Math in the Media

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Bayes's Theorem in the New York Times

Siobhan Roberts' article "Thinking Like an Epidemiologist" ran on August 4, 2020 in the Science section of the Times. The sub-head reads "Don't worry, a little Bayesian analysis won't hurt you" and in fact the piece is about Bayes's Theorem and its applications, most explicitly to COVID testing.
Bayes's Theorem concerns relative probability: $P(A)$ represents the probability of (condition, event, ...) $A$, whereas $P(A|B)$ represents the probability of $A$ given that (condition, event, ...) $B$ (holds, has occurred, ...). Bayes's Theorem relates $P(A|B)$ to $P(B|A)$. This may seem counterintuitive but it follows directly from the common-sense equation $P(A ~\mbox{and}~ B)= $$~P(B)\times P(A|B) = $$~P(A)\times P(B|A)$: $$\mbox{Bayes's Theorem:}~~~ P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}.$$
Roberts: "Take diagnostic testing. In this scenario, the setup of Bayes's theorem might use events labeled 'T' for a positive test result and 'C' for the presence of Covid-19 antibodies:
$$P(C|T)=\frac{P(T|C)\times P(C)}{P(T)}.$$
Now suppose the prevalence of cases is 10 percent (that was so in New York City in the spring), and you have a positive result from a test with accuracy of 87.5 percent sensitivity and 97.5 percent specificity. Running numbers through the Bayesian gears, the probability that the result is correct, and that you do indeed have antibodies is 79.5%."
Here's how "running the numbers ..." works. The $87.5\%$ sensitivity means that $P(T|C)=.875$. The $97.5\%$ specificity means that the probability of a false positive, i.e. $P(T|\mbox{not}~C)$, is $1-.975 = .025$. This allows us to calculate $P(T) $$= P(T|C)\times P(C)+P(T|\mbox{not}~C)\times P(\mbox{not}~C) =$$ .875\times .1 +.025\times .9=.11.$ Then $$P(C|T)=\frac{P(T|C)\times P(C)}{P(T)} = \frac{.875\times .1}{.11} = .795.$$
The diagnostic test example can serve to illustrate how the article's opening statements are actually related to Bayes's Theorem. Roberts starts: "There is a statistician's rejoinder . . . that could hardly be a better motto for our times: 'Update your priors!' In stats lingo, 'priors' are your prior knowledge and beliefs, inevitably fuzzy and uncertain, before seeing evidence. Evidence prompts an updating; and then more evidence prompts further updating, so forth and so on. This iterative process hones greater certainty and generates a coherent accumulation of knowledge."
In the example, before you take the test you have "prior" probability $P(C)=.1$ of having COVID antibodies. Evidence comes from the test result $T$. Given that result, you must update that probability to $P(C|T)=.795$.
For the print version of this article the Times's typesetters left-justified Roberts's equations, which became, unfortunately, unintelligible. This has been rectified online.

$SL(2,\mathbb{R})$ in fusion reactors.

tokamak Tokamak fusion reactors were initially envisioned by Soviet physicists in the 1950s (their name is a Russian acronym; here is one on a 1987 USSR stamp), and have been under continuous development ever since, with the goal of an inexhaustible low-cost source of energy. A tokamak uses magnetic fields to confine a hot plasma to a ring. In an article published in Nuclear Fusion on July 16, 2020, C. B. Smiet, G. J. Kramer and S. R. Hudson (Princeton Plasma Physics Laboratory) address the control of the sawtooth oscillation instability, which "was first observed in 1974 and has been observed in almost every tokamak since. Understanding and mitigating this temperature-limiting instability is crucial for the success of future tokamak reactors such as ITER [the International Thermonuclear Experimental Reactor, scheduled to start working in 2035]." They explain: "The sawtooth oscillation in tokamaks consists of a slow rise in core temperature lasting several to hundreds of ms, followed by a rapid crash lasting 50,200 μs." The occurrence of this instability is linked to a quantity $q_0$, the central safety factor, which is the ratio of the number of times a magnetic field line winds horizontally around the torus to the number of times it winds vertically around the central core ("the ratio of toroidal to poloidal winding"). "In this paper we present a sawtooth model that predicts a crash to occur by a change in topology of the magnetic field in the core when $q_0 = 2/3$." The mathematical core of their model is a study of the field line mapping. This mapping, also called the Poincaré mapping, "is constructed by integrating once around the torus along the magnetic field. Integration is started from a point on a prescribed surface, called the Poincaré surface, that is transverse to the magnetic field. We choose the Poincaré section to be the plane $\phi=0$ and write the mapping as $f(R_0, Z_0) = (R_1, Z_1)$. The region of the Poincaré section that is bounded by a closed flux surface (a region that is topologically a disk), is mapped to itself under the field line mapping."
 
Poincare map

The Poincaré map (yellow arrow) is defined by following a magnetic field line (purple) around the torus until it comes back to the same initial cross-section. Larger image. Images courtesy of Christopher Smiet.

The magnetic axis (red in the figure) is a fixed point for $f$; from physical considerations the authors argue that the derivative map $\partial(R_1, Z_1)/\partial(R_0, Z_0)$ at that point (which gives a first approximation to $f$) must be area-preserving and thus corresponds to a $2\times 2$ matrix of determinant 1, i.e. an element of the Special Linear Group $SL(2,\mathbb{R})$.
 
SL2R images

The three stable configurations for a $2\times 2$ matrix $M$ of determinant 1 depend on its trace $\mbox{Tr}(M)$. Larger image. From left to right: $M$ is alternating hyperbolic if $\mbox{Tr}(M)<-2$; $M$ is elliptic if $-2<\mbox{Tr}(M)<2$; $M$ is hyperbolic if $2 <\mbox{Tr}(M)$. Here the two ends of a colored arrow correspond to a point ${\bf x}$ and its image $M{\bf x}$.

The link with the sawtooth instability comes through the equation $\cos(2\pi/q_0) = (1/2)\mbox{Tr}(M)$ relating the central safety factor and the trace of $M$, which implies that $\mbox{Tr}(M) = -2$ corresponds to $q_0 = 2\,/(1+2n)$. "When $\mbox{Tr}(M)=-2$ an elliptic fixed point can become alternating-hyperbolic. ... If this occurs at the magnetic axis, the axis becomes an alternating-hyperbolic X-point. This can happen when $q_0 = 2/3$, which is very close to $q_0 = 0.7 \pm 0.1$, the value at which experiments have shown some crashes to be triggered." The PPPL press release on the topic was picked up by Phys.org on June 6, 2020. Raphael Rosen tells us: "Smiet hopes to verify the new model by running experiments on a tokamak. 'The mathematics has shown us what to look for,' he said, 'so now we should be able to see it.'"

Topology and physics, the recent history.

"The mathematician who helped to reshape physics" was Davide Castelvecchi's title for his interview of Barry Simon (CalTech) for the August 4, 2020 Nature. His sub-head: "Barry Simon linked a phenomenon that had shocked physicists to topology, the branch of mathematics that studies shapes." The phenomenon in question is the quantum Hall effect, "first described by German physicist Klaus von Klitzing 40 years ago this month. Von Klitzing had seen electrons behave in a surprisingly orderly way when confined to a 2D layer of a semiconductor kept just above absolute zero and exposed to a strong magnetic field. When the voltage on the semiconductor ramped up, the electrical resistance did not change continuously. Instead, it jumped between values that were predictable. And this wasn't affected by temperature fluctuations, say, or by impurities in the material." He goes on: "... it took Simon ..., alongside collaborators, to recognize that equations created to describe the quantum Hall effect were a manifestation of topology. It was topology that was making the material's resistance robust to small changes, allowing it to change in only discrete jumps." Along with questions about details of the discovery, Castelvecchi asks about the relation between mathematicians and physicists; this includes:
  • DC: How would you describe the relationship between the two communities?
  • BS: It really depends on the subfields. The condensed-matter physicists were so used to being looked down upon by the high-energy physics community -- particle physicist Murray Gell-Mann described condensed matter as "squalid-state physics" -- that they didn't look down on other people. There's a tradition among high-energy physicists and string theorists, that really goes back to Enrico Fermi, that's not very positive towards maths. Sometimes there's a lack of mutual respect.
  • DC: Have interactions between the two communities improved since the 1980s?
  • BS: There are still separate camps, but the landscape has changed enormously. There is much more attention in both directions now than there was 40 years ago. It amazes me what has happened to the use of topological ideas in condensed-matter physics. It's really, really striking.