Topology and Homology Address the Form of Data

Nice short piece on Topology.   I was introduced to it at an early age, and liked aspects of how it linked to the real world in a special ways.   But I never saw very much direct use for it.   Except, as the article suggests, for categorizing shapes in an interesting way. But now something called Homology, which lets you think about shapes in data. Now thinking of how it might be used.    In Quanta Mag, a largely, but subtly non-technical description: 

How Mathematicians Use Homology to Make Sense of Topology

Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

In Quanta Magazine By Kelsey Houston-Edwards    Contributing Writer  ..... '

Infinite Pool Tables

We actually used this idea for solving cleaning coverage problems.   A rare case where advanced topology math principles came into play in industry.  This was the kind of math that I always liked, not too abstractly symbolic, but visually interesting.

New Shapes Solve the Infinite Pool-Table Problem

Quest for a Topological Quantum Computer

In the CACM.  Pointing to a Nature article.  Was part of a group that looked at potential practical applications in complex supply chain applications.

' ... The race is on build a "universal" quantum computer. Such a device could be programmed to speedily solve problems that classical computers cannot crack, potentially revolutionizing fields from pharmaceuticals to cryptography. ... " 

Graph Visualization and Analytics Tools

Piece on commonly used graph visualization tools.    One less known and not mentioned is Polinode, which I was recently introduced to.   Every data scientist should such a tool available for understanding the topology of complex networks.

The Shape of Data

In O'Reilly:  The Application of mathemtical topology to data architecture.  " ... Data has a shape ... Using topology to uncover the shape of your data: An interview with Gurjeet Singh. ... " 

Robots, Sensors and Math at Penn

Good, thoughtful but non technical article,
 I have posted about the GRASP Laboratory here before.  In the Alumni magazine.

" ... Last March seven researchers from the University of Pennsylvania landed in Dayton, Ohio for two days of meetings at the Air Force Research Laboratory. The lab, housed on Wright-Patterson Air Force Base, has been home over the years to breakthroughs in everything from lasers to propulsion technology, and the agenda for these meetings was no less forward-looking: to consider ways of engineering fully autonomous flying robots able to work together to search buildings, track targets, and gather information about areas that are too dangerous for US soldiers to enter.

Among the representatives from Penn was Andrea Mitchell University Professor Robert Ghrist, a Penn Integrates Knowledge (PIK) professor with appointments in the departments of mathematics and electrical/systems engineering. Not long ago, the presence of an applied mathematician like Ghrist at this kind of meeting would have been surprising. His specialty is algebraic topology, an abstract branch of mathematics that studies the properties of different kinds of spaces and has existed as a purely academic pursuit for most of its history.

Yet over the last decade, Ghrist has found a number of innovative ways to use algebraic topology to solve real-world problems in robotics and sensor networks. These discoveries have helped make him one of the best-funded mathematicians in the world. Over the last decade he’s been a principal investigator on more than $20 million in grants from military organizations including the Defense Advanced Research Projects Agency, the Department of Defense, and the Office of Naval Research. 
 ... " 
" ... Since coming to Penn, Ghrist has spent a lot of time collaborating with engineers in the General Robotics, Automation, Sensing & Perception, or GRASP, Laboratory. Much of his work there has focused on technology for autonomous vehicles, which, rather than taking commands from a remote human operator, as many drones do, are able navigate an environment completely on their own. This ability is especially useful in places where good maps are lacking and GPS signals are unreliable, like the interior of a building or beneath the ocean.  .... " 

Topology Mapping for Analytics

Seems Vincent Granville has been thinking similarly lately.  See his article and code in Data Science Central:   A little known component that should be part of most data science algorithms. Beware, this has a general introduction, and then gets technical with R code. And goes further:

" ...  This is a component often missing, yet valuable for most systems, algorithms and architectures that are dealing with online or mobile data, known as digital data: be it transaction scoring, fraud detection, online marketing, marketing mix and advertising optimization, online search, plagiarism and spam detection, etc. .... I will call it an Internet Topology Mapping. It might not be stored as a traditional database (it could be a graph database, a file system, or a set of look-up tables). It must be pre-built (e.g. as look-up tables, with regular updates) to be efficiently used. ... "

I came to think of this when I experimented with the free, open source GePhi network management system and reported on it here. Though the most common network we might work topologically is the Internet, or a social network. These are not the only networks, things like influence maps come to mind.     Have not thought how R in particular links with GePhi, but exploring.  I have a real application in mind. Any thoughts out there?

Mathematical Shape of Things to Come

New challenges on how to deal with bigger, more complex, dynamic and disorganized data.  Some great nontechnical examples here that can give you an appreciation  of the problems being addressed. Will the mathematics of topology point the way?  See   my recent posts on GePhi as one way to explore these problems. Visualization is always the place to start doing analytics.

" ... Scientific data sets are becoming more dynamic, requiring new mathematical techniques on par with the invention of calculus.... 

In Quanta:

 ... DeDeo is not the only researcher grappling with these challenges. Across every discipline, data sets are getting bigger and more complex, whether one is dealing with medical records, genomic sequencing, neural networks in the brain, astrophysics, historical archives or social networks. Alessandro Vespignani, a physicist at Northeastern University who specializes in harnessing the power of social networking to model disease outbreaks, stock market behavior, collective social dynamics, and election outcomes, has collected many terabytes of data from social networks such as Twitter, nearly all of it raw and unstructured. “We didn’t define the conditions of the experiments, so we don’t know what we are capturing,” he said. ... "