Special issue on "Algorithms and Software for Computational Topology"

...in the Journal of Symbolic Computation

Aim and Scope:

The interest in algorithms on topological problems and their implementation has rapidly grown during the last decade. One driving force is the emergence of ``topological data analysis'' which connects topological concepts like Morse theory and homology to the investigation of real-world data. Another recent track of research substantially expands the realm of possibility for computational approaches in 3-manifold and knot theory. Common to these and other developments is the ability to handle large data collections through an efficient algorithmic framework as well as mature software implementations of those. A workshop session at the International Congress of Mathematical Software (ICMS) in August 2014 was dedicated to this topic (http://icms14.appliedtopology.org/).

The Journal of Symbolic Computation (JSC) invites high-quality contributions from researchers in the area of Computational Topology reporting on original research achievements towards algorithms, software, and applications. The list of topics includes, but is not limited to

  • (Persistent) homology
  • Topological data analysis
  • 3-manifold topology and knot theory
  • (Discrete) Morse theory

Researchers which are unsure whether their contribution is suitable are encouraged to contact the guest editor.

Submission instructions:

It is recommended to prepare submission in the same format as regular submission to JSC (see the "Guide for Authors" at http://www.journals.elsevier.com/journal-of-symbolic-computation/).

The paper must start with a introduction that

  • clearly states the considered problem
  • discusses its relevance and related work
  • explains the main contribution of the paper
  • explains why the contribution is original and non-trivial

There is no page limit on submitted manuscripts. It is required, however, that

  • all related work is completely and carefully discussed
  • all theorems are rigorously proved
  • important definitions/theorems/algorithms are illustrated by well-chosen examples.

All submitted papers will be refereed according to the high standards of JSC.

Guest editor:

Michael Kerber (Max Planck Institute for Informatics) - mkerber@mpi-inf.mpg.de

Deadline:

The submission deadline is January 31 2015. The special issue is planned to appear in Fall 2015

AATRN Seminar: Robert Ghrist

Today, the promised AATRN seminar series got started with Robert Ghrist as the inaugural speaker. His lecture, through WebEx, builds up the cellular sheaf perspective on networks with capacity, Max Flow / Min Cut, and the work done by Ghrist, Yasu Hiraoka, and Sanjeevi Krishnan on categorifying and sheafifying MF/MC.

Among the novel insights coming from this talk even if one has been following the UPenn developments for a while was the connection to Poincare Duality: “Flow duality is a form of Poincare duality” — S. Krishnan

The approach detailed by Krishnan in his preprint (on http://www.math.upenn.edu/~sanjeevi/papers/mfmc.pdf ) encodes a flow network as a sheaf of semi-modules of capacities over a semi-ring over the directed graph of the network. Flows correspond to homology, cuts to cohomology.

MF/MC translates to:

Theorem (S. Krishnan)

Given a network X, a distinguished (virtual) edge e and a capacity sheaf F, (all) flowvalues at e of flows on F correspond to the homotopy limit over all cuts of cutvalues of cuts on F.

This approach, and connecting flows and cuts to a lattice structure on the constraints produces a setting where multi-kind flows can easily be analyzed with the same tools as ordinary flows, and where the algebra fixes the duality gaps that show up when naively searching for minima and maxima.

The talk culminated in a primer on the homology and cohomology of directed sheaves, to set us up to read Krishnan's paper, constructing compact support cohomology and Borel-Moore homology for sheaves over directed spaces.

The added abstraction levels seem to enable MF/MC theorems for, for instance, probability distributions. It also carries a promise for insights into duality gaps in MF/MC type problems.

Applied Algebraic Topology Research Network

Peter Bubenik writes:

[…] Robert Ghrist, Konstantin
Mischaikow, Fadil Santosa and I are starting a Research Network in Applied Algebraic Topology. To start, the main activity of the network will be a weekly interactive online seminar. We have plans to expand our activities in the future.

Rob Ghrist will give the first talk on Tue Sept 23.

Please check out our web site at https://www.ima.umn.edu/topology/ and become a member.