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Hopf algebra methods are applied to study Drinfeld twists of (3+1)-diffeomorphisms and deformed general relativity on \emph{commutative} manifolds. A classical nonlocality length scale is produced above which microcausality emerges. Matter…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
Our results are of three types. First we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions.A host of examples of such spectra is provided by…
The goal of this paper is to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup $\Gamma_{0}(9)$. The first property is that the…
We introduce and study the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups. We realize some results of chromatic homotopy theory in terms of the geometry of this stack. For…
We study some model-theoretic notions in NIP by means of spectral topology. In the o-minimal setting we relate the o-minimal spectrum with other topological spaces such as the real spectrum and the space of infinitesimal types of Peterzil…
We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1…
We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension…
We show by a direct computation that, for any Hopf algebra with a modulus-like character, the formulas first introduced in [CM] in the context of characteristic classes for actions of Hopf algebras, do define a cyclic module. This provides…
We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology…
Motivated by the study of persistence modules over the real line, we investigate the category of linear representations of a totally ordered set. We show that this category is locally coherent and we classify the indecomposable injective…
This paper introduces a reformulation of the classical convergence theorem for spectral sequences of filtered complexes which provides an algorithm to effectively compute the induced filtration on the total (co)homology, as soon as the…
Given a homotopy equivalence f between two topological spaces we assemble well known pieces and unfold them into an explicit formula for a strong deformation retraction of the mapping cylinder of f onto its top.
This article is a survey on recent contributions to an effective version of Bautin's theory about the bifurcation of periodic orbits (limit cycles). The analysis of Hopf bifurcations of higher order is possible by use of the return mapping.…
We provide an informal discussion of pattern formation in a finite universe. The global size and shape of the universe is revealed in the pattern of hot and cold spots in the cosmic microwave background. Topological pattern formation can be…
We connect and solve two longstanding open problems in quite different areas: the model-theoretic question of whether $SOP_2$ is maximal in Keisler's order, and the question from set theory/general topology of whether $\mathfrak{p} =…
Defect models have recently been declared dead (Watson 1997), because they predict microwave background and matter fluctuations grossly out of line with what we see. In this talk we apply the fact that many defects are automatically…
We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $H_\pi$ of the general linear group GL(n) which stabilize a tensor of Young symmetry $\{\pi\}$. It turns out that the representation ring of the…
We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples…