Tag - Topology
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.
Homological stability is now well established as an organizing principle and computational tool in algebraic topology and other areas. In many cases it is of interest to obtain homological stability with twisted coefficients, and the standard choice of such coefficients are the polynomial coefficient systems. All known approaches to homological stability with polynomial coefficients produce a stable range depending on the degree of polynomiality. I will explain a method of obtaining uniform stable ranges for some classes of groups and coefficients of natural interest. This has important consequences in arithmetic statistics, discussed in the number theory seminar on Nov 2.
The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.
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.
We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The diagrams can even be arranged to have additional nice properties, such as being alternating with minimal crossing number. Moreover, the reduction is polynomially uniform in the self-braiding exponent of the colouring object. Various complexity-theoretic hardness results regarding the calculation of quantum invariants of knots follow as corollaries. In particular, we argue that the hyperbolic geometry of knots is unlikely to be useful for topological quantum computation.
It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, extending an earlier work of the speaker with Conway and Etnyre, we will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (n) surgery along a Legendrian knot to yield a weakly fillable contact manifold, for some integer n > 0. When specialized to knots in the three sphere with its standard tight structure, this result can be effectively used to find many examples of fillable surgeries along with various obstructions and surprising topological applications. For example, we prove that a knot admitting lens space surgery must have slice genus equal to its 4-dimensional clasp number.
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