Tony’s Take December 2021

This month’s topics:

The exponential function in Slate

The online magazine Slate invited Gary Cornell to explain to all of us why the Omicron variant of Covid-19 was spreading so fast. (This was back on December 17, 2021). He contributed “The Math That Explains Why Omicron Is Suddenly Everywhere” with an illustration containing a graph like this one:

A handwritten graph of the exponential function
The exponential function, adapted from Slate.

Cornell explains that the current pattern of Omicron cases doubling every two days is an example of exponential growth, and that we humans are not wired to process that phenomenon reliably: “when it comes to exponential growth, your gut feelings are going to be wrong and you need to stop and do some (elementary) math.” The characteristic of exponential growth is that the increase during the next time period is proportional to the value right now. To illustrate this phenomenon Cornell retells the story of the king and the chessboard, which ends up with the king owing his opponent one grain of rice on the first square, two on the second, four on the third, and so on, totaling “more than 18 quintillion grains of rice, which would roughly cover the planet and would be the world’s output of rice for about 1,000 years.” He reminds us that cases of the Omicron variant, which were doubling every two days, have the same growth potential. And even if the hospitalization rate is as low as 1%, after less than two weeks the number of hospitalizations would be greater, day by day, than the number of infections from two weeks before.

New mathematical ideas from artificial intelligence

“For the first time, machine learning has spotted mathematical connections that humans had missed.” This is the beginning of Davide Castelvecchi’s news piece in the December 9, 2021 issue of Nature. The article Castelvecchi refers to, “Advancing mathematics by guiding human intuition with AI,” was published in the same journal a week before. The authors were 11 members of Alphabet Inc.’s Deep Mind laboratory in London working with three academic mathematicians from Oxford and Sydney. The team, led by Deep Mind’s Alex Davies and Pushmeet Kohli, states: “We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures.” They present two examples, one involving knots and one representation theory, of their process in action.

The authors set up a general framework for approaching the process, as follows. They take $z$ to be a variable element of some class of mathematical entities (knots, polyhedra, groups, …) and consider two “mathematical objects” $X(z)$ and $Y(z)$ associated with $z$. Are these two objects related, and how? They think of such a relationship as a function, and ask if there is an $f$ such that $f(X(z))\approx Y(z)$ for all $z$.

They give an example of what they have in mind. Let $z$ range over the set of 2-dimensional polyhedra; they set $X(z)$ to be the vector $(V(z), E(z), Vol(z), Surf(z))$, where $V$ and $E$ are the numbers of vertices and edges, $Vol$ and $Surf$ the volume and the area of the polyhedron; and $Y(z)$ to be the number $F(z)$ of faces of $z$. In this case Euler’s formula $V-E+F=2$ means that the function $f(X) = (V, E, Vol, Surf)\cdot (-1, 1, 0, 0) +2$, where $\cdot$ is the dot-product of vectors, does the trick, since in fact $-V+E+2 = F$. Notice that in this example the volume and surface area are not part of the calculation. Eliminating spurious information is part of the “attribution techniques” mentioned above, used in an iterative process of refining the form of the relation to be found.

The training process relies on a large set of specimen values of $z$, a large bank of possible functions as candidates for $f$, and an initial guess of which objects $X(z)$ and $Y(z)$ have the potential of being part of a true mathematical statement of the form $f(X(z))= Y(z)$. If the machine finds an $f$ that works more often than expected by chance, the human partners examine the form of $f$ to see how it can be improved. Besides discarding spurious variables as above, a technique the authors use is gradient saliency: imitating optimization in calculus by examining the derivative of outputs of $f$ with respect to the inputs. “This allows a mathematician to identify and prioritize aspects of the problem that are most likely to be relevant for the relationship,” they write. They emphasize the interactive aspect of the iterative process, where “the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.”

Applied to knot theory, the process yielded a theorem the authors describe as “one of the first results that connect the algebraic and geometric invariants of knots.” They comment: “It is surprising that a simple yet profound connection such as this has been overlooked in an area that has been extensively studied,” echoing Euler’s remark about his $V-E+F=2$ discovery: “It seems extremely amazing that, while Stereometry along with Geometry has been studied for so many centuries, nevertheless some of its most basic elements have been unknown until now” (Demonstratio … , p. 141).

Castelvecchi tells us that Alex Davies, one of the project leaders from Deep Mind, “told reporters that the project has given him a ‘real appreciation’ for the nature of mathematical research. Learning maths at school is akin to playing scales on a piano, he added, whereas real mathematicians’ work is more like jazz improvisations.”

Math history online

“Online exhibit adds up the history of mathematics” by Erin Blackmore ran in the Washington Post on November 28, 2021. Blackmore reports on a collaboration among the National Museum of Mathematics, Wolfram Research, and the Overdeck Family Foundation. The History of Mathematics Project has nine interactive exhibits (Counting, Arithmetic, Algebra, Geometry, etc.), each with around six items. Each item has extra graphics, interpretations, and an interactive component. For example, the Rhind Papyrus appears in the Arithmetic exhibit along with a Wolfram-powered app that visualizes how each of the fractions 2/(odd number between 3 and 101) can be expressed, Egyptian-style, as a sum of fractions with numerator 1. If you want, you can see the new denominators in the Egyptian hieratic script used on the Papyrus. “Whether you come to try your hand at some ancient math homework or to enjoy imagery from artifacts from around the world,” Blackmore writes, “you’ll come away with a greater appreciation of how math developed — and how much modern math owes to our brainy ancestors.”