Tag - Homotopy theory
I will discuss a joint work with Ben-Moshe, Schlank, and Yanovski, proving the compatibility of T(n+1)-local algebraic K-theory with the formation of homotopy limits with respect to p-local π-finite group actions on T(n)-local categories. This is a generalization of the results of Thomason for height 0 and Clausen, Mathew, Naumann, and Noel for actions of discrete p-groups in arbitrary chromatic height. I will then discuss the compatibility of K-theory with the chromatic cyclotomic extensions, chromatic Fourier transform, and higher Kummer theory from previous works with Barthel, Schlank, and Yanovski, phenomena we refer to as "cyclotomic redshift''. Finally, I will explain how cyclotomic redshift gives hyperdescent for K-theory along the cyclotomic tower after K(n+1)-localization.
I will report on joint work with Robert Burklund. We prove that the canonical functor from p-complete, nilpotent spaces to E∞-coalgebras over the algebraic closure of 𝔽p is fully faithful. This generalizes a theorem of Mandell.
This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. (This will be explained in Craig Westerland's lecture on Nov 2.) We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. (This will be explained in my lecture on Nov 3.) Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.
It is well-known that strongly homotopy structures can be transferred over chain homotopy equivalences. Using the uniqueness results of Markl and Rogers we show that the transfers could be organized into a discrete Grothendieck bifibration. An immediate aplication is e.g. functoriality up to isotopy.
By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.
How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X → Y from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g., the geometric Batyerv-Manin type conjectures), algebraic geometry (e.g., moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc) and if time permits, homotopy theory (e.g., derived indecomposables of modules over monoids). I will show how one can fruitfully attack such problems by incorporating techniques from topology to the holomorphic/algebraic world (e.g., by constructing a new spectral sequence).
We prove a variant of Quillen's stratification theorem in equivariant homotopy theory for a finite group working with arbitrary commutative equivariant ring spectra as coefficients, and suitably categorifying it. We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing tensor-ideals of the category of equivariant modules over Lubin-Tate theory, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.
Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy invariant. This is a shame, because it means that the stable motivic homotopy theory of Morel-Voevodsky cannot be employed in studying the deeper aspects of such theories, such as cohomology operations that act on the cohomology groups. In this talk, I will discuss ongoing efforts, joint with Ryomei Iwasa and Marc Hoyois, to set up a workable theory of non-homotopy invariant stable motivic homotopy theory, with the goal of providing effective tools of studying cohomology theories in algebraic geometry by geometric means.
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