All talks take place room HG00.303 of the Huygensgebouw, Heyendaalseweg 135, Nijmegen, The Netherlands
This is the fifteenth in a series of short workshops jointly organized by geometers and topologists from Bonn, Nijmegen and Utrecht, all situated along the Lower Rhine. The focus lies on the development and application of new structures in geometry and topology such as Lie groupoids, differentiable stacks, Lie algebroids, generalized complex geometry, topological quantum field theories, higher categories, homotopy algebraic structures, higher operads, derived categories, and related topics.
The workshop series is organized by Christian Blohmann, Marius Crainic, Ioan Mărcuț, Ieke Moerdijk and Steffen Sagave, and the local organizers for this workshop are Magdalena Kędziorek and Steffen Sagave.
The talk is based on joint work with Marius Crainic. Our starting point is the remark that geometric structures on differentiable manifolds correspond to a certain class of Lie groupoid valued cocycles. This point of view allows one to discover the existence of a "Chern-Weil" characteristic map for geometric structures on manifolds. Our work is inspired by — and provides a generalization of — the work of André Haefliger on characteristic classes of foliations. We make use of input from the theory of multiplicative structures on Lie groupoids and from the vast literature on the geometry of PDEs.
The Lie algebra of vector fields on a manifold acts on differential forms by Lie derivatives and contractions, and these operations are related by the Cartan relations. We will explain an interpretation of these relations from the point of view of Lie theory, and describe how this leads to a categorification of the Chern-Weil homomorphism.
For a Lie group \(G\), we consider the space of smooth singular chains \(C(G)\), which is a differential graded Hopf algebra. We show that the category of sufficiently local modules over \(C(G)\) can be described infinitesimally, as the category of representations of a dg-Lie algebra which is universal for the Cartan relations. If \(G\) is compact and simply connected, the equivalence of categories can be promoted to an A-infinity equivalence of dg-categories, which are also A-infinity equivalent to the category of infinity local systems on the classifying space \(BG\). The equivalence can be realized explicitly to provide a categorification of the Chern-Weil homomorphism.
The talk is based on joint works with A. Quintero and S. Pineda, and work in progress with M. Rivera.
This is a report on joint work with G. Angelini-Knoll, Ch. Ausoni, D. Culver and J. Rognes.
We compute the mod \((p, v_1, v_2)\) homotopy of the topological cyclic homology of the second truncated Brown-Peterson spectrum at primes \(p \geq 7\). We in particular see that \(V(2)_*TC(BP\langle2\rangle)\) is free and finitely generated as \(P(v_3)\)-module. This confirms a strong form of the chromatic redshift conjecture in this example. Our result also determines the mod \((p, v_1, v_2)\) homotopy of the algebraic K-theory of \(BP\langle2\rangle\).
We will first recall how the category Omega of trees can be used to model infinity-operads and give a definition of linear infinity-operads. We then construct a homology theory for those, using ideas very similar to the bar construction of Ginzburg-Kapranov. We obtain a bar-cobar duality theorem for linear infinity-operads, and in the end, we will sketch how to extend this theory to algebras over such operads.
This is a joint work with Ieke Moerdijk.
The Hochschild-Kostant-Rosenberg theorem exhibits a close relationship between differential forms and Hochschild homology. The latter frequently appears in (derived) algebraic geometry and homotopy theory, and the HKR-theorem often serves as a key input in its study.
Logarithmic geometry is a variant of algebraic geometry in which the notion of smoothness is more flexible. This theory comes with a notion of log differential forms. Moreover, Rognes has extended the definition of Hochschild homology to allow for log rings — the affine schemes of log geometry — as input.
I will explain the main results of joint work with Binda, Park, and Østvær, which includes a generalization of the HKR-theorem to logarithmic geometry. These results will suggest how Hochschild homology and its logarithmic variant can be applied to measure ramification. I will show how this intuition extends to higher algebra, using real and complex K-theory spectra as guiding examples.
This talk will describe Hopf ring structure in the \(RO(C_2)\)-graded bar spectral sequence leading to a complete computation of the \(RO(C_2)\)-graded homology of \(C_2\)-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor \(\underline{\mathbb{F}}_2.\)
Verdier duality is a key feature of derived categories of constructible sheaves on well-behaved stratified spaces. In this talk we will explain how to extend the duality theorem to constructible sheaves on conically smooth stratified spaces and with values in a general stable bicomplete infinity-category. Our proof relies on two main ingredients, one categorical and one geometric. The first one is an equivalence between sheaves and cosheaves proven by Lurie in Higher Algebra. Lurie's theorem will appear in our discussion both as a fundamental building block for the six functor formalism in a very general setting and as a factor of the duality functor on constructible sheaves. The second is the unzip construction introduced by Ayala, Francis and Tanaka, which provides a functorial resolution of singularities to smooth manifolds with corners. This will be used to prove that the exit path infinity-category of any compact conically smooth stratified space is finite.