. In this example it is possible to understand certain usually quite abstract constructs in a very explicit manner.	Ulrich Bunke
12	16.1.	Dualizability in the higher Morita category After introducing the (∞,n+k)-Morita category of En-algebras (constructed as an (n+k)-fold Segal space). I will explain dualizability results therein, in more detail. If time permits, I will go into consequences for (relative) extended topological field theories.	Claudia Scheimbauer (D.-C. Cisinski)
13	23.1.	Tannaka categories associated to certain triangulated categories (informal discussion)	Daniel Schaeppi
14	30.1.
15	6.2.	On Codes	Masoud Zargar