A possible error with mathematical ramifications

 Non-technical fact-check source

Dan Brown is the bestselling author of the novel The Da Vinci Code. His most recent bestseller, published in 2013, is Inferno. Like two of his earlier blockbusters it has been made into a movie. It stars Tom Hanks and Felicity Jones and is slated for release on October 28.

Today I want to talk about a curious aspect of the book Inferno, since it raises an interesting mathematical question.

Brown’s books are famous for their themes: cryptography, keys, symbols, codes, and conspiracy theories. The first four of these have a distinctive flavor of our field. Although we avoid the last in our work, it is easy to think of possible conspiracies that involve computational theory. How about these: certain groups already can factor large numbers, certain groups have real quantum computers, certain groups have trapdoors in cryptocurrencies, or …

The book has been out for awhile, but I only tried to read it the other day. It was tough to finish so I jumped to the end where the “secret” was exposed. Brown’s works have sold countless copies and yet have been attacked as being poorly written. He must be doing something very right. His prose may not be magical—whose is?—but his plots and the use of his themes usually makes for a terrific “cannot put down” book.

Well I put it down. But I must be the exception. If you haven’t read the book and wish to do so without “spoilers” then you can put down this column.

## In The Inferno

The Inferno is about the release of a powerful virus that changes the world. Before I go into the mathematical issues this virus raises I must point out that Brown’s work has often been criticized for making scientific errors and overstepping the bounds of “plausible suspension of disbelief.” I think it is a great honor—really—that so many posts and discussions are around mistakes that he has made. Clearly there is huge interest in his books.

Examples of such criticism of The Inferno have addressed the DNA science involved, the kind of virus used, the hows of genetic engineering and virus detection, and the population projections, some of which we get into below. There is also an entire book about Brown’s novel, Secrets of Inferno

However, none of these seems to address a simple point that we hadn’t found anywhere, until Ken noticed it raised here on the often-helpful FourmiLab site maintained by the popular science writer John Walker. It appears when you click “Show Spoilers” on that page, so again you may stop reading if you don’t wish to know.

## How The Virus Works—or Doesn’t?

How does the virus work? The goal of the virus is to stop population explosion.

The book hints that it is airborne, so we may assume that everyone in the world is infected by it—all women in particular. Brown says that 1/3 are made infertile. There are two ways to think about this statement. It depends on the exact definition of the mechanism causing infertility.

The first way is that when you get infected by the virus a coin is flipped and with probability 1/3 you are unable to have children. That is, when the virus attacks your original DNA there is a 1/3 chance the altered genes render you infertile. In the 2/3-case that the virus embeds in a way that does not cause infertility, that gets passed on to children and there is no further effect. In the 1/3-case that the alteration causes infertility, that property too gets passed on. Except, that is, for the issue in this famous quote:

Having Children Is Hereditary: If Your Parents Didn’t Have Any, Then You Probably Won’t Either.

Thus the effect “dies out” almost immediately; it would necessarily be just one-shot on the current generation.

The second way is that the virus allows the initial receiver to be fertile but has its effect when (female) children are born. In one third of cases the woman becomes infertile, and otherwise is able to have children when she grows up.

In this case the effect seems to work as claimed in the book. Children all get the virus and it keeps flipping coins forever. Walker still isn’t sure—we won’t reveal here the words he hides but you can find them. In any event, the point remains that this would become a much more complex virus. And Brown does not explain this point in his book—at least I am unsure if he even sees the necessary distinctions.

The other discussions focus on issues like how society would react to this reduction in fertility. Except for part of one we noted above, however, none seems to address the novel’s mathematical presumptions.

## The Math and Aftermath

The purpose of the virus is to reduce the growth rate in the world’s population. By how much is not clear in the book. The over-arching issue is that it is hard to find conditions under which the projection of the effect is stable.

For example, suppose we can divide time into discrete units of generations so that the world population of women after ${t}$ generations follows the exponential growth curve ${N = N_0 r^t}$. Ignoring the natural rate of infertility and male-female imbalance and other factors for simplicity, this envisions ${N}$ women having ${r}$ female children on average. The intent seems to be to replace this with ${2N/3}$ women having ${r}$ female children each, for ${N' = 2rN/3}$ in the next generation. This means multiplying ${N}$ by ${\frac{2}{3}r}$, so

$\displaystyle N'_t = N_0 \left(\frac{2}{3}r\right)^t$

becomes the new curve. The problem is that this tends to zero unless ${r \geq 3/2}$, whereas the estimates of ${r}$ that you can get from tables such as this are uniformly lower at least since 2000.

The point is that the blunt “1/3” factor of the virus is thinking only in such simplistic terms about “exponential growth”—yet in the same terms there is no region of stability. Either growth remains exponential or humanity crashes. Maybe the latter possibility is implicit in the dark allusions to Dante Alighieri’s Inferno that permeate the plot.

In reality, as our source points out, it would not take much for humanity to compensate. If a generation is 30 years and we are missing 33% of women, then what’s needed is for just over 3% of the remaining women to change their minds about not having a child in any given year. We don’t want to trivialize the effect of infertility, but there is much more to adaptability than the book’s tenet presumes.

## Open Problems

Have you read the book? What do you think about the math?

Some CS reflections for our 700th post

 MacArthur Fellowship source

Lin-Manuel Miranda is both the composer and lyricist of the phenomenal Broadway musical Hamilton. A segment of Act I covers the friendship between Alexander Hamilton and Gilbert du Motier, the Marquis de Lafayette. This presages the French co-operation in the 1781 Battle of Yorktown, after which the British forces played the ballad “The World Turned Upside Down” as they surrendered. The musical’s track by the same name has different words and melodies.

Today we discuss some aspects of computing that seem turned upside down from when we first learned and taught them.

We revisit a paper from 1994

Richard Lipton is, among so many other things, a newlywed. He and Kathryn Farley were married on June 4th in Atlanta. The wedding was attended by family and friends including many faculty from Georgia Tech, some from around the country, and even one of Dick’s former students coming from Greece. Their engagement was noted here last St. Patrick’s Day, and Kathryn was previously mentioned in a relevantly-titled post on cryptography.

Today we congratulate him and Kathryn, and as part of our tribute, revisit a paper of his on factoring from 1994.

What is the role of theory today?

Anna Gilbert and Atri Rudra are top theorists who are well known for their work in unraveling secrets of computation. They are experts on anything to do with coding theory—see this for a book draft by Atri with Venkatesan Guruswami and Madhu Sudan called Essential Coding Theory. They also do great theory research involving not only linear algebra but also much non-linear algebra of continuous functions and approximative numerical methods.

Today we want to focus on a recent piece of research they have done that is different from their usual work: It contains no proofs, no conjectures, nor even any mathematical symbols.

How some longstanding open problems were made to disappear

Ernie Croot, Vsevolod Lev, and Péter Pach (CLP) found a new application of polynomials last month. They proved that every set ${A \subseteq U = \mathbb{Z}_4^n}$ of size at least ${|U|^{1-\epsilon}}$ has three distinct elements ${a,b,c}$ such that ${2b = a + c}$. Jordan Ellenberg and Dion Gijswijt extended this to ${\mathbb{F}_q^n}$ for prime powers ${q}$. Previous bounds had the form ${|U| n^{-c}}$ at best. Our friend Gil Kalai and others observed impacts on other mathematical problems including conjectures about sizes of sunflowers.

Today we congratulate them—Croot is a colleague of Dick’s in Mathematics at Georgia Tech—and wonder what the breakthroughs involving polynomials might mean for complexity theory.

New results on computing with modular gates

Shiteng Chen and Periklis Papakonstaninou have just written an interesting paper on modular computation. Its title, “Depth Reduction for Composites,” means converting a depth-${d}$, size-${s}$ ${\mathsf{ACC^0}}$ circuit into a depth-2 circuit that is not too much larger in terms of ${d}$ as well as ${s}$.

Today Ken and I wish to talk about their paper on the power of modular computation.

A way to recover and enforce privacy

 McNealy bio source

Scott McNealy, when he was the CEO of Sun Microsystems, famously said nearly 15 years ago, “You have zero privacy anyway. Get over it.”

Today I want to talk about how to enforce privacy by changing what we mean by “privacy.”