Emmy Noether and algebraic topology

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

Emmy-Noether Group Leader, MPI for Biological Cybernetics Sensorimotor learning and decision-making in complex environments

Algebraic and combinatorial Brill-Noether theory

The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether locus W r d (Γ) of a genus-g (non-metric) graph Γ is shown to be non-empty if the BrillNoether number ρd(g) is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph Γ for which W r d (Γ) is empty implies the emptiness of W ...

categorically-algebraic topology and its applications

this paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. it incorporates the most important settings of lattice-valued topology, including poslat topology of s.~e.~rodabaugh, $(l,m)$-fuzzy topology of t.~kubiak and a.~v{s}ostak, and $m$-fuzzy topology on $l$-fuzzy sets of c.~guido. moreover, its respe...

Algebraic Topology

This manuscript will be published as Chapter 5 in Wiley’s textbook Mathematical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebr...