Mathematics

The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava E -cohomology of finite abelian groups has played an important role in the study of power operations for Morava E -theory. The goal of this paper is to explore the relationship between level structures on the p -divisible group given by the trivial extension of the universal deformation by a constant p -divisible group and the Morava E -cohomology of the iterated free loop space of the classifying space of a finite abelian group.

An ideal invariant for multiparameter persistence would be discriminative, computable and stable. In this work we analyse the discriminative power of a stable, computable invariant of multiparameter persistence modules: the fibered bar code. The fibered bar code is equivalent to the rank invariant and encodes the bar codes of the 1-parameter submodules of a multiparameter module. This invariant is well known to be globally incomplete. However in this work we show that the fibered bar code is locally complete for finitely presented modules by showing a local equivalence of metrics between the interleaving distance (which is complete on finitely-presented modules) and the matching distance on fibered bar codes. More precisely, we show that: for a finitely-presented multiparameter module M there is a neighbourhood of M , in the interleaving distance d I , for which the matching distance, d 0 , satisfies the following bi-Lipschitz inequalities 1 34 d I (M,N)≤ d 0 (M,N)≤ d I (M,N) for all N in this neighbourhood about M . As a consequence no other module in this neighbourhood has the same fibered bar code as M .

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of k -algebras. Our main examples are of the form R= C ∗ (X;k) , the ring spectrum of cochains on a space X for a field k . In particular, we establish local Gorenstein duality in characteristic p for p -compact groups and p -local finite groups as well as for $k = \Q$ and X a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.

We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology.

In this article, we prove that Buchstaber invariant of 4-dimensional real universal complex is no less than 24 as a follow-up to the work of Ayzenberg [2] and Sun [14]. Moreover, a lower bound for Buchstaber invariants of n -dimensional real universal complexes is given as an improvement of Erokhovet's result in [7].

We define a simplicial enrichment on the category of differential graded Hopf cooperads (the category of dg Hopf cooperads for short). We prove that our simplicial enrichment satisfies, in part, the axioms of a simplicial model category structure on the category of dg Hopf cooperads. We use this simplicial model structure to define a model of mapping spaces in the category of dg Hopf cooperads and to upgrade results of the literature about the homotopy automorphism spaces of dg Hopf cooperads by dealing with simplicial monoid structures. The rational homotopy theory of operads implies that the homotopy automorphism spaces of dg Hopf cooperads can be regarded as models for the homotopy automorphism spaces of the rationalization of operads in topological spaces (or in simplicial sets). We prove, as a main application, that the spaces of Maurer--Cartan forms on the Kontsevich graph complex Lie algebras are homotopy equivalent, in the category of simplicial monoids, to the homotopy automorphism spaces of the rationalization of the operads of little discs.

The E 2 -term of the Adams spectral sequence for Y may be described in terms of its cohomology E ∗ Y , together with the action of the primary operations E ∗ E on it, for ring spectra such as E=H F p . We show how the higher terms of the spectral sequence can be similarly described in terms of the higher order truncated E -mapping algebra for Y − that is truncations of the function spectra Fun(Y,M) for various E -modules M , equipped with the action of Fun(M, M ′ ) on them.

In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse functions based on the induced merge tree. We then relate the matching number of a tree to a certain invariant of the induced merge tree. Finally, we count the number of merge trees that can be induced on a star graph and characterize the induced merge tree.

We present a family of model structures on the category of multicomplexes. There is a cofibrantly generated model structure in which the weak equivalences are the morphisms inducing an isomorphism at a fixed stage of an associated spectral sequence. Corresponding model structures are given for truncated versions of multicomplexes, interpolating between bicomplexes and multicomplexes. For a fixed stage of the spectral sequence, the model structures on all these categories are shown to be Quillen equivalent.

In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models \emph{a priori}. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way. Using well-developed concepts from simplicial algebraic topology, we are able to define a distance based on the persistent homology of the simplicial complexes. The persistent homology represents the structure of the models, so in this way we can identify shared topological features of different models. We then expand on this measure of distance between simplicial complexes to study the concept of equivalence between models in order to determine their conceptual similarity. We apply our methodology to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.