Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeMar 8th 2013

    I have added some more material and some slides at persistent homology.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeOct 6th 2014

    Joao Pita Costa: Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology

    Seminar za algebro, Ljubljana

    Sreda (Wednesday), 8. oktobra 2014, ob 10. uri v Plemljevem seminarju, Jadranska 19/III, Ljubljana

    João Pita Costa, Mikael Vejdemo Johansson, Primož Škraba, Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology http://arxiv.org/abs/1409.8613

    A topos theoretic generalisation of the category of sets permits ideas as for sets varying according to time intervals. In general it provides tools for unification of techniques for mathematics having had a great importance in the recent developments of Quantum Theory. Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. The basic technique is extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars through the algebraic properties of its underlying lattice structure. In this paper we shall look at the topos of sheaves over such algebra, discuss its construction and potential for a generalised simplicial homology over it. In particular we are interested in es tablishing a topos theoretic unifying theory for the various flavours of persistent homology that have emerged so far, providing a global perspective over the algebraic foundations of applied and computational algebraic topology.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)