Mathematics

Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André-Quillen homology. In the classical algebraic context it is important to distinguish between tame and wild ramification. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the trace map. We transfer the latter fact to ring spectra and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. We study ramification in examples in the context of topological K-theory and topological modular forms.

A dga model for the integral singular cochains on a moment-angle complex is given by the twisted tensor product of the corresponding Stanley-Reisner ring and an exterior algebra. We present a short proof of this fact and extend it to real moment-angle complexes. We also compare various descriptions of the cohomology rings of these spaces, including one stated without proof by Gitler and López de Medrano.

We derive the relationship between the persistent homology barcodes of two dual filtered CW complexes. Applied to greyscale digital images, we obtain an algorithm to convert barcodes between the two different (dual) topological models of pixel connectivity.

This is an expository article about power operations and their connection with the study of highly structured ring spectra. In particular, we discuss Dyer-Lashof operations and their evolving role in the study of iterated loop spaces, E n -algebras, and E n -ring spectra. We will make use of these operations to show that structured ring spectra are heavily constrained. We also discuss some ongoing directions for study. This is a preliminary version of a chapter written for the Handbook of Homotopy Theory.

We study the general linear groups of infinite fields (or more generally connected semi-local rings with infinite residue fields) from the perspective of E ∞ -algebras. We prove that there is a vanishing line of slope 2 for their E ∞ -homology, and analyse the groups on this line by determining all invariant bilinear forms on Steinberg modules. We deduce from this a number of consequences regarding the unstable homology of general linear groups, in particular answering questions of Rognes, Suslin, Mirzaii, and others.

The aim of this paper is to construct an E ∞ -operad inducing an E ∞ -coalgebra structure on chain complexes with coefficients in Z , which is an alternative description to the E ∞ -coalgebra by the Barrat-Eccles operad.

We introduce the effectual topological complexity (ETC) of a G -space X . This is a G -equivariant homotopy invariant sitting in between the effective topological complexity of the pair (X,G) and the (regular) topological complexity of the orbit space X/G . We study ETC for spheres and surfaces with antipodal involution, obtaining a full computation in the case of the torus. This allows us to prove the vanishing of twice the non-trivial obstruction responsible for the fact that the topological complexity of the Klein bottle is 4. In addition, this gives a counterexample to the possibility -- suggested in Paveši?'s work on the topological complexity of a map -- that ETC of (X,G) would agree with Farber's TC(X) whenever the projection map X?�X/G is finitely sheeted. We conjecture that ETC of spheres with antipodal action recasts the Hopf invariant one problem, and describe (conjecturally optimal) effectual motion planners.

We prove that for any ℓ≥0 , there exists an algorithm which takes as input a description of a semi-algebraic subset S⊂ R k given by a quantifier-free first order formula ϕ in the language of the reals, and produces as output a simplicial complex Δ , whose geometric realization, |Δ| is ℓ -equivalent to S . The complexity of our algorithm is bounded by (sd ) k O(ℓ) , where s is the number of polynomials appearing in the formula ϕ , and d a bound on their degrees. For fixed ℓ , this bound is \emph{singly exponential} in k . In particular, since ℓ -equivalence implies that the \emph{homotopy groups} up to dimension ℓ of |Δ| are isomorphic to those of S , we obtain a reduction (having singly exponential complexity) of the problem of computing the first ℓ homotopy groups of S to the combinatorial problem of computing the first ℓ homotopy groups of a finite simplicial complex of size bounded by (sd ) k O(ℓ) . As an application we give an algorithm with singly exponential complexity for computing the \emph{persistence barcodes} up to dimension ℓ (for any fixed ℓ≥0 ), of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity, and generalizes the corresponding results for computing the Betti numbers up to dimension ℓ of semi-algebraic sets with no filtration present.

We introduce eight versions of cyclic homology of a mixed complex and study their properties. In particular, we determine their behaviour with respect to Chen iterated integrals.

An augmented metric space is a metric space (X, d X ) equipped with a function f X :X→R . This type of data arises commonly in practice, e.g, a point cloud X in R d where each point x∈X has a density function value f X (x) associated to it. An augmented metric space (X, d X , f X ) naturally gives rise to a 2-parameter filtration K . However, the resulting 2-parameter persistent homology H ∙ (K) could still be of wild representation type, and may not have simple indecomposables. In this paper, motivated by the elder-rule for the zeroth homology of 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode H 0 (K) . Specifically, if n=|X| , the elder-rule-staircode consists of n number of staircase-like blocks in the plane. We show that if H 0 (K) is interval decomposable, then the barcode of H 0 (K) is equal to the elder-rule-staircode. Furthermore, regardless of the interval decomposability, the fibered barcode, the dimension function (a.k.a. the Hilbert function), and the graded Betti numbers of H 0 (K) can all be efficiently computed once the elder-rule-staircode is given. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O( n 2 logn) time, which can be improved to O( n 2 α(n)) if X is from a fixed dimensional Euclidean space R d , where α(n) is the inverse Ackermann function.