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Topology & Neuroscience
When doing data analysis, we usually measure and collect discrete data. For example, if we measure temperature and pressure of a certain place over time, we will end up with a bunch of points in the euclidean plane. How can we use geometry to study such object? From a topological perspective, the mutual distances do not matter, and this is just equivalent to any other bunch of points with the same size. The striking idea of topological data analysis is to allow a persistent parameter, and studying instead a family of spaces associated to this point cloud. The most common approach is to allow balls centered in our points to grow over time. By studying how the topology of the union of balls changes, we can capture very interesting properties of the point cloud. Amazing!
This approach can be fruitfully applied to Neuroscience in many ways. One of the most visually appealing applications is depicted here, and concerns the analysis of brain dynamics. Here, each point of our cloud corresponding to a timeframe of a dynamic scan, called functional Magnetic Resonance Imaging (fMRI). The distance between two points reflects the difference in "brain activation patterns" between the two timeframes. The persistent topology of this point cloud turns out to encode many interesting properties of the brain dynamics. In this picture, two different subjects were given a variety of tests (working memory, math...) during a scan. As you can see, the topology of brain patterns is different depending on the performance: the high performing subjects have different brain patterns depending on the task (color), while the low-performing subjects seems to use similar ares to address different problems. Incredibly enough, we are able to capture this "specialization difference" at a glance, thanks to topology!
Knot Theory
A knot is a circle embedded in the space, like a rope with its ends tied together. They can be represented in 2D by making clear which strand is passing over at crossings. One-million dollars question: given a knot diagram, can it be untied?
Reidemeister moves describe all the possible ways in which we can manipulate a knot diagram. This colored version proves that tricolorability of a diagram remains unchanged when we manipulate the knot.
Devising invariants to distinguish knots is a vast area of research. A broad class is obtained by considering knots with self-intersections and the associated "chord diagrams": a circle with points corresponding to times of self-intersection, and lines joining times mapped to the same point.
Long knots can be studied by means of embedding calculus, by restring to a finite number of intervals and then gluing back this information via homotopy limits. Better suited for knots in dimensions 4 or greater, it can be used to construct the Sinha spectral sequence.
It turns out that the essential homotopical information in the intervals provided by embedding calculus is given by their centers. The resulting object is a tuple of distinct points, called a configuration. Fox-Neuwirth trees index a stratification on the space of configurations, that can be used to study (co)homological properties of the space of knots.
Algebraic Machinery
In the last decades, there has been a conceptual shift in mathematics (especially in algebra) from studying objects to study relations between them. The framework corresponding to this style of doing math is called a category, that is made up of objects and arrows (abstract parallel of a function) between them. Playing around with commutative diagrams is the daily bread of many working in algebra and geometry.
If that wasn't enough, mathematicians have become increasingly interested in relations between relations, and relations between relations between relations... As crazy as it might seem, it has a geometrically easy counterpart: if objects are points, and relations are arrows... relations between relations are triangles! An ∞-category is then a geometric structure (simplicial set) made up of pyramids of all dimensions, providing an amazing bridge between structures algebraic relations and infinity dimensional geometry.
A cool example is given by the study of diagrams arising from geometry. Sometimes, the diagrams involved in encoding algebraic structures do not commute on the spot, but up to the deformation. As you can see from the picture, the different outputs that we expect to agree are not equal, but they can be somehow connected by edges and triangles making up a square. When we are able to construct such fillings in a coherent fashion, we obtain what is called a homotopy-coherent diagram.
Operads are ubitous in modern algebraic topology. They are algebraic objects encoding the n-ary operations of a given structure, and how these operations compose with each other. For example, there is an operad Comm encoding commutative algebras, generated by one 2-ary operation m (multiplication) and relations m(x,y)=m(y,x), m(m(x,y),z)=m(x,m(y,z)). Little disk operads, depicted here, are a homotopical generalization of commutative algebras, where the two relations above only hold in some sense up to deformation. An operation is given a collection of disjoint disks inside a disk, and look at how easy and stunning is the composition of operations—just plugging nested disks into nested disks and removing the middle circles!
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