Tony Phillips' Take Blog on Math Blogs

 Mail to a friend · Print this article · Previous Columns Tony Phillips' Take on Math in the Media A monthly survey of math news

This month's topics:

Yitang Zhang and twin primes in the New Yorker

The New Yorker dated February 2, 2015 has "The Pursuit of Beauty", a profile of Yitang Zhang by their staff writer Alec Wilkinson. As Wilkinson reminds us, "Yitang Zhang, a solitary, part-time calculus teacher at the University of New Hampshire ... received several prizes, including a MacArthur award in September, for solving a problem that had been open for more than a hundred and fifty years." The problem is a weaker form of the twin prime conjecture, which states that there are pairs of primes exactly two units apart arbitrarily far out on the number line; in other words, that there exist infinitely many twin primes. Wilkinson explains what Zhang proved in terms of the existence of a ruler of fixed length which can be slid to positions arbitrarily far out where it will encompass two primes: "Zhang chose a ruler with a length of seventy million, because a number that large made it easier to prove his conjecture. (If he had been able to prove the twin-prime conjecture, the number for the ruler would have been two.)" Wilkinson goes on and applies the pigeon-hole principle to show us how to deduce from Zhang's result that "there is a number smaller than seventy million which precisely defines a gap separating an infinite number of pairs of primes." In fact, as he also tells us, the bound was subsequently reduced to 246 by the Polymath8 consortium of mathematicians.

There are many very nice details in this profile. Among them: Zhang's story of the moment he knew he had a proof (he was smoking a cigarette in a yard in Colorado), with context given by a quote from Jacques Hadamard's "The Psychology of Invention in the Mathematical Field." Recollections from Zhang's advisor, T. T. Moh, of how he appeared as a graduate student: "When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the North Pole." An excerpt from the referees' report to the Annals of Mathematics on Zhang's paper: "We have completed our study of the paper 'Bounded Gaps Between Primes' by Yitang Zhang. ... The main results are of the first rank. The author has succeeded to prove a landmark theorem in the distribution of prime numbers." And a delirious paragraph on all the special kinds of prime numbers that have been catalogued through the years, including beastly primes (with 666 at the center): "The number 700666007 is a beastly palindromic prime, since it reads the same forward and backward."

Grothendieck obituary in Nature

Nature published on January 14, 2015 an obituary for Alexander Grothendieck, written by (Fields Medalist) David Mumford and (Abel Prizewinner) John Tate, two of the most distinguished American mathematicians. Besides setting out the basic facts and dates of Grothendieck's life, Mumford and Tate endeavor to give the general scientific reader some idea of the basis for his enormous impact on mathematics.

• "Algebraic geometry is the field that studies the solutions of sets of polynomial equations by looking at their geometric properties. For instance, a circle is the set of solutions of $x^2 + y^2 = 1$, and in general such a set of points is called a variety.
• Traditionally, algebraic geometry was limited to polynomials with real or complex coefficients, but just before Grothendieck's work, André Weil and Oscar Zariski had realized that it could be connected to number theory if you allowed the polynomials to have coefficients in a finite field. These are a type of number that are added like the hours on a clock - a finite set arranged in a circle - and it creates a new discrete type of variety, one variant for each prime number $p$.
• But the proper foundations of this enlarged view were unclear, and this is where Grothendieck made his first hugely significant innovation. He proposed that a geometric object called a scheme was associated to any commutative ring - that is, a set in which addition and multiplication are defined and multiplication is commutative, $a \times b = b \times a$. Before Grothendieck, mathematicians considered only the case in which the ring is the set of functions on the variety that are expressible as polynomials in the coordinates. ...
• Probably his best known work was his discovery of how all schemes have a topology. Topology had been thought to belong exclusively to real objects, such as spheres and other surfaces in space. But Grothendieck found not one but two ways to endow all schemes, even the discrete ones, with a topology, and especially with the fundamental invariant called cohomology. With a brilliant group of collaborators, he gained deep insight into theories of cohomology, and established them as some of the most important tools in modern mathematics."

[This text is a revised version of an obituary that was invited by Nature, and then turned down for being too technical. Some of the story, along with the text of the original submission, is given in the posting "The two cultures of mathematics and biology" on the blog maintained by Lior Pachter (Math and Molecular Biology, Berkeley). Pachter refers to the posting Can one explain schemes to biologists on Mumford's blog, for the authors' point of view. I think I understand Mumford and Tate's desire to give the general scientific public some appreciation of what Grothendieck accomplished, but the unfortunate truth is that talking about schemes to people who need to have a finite field defined in terms of clock arithmetic is a completely futile exercise, which can only lead to mystification or even resentment on the part of the audience. I do not think Grothendieck would have approved. -TP]

Mathematics and Ebola

A "Perspectives" piece by Andy Dobson (Princeton) on "Mathematical models for emerging disease" appeared in Science, December 14, 2014. Dobson describes the "$SEIQR$" model, the following set of coupled differential equations relating the populations of susceptible ($S$), exposed ($E$), infected ($I$), quarantined ($Q$) and recovered ($R$) individuals in a population of $N=S+E+I+Q+R$ ($\dot{S}=dS/dt$, etc., with $t$ measured in days): $$\begin{array}{ccl} \dot{S} &=& \beta S I\\ \dot{E} &=& \beta S I -(\epsilon + q)E\\ \dot{I}&=&\epsilon E -(\delta +\alpha+q)I \\ \dot{Q}&=& q(E+I)-(\alpha + \delta)Q \\ \dot{R}&=&\alpha(I+Q)-\mu R \end{array}.$$ Dobson applies the model to the Ebola epidemic in a hypothetical African town with $N=50,000$ inhabitants; the constants are chosen to match current data from the epidemic: "$\beta$ is the transmission rate between Infected and Susceptible hosts, $\epsilon$ is the rate at which exposed hosts become infectious ($1/\epsilon = 6$ days), $q$ is the rate at which exposed contacts are moved into quarantine ($1/q =$ number of days to locate contact), $\delta$ is rate at which sick individuals recover ($1/\delta = 14$ days), and $\alpha$ is mortality rate of infected hosts (here set to $1/2$); the death rate parameter in $R$ is set to be very small ($\mu = 0.01$), as it is used to count individuals who have died of Ebola in the course of the outbreak. The equations can then be rearranged to give an expression for the basic reproductive number" $$R_0 = \frac{\beta N \epsilon}{(\epsilon + q)(\delta+\alpha+q)}.$$ ($R_0$, "the central concept of mathematical disease modeling," is the average number of secondary cases produced by a single infected individual); "as $R_0$ is around $2.1$ in the current epidemic, this allows us to set $\beta = 0.000033$."

Dobson emphasizes the significance of the non-linearity of the model, especially with respect to the quarantine rate $q$. "Thus, where resources are limited for quarantine and contact tracing (finding everyone who comes in direct contact with a sick Ebola patient), large epidemics quickly result. As quarantine delays get longer, $R_0$ increases to its peak value. Once $R_0$ is bigger than 1, the numbers of people dying in the epidemics rise quickly. Plainly, the key intervention is to quarantine people as quickly as possible, preferably faster than the average time to develop symptoms. Delays in quarantine quickly increase the final size of the epidemic. " He remarks in the conclusion: "Mathematical models are the only way to formally test the interaction between what is known and what is only partially understood about any evolving or emerging pathogen and its host population."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu