arXiv subjects

Malte Lackmann

Publications and source records attributed to Malte Lackmann.

4 recordsLinked to original sources

arXiv2020

Sufficient and necessary conditions for hereditary of infinite category algebras

We describe necessary and sufficient conditions for the hereditarity of the category algebra of an infinite EI category satisfying certain combinatorial assumptions. More generally, we discuss conditions such that the left global dimension of a category algebra equals the maximal left global dimension of the endomorphism algebras of its objects, and classify its projective modules in this case. As applications, we completely classify transporter categories, orbit categories, and Quillen categories with left hereditary category algebras over a field.

math.RT↗

arXiv2019

The octonionic projective plane

This small note, without claim of originality, constructs the projective plane over the octonionic numbers and recalls how this can be used to rule out the existence of higher-dimensional real division algebras, using Adams' solution of the Hopf invariant $1$ problem.

math.AT↗

arXiv2019

External Spanier-Whitehead duality and homology representation theorems for diagram spaces

We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to finite $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of a balanced smash product with a fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. We use this to construct Chern characters for certain rational $\mathcal{C}$-homology theories.

math.KT↗

arXiv2010

A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that intersects the basins under Newton's method of \emph{all} roots of \emph{all} (suitably normalized) complex polynomials of fixed degrees $d$, with arbitrarily high probability. This set is an efficient and universal \emph{probabilistic} set of starting points to find all roots of polynomials of degree $d$ using Newton's method; the best known \emph{deterministic} set of starting points consists of $\lceil 1.1d(\log d)^2\rceil$ points.

math.DS↗