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Robin Stoll

Publications and source records attributed to Robin Stoll.

9 recordsLinked to original sources
arXiv2026

A rational model for the fiberwise THH transfer II: $A_\infty$-algebras

In Part I, we proved that a rational model for the fiberwise THH transfer of a map $f$ of fibrations over a base space is given by the Hochschild homology transfer of a cdga model of $f$. In this paper, we provide an explicit description of this Hochschild homology transfer in terms of $A_\infty$-algebras, generalizing work of Bouc. Using a result of Lind-Malkiewich, we deduce a rational model for the Becker-Gottlieb transfer. We furthermore use our results for the following applications to manifold topology. Firstly, we consider the rational characteristic classes constructed by Berglund for fibrations with fiber a Poincar\'e complex (which generalize classes found by Berglund-Madsen); they are defined via the Lie graph complex, and we prove that the classes corresponding to non-trivalent graphs with exactly one loop vanish when evaluated on fiber bundles with fiber a compact simply connected topological manifold. Secondly, we provide a rational model for the space of fiberwise THH-simple structures, which is a step towards obtaining rational models for the classifying spaces of diffeomorphisms and homeomorphisms of a compact simply connected manifold in the rational concordance stable range.

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arXiv2026

A rational model for the fiberwise THH transfer I: Sullivan algebras

Given a map $f$ of fibrations over a space $B$ such that the fiber of $f$ is simply connected and finitely dominated, we prove that its fiberwise THH transfer, considered as a map of parametrized spectra over $B$, is rationally modeled by the Hochschild homology transfer of a Sullivan model of $f$. The proof goes in two steps. Firstly, we use the machinery of higher categorical traces to show that the fiberwise THH transfer can be computed internally to parametrized spectra. Secondly, we model the resulting description rationally using work of Braunack-Mayer, who proved that parametrized spectra can be modeled by modules over Sullivan algebras. In Part II, we will use our result to obtain a rational model of the Becker-Gottlieb transfer, and for applications to manifold topology.

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arXiv2025

Equivariant algebraic models for relative self-equivalences and block diffeomorphisms

We construct rational models for classifying spaces of self-equivalences of bundles over simply connected finite CW-complexes relative to a given simply connected subcomplex. Via work of Berglund-Madsen and Krannich this specializes to rational models for classifying spaces of block diffeomorphism groups of simply connected smooth manifolds of dimension at least 6 with simply connected boundary. The main application is a formula for the rational cohomology of these classifying spaces in terms of the cohomology of arithmetic groups and dg Lie algebras. We furthermore prove that our models are compatible with gluing constructions, and deduce that the model for block diffeomorphisms is compatible with boundary connected sums of manifolds whose boundary is a sphere. As in preceding work of Berglund-Zeman on spaces of self-homotopy equivalences, a key idea is to study equivariant algebraic models for nilpotent coverings of the classifying spaces.

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arXiv2023

Higher structures in rational homotopy theory

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or alternatively an $L_\infty$-algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it.

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arXiv2022

Functor calculus via non-cubes

We study versions of Goodwillie's calculus of functors for indexing diagrams other than cubes. We in particular construct universal excisive approximations for a larger class of diagrams, which yields an extension of the Taylor tower. We prove that the limit of this extension agrees with the limit of the Taylor tower using criteria for the existence of maps between excisive approximations. Lastly we investigate in which cases our new notions of excision coincide with classical ones.

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arXiv2022

Modular operads as modules over the Brauer properad

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

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arXiv2017

Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.

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arXiv2015

Newton's method in practice: finding all roots of polynomials of degree one million efficiently

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm that can be applied routinely to find all roots without deflation and with the inherent numerical stability of Newton's method. We specify an algorithm that provably terminates and finds all roots of any polynomial of arbitrary degree, provided all roots are distinct and exact computation is available. It is known that Newton's method is inherently stable, so computing errors do not accumulate; we provide an exact bound on how much numerical precision is sufficient.

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