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The 11 March 2011 Tohoku-Oki earthquake was the largest and most destructive in the history of Japan. Scientists are hard at work trying to understand just what made it so devastating. Science Daily Earth and Climate News
In 1847, Gabriel Lamé proved Fermat’s Last Theorem. Or so he thought. Lamé was a French mathematician who had made many important discoveries. In March of that year he sensed he’d made perhaps his biggest: an elegant proof of a problem that had rebuffed the most brilliant minds for more than 200 years.
His method had been hiding in plain sight. Fermat’s Last Theorem, which states that there are no positive integer solutions to equations of the form an + bn = cn if n is greater than 2, had proved to be intractable. Lamé realized that he could prove the theorem if he just expanded his number system to include a few exotic values.
Adding (or “adjoining”) new values to the old numbers is not hard to do — there’s a straightforward mathematical recipe for how to incorporate the square root of 5 as a normal number between 2 and 3, for example, after which you can carry on with the business of arithmetic as usual. All you do is write each value in the new number system as a + b√5, where a and b are integers. This might seem like an ungainly way to write a number, but it serves as a coherent basis for creating a new number system that functions just like the old one. Mathematicians call this new system a number “ring”; they can create an infinite variety of them, depending on the new values they choose to incorporate.
Of course, it’s hard to tinker with something as intricate as a number system without producing unintended consequences. When Lamé started adjoining these funny numbers, everything looked great at first. But then other mathematicians pointed out that this newly gained flexibility came at a high cost: The new number system lacked unique prime factorization, by which a number — for example, 12 — can be expressed uniquely as a product of primes: 2 x 2 x 3. This violated a bedrock principle of conventional arithmetic.
Unique prime factorization ensures that each number in a number system can be built up from prime numbers in exactly one way. In a number ring that includes a √-5 (in practice, mathematicians often employ number systems that use the square roots of negative numbers), duplicity creeps in: 6 is both 2 x 3 and also (1 + √-5) x (1 – √-5). All four of those factors are prime in the new number ring, giving 6 a dual existence that just won’t do when you’re trying to nail things down mathematically.
“In algebra classes when we first teach that unique prime factorization sometimes doesn’t hold, students gasp, they say, ‘Oh my, what happened?’ We always take for granted that everything can be uniquely decomposed into primes,” said Manjul Bhargava, a professor at Princeton University and winner of the Fields Medal, math’s highest honor.
Unique prime factorization is a way of constructing a number system from fundamental building blocks. Without it, proofs can turn leaky. Mixing roots with the regular numbers failed as an attack on Fermat’s Last Theorem, but as often happens in math, the way in which it failed was provocative. It launched an area of inquiry unto itself called algebraic number theory.
Today mathematicians are actively engaged in the study of “class numbers” of number systems. In their crudest form, they’re a rating of how badly a number system fails the test of unique prime factorization, depending on which roots get mixed in: A number system that gets a “1” has unique prime factorization; a system that gets a “2” misses unique prime factorization by a little; a system that gets a “7” misses it by a lot more.
On their face, you’d expect class numbers to be randomly distributed — that class number 5 occurs with the same frequency as class number 6, or that half of all class numbers are even. That’s not the case, though, and current research in the subject aims to understand why. Today mathematicians are circling in on the structure that underlies class numbers and inching closer to establishing the truth about long-conjectured values. It’s an effort that has generated insights about how numbers behave that go far beyond a proof of any one problem.
Around the same time Lamé gave his failed proof, the German mathematician Ernst Kummer developed a way to fix the loss of prime factorization with what he called “ideal numbers.” They’re not numbers in any conventional sense. Rather, they’re sprawling constructions in set theory that perform a number-like function.
For example, the simplest ideal is the infinite set of all multiples of a given integer — 5, 10, 15, 20 and so on. Ideals can be added into an already expanded number ring to restore unique factorization. They allow mathematicians to reconcile competing prime factorizations into a single set of prime factors.
Ideals can be categorized into various classes. The number of different classes of ideals you need to add to a number ring in order to restore unique factorization is the ring’s “class number.”
The study of class numbers goes at least as far back as Carl Friedrich Gauss in the early 19th century. In a sign of how hard it’s been to make progress in this area, many of his results are still state of the art. Among his contributions, Gauss conjectured that there are infinitely many positive square roots that can be adjoined to the whole numbers without losing unique factorization — a proof of which remains the most sought-after result in class numbers (and is rumored to have frustrated Kurt Gödel enough to make him give up number theory for logic). Gauss also conjectured that there are only nine negative square roots that preserve prime factorization. √-163 is the very last one.
Today, research on class numbers remains inspired by Gauss, but much of it takes place in a context established in the late 1970s by the mathematicians Henri Cohen, emeritus professor of mathematics at the University of Bordeaux, and Hendrik Lenstra, who recently retired from Leiden University in the Netherlands. Together they formulated the Cohen-Lenstra heuristics, which are a series of predictions about how frequently particular kinds of class numbers should appear. For example, the heuristics predict that 43 percent of class numbers are divisible by 3 in situations where you’re adjoining square roots of negative numbers.
“That’s interesting because it tells you this way in which class numbers are behaving unexpectedly. If you go and look at a list of telephone numbers or something, then generally speaking one in three of them should be divisible by 3,” said Akshay Venkatesh, a mathematician at Stanford University.
Gauss had to compute class numbers by hand. By the time Cohen and Lenstra made their predictions, computers made it possible to rapidly calculate class numbers for billions of different number rings. As a result, there is good experimental evidence to support the Cohen-Lenstra heuristics. However, knowing something with confidence is entirely different from proving it.
“Probably in other sciences this is where you’d be done. However, in math that’s just the beginning. Now we want to know for sure,” said Melanie Wood, a mathematician at the University of Wisconsin-Madison.
The fact that class numbers are not distributed randomly suggests something interesting is going on beneath the surface. A class number, remember, tells us something about a given number ring: the number of classes of ideals required to restore unique factorization. Those ideal classes form the “class group” of that number ring. Groups have all sorts of interesting structural properties that are not evident just from knowing the number of elements they contain, in the same way that knowing the number of people in a family doesn’t tell you much about how those people are related.
In order to understand why class numbers are distributed as they are, mathematicians need to study the structure of the class groups that give rise to the class numbers. In particular, they’re interested in the amount of symmetry in one group versus another, with the understanding that groups that have more symmetry will occur proportionally less often than groups with less symmetry.
To see the relationship between the amount of symmetry something has and the frequency with which it occurs, consider a geometric example. Start with three points arranged to make a triangle. (These points are analogous to elements of a group, but they’re not a group in any real mathematical sense.) Now think of all possible ways of connecting those points with lines, which are a stand-in for mathematical relationships. There are eight possible configurations:
- One with three lines that make a triangle.
- Three with two lines that make an ‘‘open jaw’’ shape.
- Three with one line that connects two points.
- One with no lines.
The triangle has six symmetries and appears once. The open jaw shape has two symmetries and appears three times. Or, put another way, the triangle has three times as much symmetry as the open jaw and appears one-third as often. This relationship — the more symmetry something has, the less often it occurs — holds throughout mathematics. It’s true because the less symmetry something has, the more ways it can appear. Consider that there are an infinite number of two-dimensional shapes with no symmetry, but only one shape that has infinite lines of symmetry — the circle.
“It’s not just a rough [correlation], it’s exact and precise: If one thing has three times as many symmetries, it appears one-third as often,” said Wood.
The same relationship between how symmetric something is and how frequently it occurs holds for the way groups are constructed. In the example above, relationships are defined by lines drawn between points. In a group, relationships are established by the way the elements of the group can be added together.
To be a group, those additive relationships have to satisfy certain axioms. The elements of class groups must obey the associative and commutative properties of addition, and must include a zero element, such that zero plus any other element leaves the element unchanged. The whole numbers are in a sense the original group because they satisfy all these axioms. But certain finite sets (like class groups) also satisfy these axioms, creating, in essence, miniature number systems.
Knowing that a group has, say, four elements doesn’t tell you everything about how those four elements are related to one another. Consider two groups — call them Group 1 and Group 2 — each with four elements. What’s different about the two groups is the additive relationships between those elements. The tables below show what happens when you add an element to another element in each group.
In this setting, a “symmetry” of the group occurs wherever it’s possible to rearrange elements of the group in a way that preserves the addition structure of the group. For Group 2, there are two such symmetries: the “identity” symmetry (in which you don’t change the places of any elements), and the symmetry that swaps x with z. (Because x + x = y and z + z = y, x and z are interchangeable.)
Group 1 has more symmetries. The elements a, b, and c are all interchangeable, since a + a = 0, b + b = 0, and c + c = 0. Given that, every way of rearranging these three elements is a symmetry (or “automorphism”) of the group. If you work through all the combinations you see there are six symmetries in all. Putting this together, Group 1 has three times as many symmetries as Group 2. You’d therefore expect to find Group 2 three times as often as you would Group 1, in keeping with the rule that arrangements occur in inverse proportion to their number of symmetries. This law is as true for groups with four simple elements like Group 1 and Group 2 as it is for other, more complicated, groups of ideals.
When mathematicians are confronted with a class number, they want to know the structure of the underlying group it represents. If they can establish the structure of the underlying group, and establish how frequently groups of a given structure arise, they can bring that information back to the surface and use it to understand how often a given class number should occur.
If you start to examine the group structure and its symmetries, then “suddenly it gives you what the distribution of class numbers should be on the nose,” said Bhargava.
A New Way to Test Structure
The two groups above are (relatively) easy to parse. Groups of ideals are much harder to pin down; it’s not easy to sketch out their addition tables. Instead, mathematicians have ways of probing the groups, testing their structure, even when they can’t see the whole thing completely. In particular, they test how far each element in the group is from zero.
Recall that every group has a zero element that, when added to any other number, leaves that number unchanged. To investigate the structure of class groups, mathematicians try to get a feel for the number of elements in a given class group that have what they call “n-torsion,” which means that when you add n copies of the element, you wind up at the zero of the group. An element is 2-torsion, for example, if x + x = 0, 3-torsion if x + x + x = 0, 4-torsion if x + x + x + x = 0 and so on.
One way to make clear the difference between the two groups above is to consider how many of their elements are 2-torsion. In Group 1, all four elements are 2-torsion, which is evident by the line of zeroes on the diagonal: 0 + 0 = 0, a + a = 0, b + b = 0, c + c = 0. In Group 2, only 0 and y are 2-torsion. The amount of different types of torsion in the group is an exact reflection of the group’s overall structure.
“If the number of n-torsion elements in two groups is the same for all n then they’re the same group. Investigating how many n-torsion elements there are is a simple strategy that probes the group and is enough to recover the group if you understand everything about torsion,” said Bhargava.
A lot of the work on the Cohen-Lenstra heuristics today has to do with establishing how many elements in a class group have different types of torsion. The Cohen-Lenstra predictions with respect to torsion are quite easy to state. For example, if you’re adjoining the square roots of negative numbers, how many ideals in their class group should have 3-torsion? Cohen-Lenstra predict that there should be on average two 3-torsion elements per number ring. How many should have 5-torsion? 7-torsion? 11-torsion? The answer again, for each prime, is two.
This constancy is striking because from a naïve perspective, you’d expect the number of elements with a given torsion to grow as the size of the class group grows. Yet even as the sizes of the class groups vary, the Cohen-Lenstra heuristics predict that the number of elements with, say, 3-torsion, will on average remain constant.
“It’s interesting that this prediction is independent of the prime number,” said Bhargava. “It’s an amazing prediction.”
It’s an amazing prediction that’s been borne out statistically in countless computer runs, yet remains hard to prove.
Lowering the Bound
The Cohen-Lenstra heuristics, further extended by Cohen and Jacques Martinet in 1987, have been around for more than 40 years. Yet you could summarize progress on them on a Post-it. Only two cases have ever been proved: one in 1971, by Harold Davenport and Hans Heilbronn, and another in 2005 by Bhargava. Otherwise, “almost nothing has been proven,” said Bhargava.
With proofs of the heuristics being hard to come by, mathematicians have adopted more modest goals. They’d like to prove that the average number of n-torsion elements for a given prime is as expected, but short of that, they’ll settle for at least putting a ceiling on the number. This is called establishing an upper bound, and mathematicians have been making gradual progress in this regard.
When you’re adjoining the square root of a negative number to your number system, the class number grows in proportion to the size of the square root. If you’re adjoining the square root of –13, you can expect the class group to be, at most, about square root of 13 elements in size. Another way of writing the square root for any number n is n0.5, and that number — the 0.5 in the exponent — is the place mathematicians start when trying to fix an upper bound. If the whole class group contains n0.5 elements, then you know from the start that there can’t be more than n0.5 elements with say, 3-torsion, because that would be every element. For that reason, n0.5 is considered the trivial bound on n-torsion in the class group.
Mathematicians typically use one of several general approaches to lowering these bounds. One is an approach called a “sieve,” which you can analogize as “panning” for n-torsion elements the way a prospector pans for gold. The two other methods involve complicated transformations through which elements with n-torsion can be counted as lattice points in a region or on a curve.
One of the first to break the trivial bound was Lillian Pierce, a mathematician at Duke University, when, in 2006, she proved that the number of 3-torsion elements in a particular number ring is at most n0.49. It was a small improvement over the trivial bound, but it started a trail that other mathematicians followed. Independently and around the same time, Venkatesh and Harald Helfgott of the University of Göttingen lowered the bound to n0.44, and the next year Venkatesh and Jordan Ellenberg of the University of Wisconsin-Madison brought the bound down even further, to n0.33. These are not expected to be the optimal bounds, but they do move the field forward. “From my point of view it’s much more important to prove anything at all in the first place,” said Venkatesh.
The most recent result in this area comes from Bhargava and five coauthors, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, and Yongqiang Zhao. In January, they posted a paper to the scientific preprint site arxiv.org that lowered the bound for 2-torsion in cubic and quartic number rings to n0.28. In that same paper they also proved that they can break the trivial bound for 2-torsion for number rings in any degree.
“It is just a small savings, but it’s chipped away at the trivial bound for the first time in infinitely many cases,” said Pierce.
Even that small savings has already paid mathematical dividends. The methods Bhargava and his collaborators used have proved useful for bounding the number of solutions to a specific class of polynomial equation called elliptic curves, which is consistent with the way that class numbers seem to be situated at the intersection of many different mathematical fields. And, while there’s a long way to go before this happens, progress on class numbers could end up redeeming the original purpose of the number rings they describe.
“A proof of Fermat’s Last Theorem has never been obtained just by studying these class numbers,” said Bhargava. “If we fully understood how class groups behave in general, it seems conceivable a proof of that kind could work for FLT and for many other equations. It’s hard to say because we still have a long way to go.”
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US intelligence agencies face a difficult task. They are supposed to provide meaningful analysis that enables officials to manage serious national security problems such as terrorism, weapons proliferation, network attacks on government infrastructure, and counterintelligence efforts. Today these are diffuse and complex threats. There are newly powerful political actors on the international stage. Organizations that are not governments and have no physical territory can inflict great harm. And individuals and diffuse coalitions are increasingly able to traffic in military technology, digital viruses, and other dangerous, potentially lethal tools. These challenges are real, and overcoming them are legitimate goals of foreign intelligence surveillance.
But the track record of the collection programs Edward Snowden revealed provides little evidence that massive surveillance will help us identify future terrorist attacks or mitigate these new risks. American spies’ allegiance to massive surveillance is based on faith, not track record. The Boston Marathon bombing in April of 2013 illustrates how broad proactive surveillance is no panacea against attacks. The NSA was conducting its massive spying at the time, and the attacks happened anyway.
In that case, two brothers allegedly built pressure cooker bombs and placed them near the finish line of the Boston Marathon. The bombings killed three people and injured scores of others. The older brother died and the younger brother was injured in the subsequent manhunt. The younger brother, Dzhokhar Tsarnaev, was sentenced to death in early 2015. Relevant information about the bombers did not come from electronic surveillance. Rather, it came from another government. A few years before the bombing, the Russian government had warned the FBI that the older brother, Tamerlan Tsarnaev, was dangerous. The FBI investigated and found nothing to link either person to terrorism, so they closed the investigation in June 2011. But later that same year, the Russians sent the same warning to the CIA.
The CIA asked the U.S. National Counterterrorism Center to add Tamerlan and his mother’s names to a terrorism watch list. That watchlist is called the Terrorist Identities Datamart Environment, or TIDE. You can be placed on the TIDE list based on an informant pointing the finger at you, your social media posts, or the conduct of your relatives. The list is used to generate other watch lists, like the No Fly list and border security lists. Being on the TIDE list can really complicate your life. But it doesn’t necessarily assist the government in identifying true threats among the more than one million people on the list. How could it, when there are so many people on the list for so many different reasons? The Tsarnaev family might have been on the radar, but even with massive NSA collection of phone call and email data, no one identified the plot.
We should not conflate massive surveillance with broad data collection used to investigate crimes that have already occurred.
After the bombing, FBI agents looked at nearly 13,000 videos and more than 120,000 photographs taken near the scene of the bombing. They found a video that seemed to show the perpetrators. They were the only people who didn’t look surprised when the first bomb went off. The FBI then released the video, asking for the public’s assistance in locating the men. Farhad Manjoo, a reporter for the Wall Street Journal, the New York Times, and National Public Radio, argues that the Boston Marathon bombing makes a good case for broad surveillance. The FBI’s access to so many video and still images is what helped them identify and eventually catch the bombers.
But we should not conflate massive surveillance with broad data collection used to investigate crimes that have already occurred. The Boston Marathon investigation photos and videos were made by private parties. The government did not collect, aggregate, or analyze them until after they knew that a crime had happened. This wasn’t a fishing expedition. The investigators knew what block of the city to focus on, what time frame, and what they were looking for. While the amount of information collected was large, the targeting was narrow. Officers were investigating a particular crime. They collected only videos and photos that would likely contain evidence of that crime.
That’s not to say that there are no problems with broad collection, even in the criminal context. For example, in the same Boston Marathon investigation, FBI agents searched for purchase records for the model of pressure cooker used to construct the bombs detonated in the attack. They were looking to narrow the field of potential suspects. It turned out, there were only a few dozen of those pressure cookers sold in the year before the attack. But what happened next is worrisome. A woman reported that law enforcement paid her a visit after she had been shopping for pressure cookers and backpacks online. Ultimately the family learned that the investigation was spurred when the husband’s former employer reported his Internet searches to local police. FBI agents confronted a Saudi student for carrying a pressure cooker to a student dinner. Interrogating people who purchase pressure cookers is not a good way to find future attackers. Millions of people purchase these devices without using them in a bombing attack.
Targeted surveillance of people known to be connected to terrorism is the best way to find terrorists. Indeed, almost every major terrorist attack on Western soil in the past fifteen years was committed by someone already on the government’s radar for one or another reason. In January of 2015, two gunmen shot twelve people dead in the Paris offices of satirical magazine Charlie Hebdo. One of the gunmen had already been sent to prison for recruiting jihadist fighters. The other had reportedly studied in Yemen with Umar Farouk Abdulmutallab, who was arrested by the FBI in 2009 after trying and failing while on an airplane to detonate explosives hidden in his underwear. The leader of the July 7, 2005 London suicide bombings had been observed by British intelligence meeting with a suspected terrorist. The men who planned the Mumbai, India attacks in 2008 were already under electronic surveillance by the United States, the United Kingdom, and India. One of the Mumbai plotters had been a DEA informant. Investigators received multiple tips from the informant’s family members, friends, and acquaintances, but the officials never effectively followed up on the information.
In another case where the government failed to understand the information it had and act accordingly, Maj. Nidal Hasan, a military psychiatrist, killed thirteen people at Fort Hood, Texas, in 2009. Intelligence agencies had intercepted multiple emails between Hasan and Anwar al-Awlaki, a notoriously militant cleric living in Yemen. In the emails, Hasan asked Awlaki whether a Muslim US soldier who committed fratricide would be considered a martyr in the eyes of Islam. Despite this and other information that could justify discharging Hasan from the military, counterterrorism investigators didn’t follow up on these emails. While the Defense Department faulted failures of leadership, the Senate investigated the military’s unwillingness to name, detect, or defend against violent Islamist extremism. Scholar Amy Zegart points the finger at the Army’s organizational incentives for promoting and disciplining subordinates as well.
Historian Peter Bergen has assessed the historical record, including the case of Umar Farouk Abdulmutallab, the man who attempted and failed to detonate a bomb in his underwear on Christmas Day 2009. A few weeks before the botched attack, Abdulmutallab’s father contacted the US Embassy in Nigeria with concerns that his son had become radicalized and might be planning an attack. This information wasn’t further investigated. While the White House concluded that the government did not have sufficient information to determine that Abdulmutallab was likely working for al-Qaeda in Yemen and that the group was looking to expand its attacks beyond Yemen, the man was nevertheless allowed to board a plane bound for the United States without any question despite his father’s warning. Bergen concludes by arguing that, “[a]ll of these serious terrorism cases argue not for the gathering of ever vaster troves of information but simply for a better understanding of the information the government has already collected and that are derived from conventional law enforcement and intelligence methods.”
Massive spying didn’t stop these attacks. Nevertheless, the intelligence community seems confident that more information collection will prevent terrorism. Superficially, it just makes sense that in order to “connect the dots” you first have to “collect the dots.” The public conversation about the effectiveness of massive surveillance seems to assume this is the case. For example, in her June 6, 2013 press conference seeking to justify the section 215 phone dragnet The Guardian had just revealed, Senator Feinstein claimed that the FBI had uncovered approximately 100 terrorist plots since 2009. She couldn’t credit the phone dragnet for success in any of those investigations, but it didn’t matter to her:
I do not know to what extent metadata was used or if it was used, but I do know this: That terrorists will come after us if they can and the only thing we have to deter this is good intelligence. To understand that a plot is being hatched and to get there before they get to us.
Feinstein may have revealed more by this statement than she intended. We don’t know what works to identify terrorist plots. But surveillance is one thing we know how to do well. So we are going to do that to stop the terrorists. It’s a little like looking under the lamppost for your keys, because that’s where the light is.
Following classified and public hearings, Senator Leahy and other members of Congress concluded that there was no evidence that the phone records collection has ever been important in fighting terrorism. A subsequent review by the Privacy and Civil Liberties Oversight Board reached the same conclusion. Hundreds of thousands of Americans have had their phone records collected for years, but it hasn’t made the country any safer. A 2009 inspector general report says that the section 215 dragnet cost taxpayers $146 million in supplemental counterterrorism funds to buy new hardware and contract support and to make payments to the phone companies for their collaboration. As set out in later chapters, the program was misused and abused for years. America is none the safer for it.
Excerpted from American Spies: Modern Surveillance, Why You Should Care, and What to Do About It, by Jennifer Stisa Granick. Used with permission of Cambridge University Press. All rights reserved.
In his address to Congress, President Donald Trump said he’s preparing “a budget that rebuilds the military, eliminates the Defense sequester, and calls for one of the largest increases in national defense spending in American history.” The day before the speech, White House officials said the administration planned to propose a “historic” $54 billion increase in the defense budget.
The administration’s shopping list is a long one: more Navy ships, more troops, and a nuclear arsenal that is “top of the pack.” The proposed spending increase, however, isn’t as impressive as it sounds. A host of prosaic but urgent requirements would probably gobble up a good chunk—even if this money represented real, new dollars. Which it mostly doesn’t.
Here’s the truth: The Trump administration measured its $54 billion increase against budget caps put in place by the 2011 Budget Control Act. But the Obama administration routinely spent above those caps, and it accounted for a large portion of that $54 billion in its last budget projection. “Just to keep what you have now, $35.5 billion are already spoken for,” says Katherine Blakeley, a research fellow at the Center for Strategic and Budgetary Assessments.
It’s not going to be $54 billion for Defense. It’s more like $18 billion. Still, that’s the entire budget of NASA.
That leaves $18 billion. Now, for sure, to most other government agencies $18 billion would be an epic windfall. (That’s basically NASA’s entire budget.) But in the Defense Department, it doesn’t go that far.
So … how should the Defense Department spend their whole NASA’s worth of dough? Defense wonks have a few ideas.
The first thing on the wish list: Basic maintenance. Equipment doesn’t stop costing money the moment it rolls off a production line. Ships, aircraft, and vehicles all need to be maintained to stay in good working order. When the Budget Control Act’s caps came into force, maintenance took a hit, and so did the availability of equipment in need of it. Defense News reported in February that up to half of the Navy’s F/A-18 fighter jet fleet has been grounded due to maintenance delays, with a growing backlog of ships also waiting to be serviced.
Meanwhile, the wars in Afghanistan, Iraq, and Syria have further strained military equipment. Delays in the availability of newer systems like the F-35 Joint Strike Fighter are pushing the military to use increasingly older systems.
“The number one area I would devote my $18 billion to would be maintenance,” says Blakeley. “Specifically, depot maintenance for the Air Force, for Navy and Marine Corps aviation and for Navy ships.” Blakeley estimates satisfying all those needs would cost around $8 billion.
Training is another area where budget cutbacks have bitten deep. Air Force brass have repeatedly warned of a shortfall of 700 pilots due to difficulties in recruitment and retention. Meanwhile, pilots have had difficulty getting the training hours to advance in career and rank. I would tell the Vice Chiefs of Staff, get your training up, get your flight hours up,” says Lawrence Korb, an assistant secretary of defense in the Reagan administration and a fellow at the Center for American Progress.
That ties back into maintenance. “The motivating factor most of the time is they aren’t flying enough because they don’t have enough planes in flying condition,” says Steve Bell, a former staffer on the Senate Budget and Appropriations committees and a fellow at the Bipartisan Policy Center. “If you just sit there and you don’t have enough training, you don’t have a plane to fly, there’s an erosion of morale.” Airlines offer better pay and plenty of flight hours, inducing pilots to simply leave the service.
If you just sit there and you don’t have enough training, you don’t have a plane to fly, there’s an erosion of morale. Steve Bell, Bipartisan Policy Center
Bolster maintenance and training, and you also bolster retention of hard-to-replace people in the armed services.
Still, spending on maintenance and training doesn’t increase “end-strength”—military parlance for people—or buy new equipment. Some defense advocates argue those priorities are just as important to start spending on now to deter potential enemies later.
“The White House and Pentagon civilian leadership are coalescing around a shortsighted investment strategy that seeks to pour money into immediate readiness needs and far-off technological bets,” says Mackenzie Eaglen, a fellow at the American Enterprise Institute, in an email. In his January budget guidance, Defense Secretary James Mattis hinted as much, laying out a plan to prioritize readiness issues while putting off investing in new troops and gear.
Eaglen says that while short term needs like maintenance and training are important, “leadership should pursue a truly balanced investment strategy that includes modest, tailored end strength growth and, more importantly, which buys existing equipment and upgrades at higher rates.”
In the coming weeks, the Trump administration will spell out line by line how it plans to spend its proposed defense budget increase. Less obvious is how it plans to get Congress to approve the budget. With top Republicans like Senator Lindsey Graham already calling Trump’s budget “dead on arrival,” the politics may be even more difficult to navigate than the math.
After Larry Biegler realized the emerald was gone from the Commonwealth International storage unit, he called the Temple City police and told the officer on duty that his emerald had been stolen and that he’d been abducted and released by the Brazilian Mafia. This triggered the arrival of Los Angeles County Sheriff’s Department detectives Miller and Gayman. Soon, Ferrara and Morrison became suspects. The detectives took a few weeks to track them down, but by December 15, 2008, Miller and Gayman were in Eagle, Idaho, in the Boise foothills, staking out Morrison’s house. They set a perimeter and shivered in their rental car for two days.
On the third day they knocked on Morrison’s door. His wife answered and said Morrison wasn’t home. As the detectives were talking to her, they saw a man walking around the side of the property and, figuring it was Morrison, tackled him. He turned out to be a cable repairman. Morrison’s wife got Morrison on the phone and he cut a deal with Miller and Gayman. He would meet the detectives in Las Vegas, where he and Ferrara had stored the emerald, and they’d turn the stone over to the Sheriff’s Department on the condition that neither Ferrara nor Morrison would be arrested. So Miller and Gayman flew home to Burbank and assembled a small army, including a dozen officers with assault rifles, and caravanned overnight out I-15 East. When they arrived at the depository at 7 am, the Las Vegas Metro Police Department was already onsite with a SWAT team and helicopter cover. Morrison showed up in a sport coat and slacks, and within the hour Miller and Gayman were wheeling a piano dolly topped with a gargantuan emerald into the desert sun. Everybody took a lot of selfies. Then the detectives loaded the Bahia emerald into a police van, drove it back over the San Gabriel Mountains, and logged it into evidence.
Ferrara spent a lot of days in settlement hearings. Only once did he lunge across the conference table and threaten to beat the shit out of somebody.
As promised, the Los Angeles Sheriff’s Department threw the question of who owned the Bahia emerald over to the Los Angeles Superior Court. From 2007 to 2015, people began endless legal battles: Conetto sued Morrison, Thomas went after Conetto, the New York gem dealer sued Biegler. Ferrara spent a lot of days in settlement hearings and a lot of nights sleeping in hidden corners of hotel lobbies so he didn’t have to pay for rooms. Only once did he lunge across the conference table and threaten to beat the shit out of somebody.
During the legal proceedings, Biegler disappeared. Conetto got distracted by a friend’s new business that turned manure into electricity. The detectives came to believe that the emerald belonged to Thomas. After all, courts found he was the only litigant who’d ever paid anything for the stone. But Thomas fared poorly at his trial. Several key facts were not on his side. One, he never called FedEx to see what happened to his $925 million package. Two, he claimed his house burned down in 2006 and incinerated his bill of sale. (The court found his claim awfully convenient.) It also turned out, though this was not revealed at trial, that there was no large emerald at the British Museum in London at all. The entire backstory of the $792 million comp was made up.
The court had great difficulty pinning down who owned the emerald or how much it was worth—or, really, any facts at all, because so many men contradicted one another under oath. This led an observer to the possibility that the stone was really a MacGuffin, in the classic Hitchcockian sense—an object that everyone’s chasing but that doesn’t really matter.
Still, in 2011, the judge rejected Thomas’ claim of ownership. Then the judge got a new job, and Thomas asked for a mistrial—which the courts granted. In 2013, a second judge heard all this insanity again. But by that point Ferrara, Morrison, and another guy had gathered into a sort of consortium, under the name FM Holdings. That way someone, any one of them, could reclaim possession of the emerald, sell it, and divide the proceeds.
The LA Superior Court awarded the Bahia emerald to FM Holdings on June 23, 2015.
But perhaps the emerald really is cursed. Before the Sheriff’s Department received the order to release the emerald to FM Holdings, the District Court of DC granted an injunction filed by the Department of Justice on behalf of the country of Brazil. Brazil claimed that the Bahia emerald had been illegally exported and really belonged to them.
“I’ll be honest,” says John Nadolenco, the primary lawyer on the case for Brazil. “When I first got the letter”—from Brazil, asking for help repatriating the Bahia emerald—“I thought it was a total hoax. I thought it was one of those Nigerian prince things where they’re going to want us to send a couple million dollars to some bank account and they’re going to take all of our money.” But Nadolenco’s partner asked him to pursue the client. Nadolenco wrote back to the Brazilians with a real proposal, though he couldn’t resist including the jokey promise that his friend Indiana Jones could help reclaim the emerald if his own efforts failed. He got the gig.
Ferrara made his point: Life is tough. People betray you and die. We all need escapes.
So today the Los Angeles County Sheriff’s Department is still—still—holding the emerald, now as evidence for a criminal case they’re building. The limbo is uncomfortable for Ferrara. He’s a big man with big, tenacious, preposterous dreams stuck in a life that feels too banal, empty, and small. My last day in Florida we met up at Cracker Barrel. Ferrara likes the tchotchkes there. During a lull in the conversation, Chrystal told me she worries what will happen if Ferrara loses the emerald for good. “It would devastate him,” she whispered. “It’s his whole life.”
Ferrara and I talked for hours and hours and hours, from the retiree breakfast rush past lunch, through every last detail of the saga. At one point, he placed the salt and pepper shakers in the middle of the table. He slid them a few inches apart and set his phone on top, like the flat roof of a house. “This is our foundation in life—your mother, your father, friends, teachers, the people that mean something to you.” (He meant the shakers to represent the people and the phone to be your life.)
He slid the shakers out from under the phone. “As these people fail you, these go away, one by one.” The phone, your life, falls.
Before I headed to the airport, we returned the truck to U-Haul and revisited Dunkin’ Donuts for some more iced coffees. We sat outside, in the horrible humid air, so Ferrara could smoke his Marlboros. He mentioned that, along with SpongeBob, he connected with Lemony Snicket’s A Series of Unfortunate Events, or at least the title. “Like I wrote one time, ‘We entered a world that was inhabited by dark shadows, the nights would never end, the mornings would never come,’” he said. He didn’t quite get the quote from his own prose correct. But he made his point: Life is tough. People betray you and die. We all need escapes.
I drove to the airport. I boarded my flight. Even before my plane touched down, Ferrara had left me a voicemail. “Call me!” he bellowed, optimistic as ever. “You will never guess what transpired today. As you left, the winds of change blew in.”
I called him back the next morning. He told me a story about the emerald, which I understood less the longer he talked. He also mentioned that he’d been approached about hosting a TV show, a reality treasure-hunter series. He would be the star. It was nice to hear his voice.
Elizabeth Weil (@lizweil) lives in San Francisco and is a contributor for The New York Times Magazine and Outside.
Additional reporting by Brendan Borrell.
This article appears in the March issue. Subscribe now.
Whenever the existence of an extra-solar planet is confirmed, there is reason to celebrate. With every new discovery, humanity increases the odds of finding life somewhere else in the Universe. And even if that life is not advanced enough (or particularly inclined) to build a radio antenna so we might be able to hear from them, even the possibility of life beyond our Solar System is exciting.
Unfortunately, determining whether or not a planet is habitable is difficult and subject to a lot of guesswork. While astronomers use various techniques to put constraints on the size, mass, and composition of extra-solar planets, there is no surefire way to know if these worlds are habitable. But according to a new study from a team of astronomers from Cornell University, looking for signs of volcanic activity could help.
Their study – titled “A Volcanic Hydrogen Habitable Zone” – was recently published in The Astrophysical Journal Letters. According to their findings, the key to zeroing in on life on other planets is to look for the telltale signs of volcanic eruptions – namely, hydrogen gas (H²). The reason being is that this, and the traditional greenhouse gases, could extend the habitable zones of stars considerably.
“On frozen planets, any potential life would be buried under layers of ice, which would make it really hard to spot with telescopes. But if the surface is warm enough – thanks to volcanic hydrogen and atmospheric warming – you could have life on the surface, generating a slew of detectable signatures.”
Planetary scientists theorize that billions of years ago, Earth’s early atmosphere had an abundant supply of hydrogen gas (H²) due to volcanic outgassing. Interaction between hydrogen and nitrogen molecules in this atmosphere are believed to have kept the Earth warm long enough for life to develop. However, over the next few million years, this hydrogen gas escaped into space.
This is believed to be the fate of all terrestrial planets, which can only hold onto their planet-warming hydrogen for so long. But according to the new study, volcanic activity could change this. As long as they are active, and their activity is intense enough, even planets that are far from their stars could experience a greenhouse effect that would be sufficient to keep their surfaces warm.
Consider the Solar System. When accounting for the traditional greenhouse effect caused by nitrogen gas (N²), carbon dioxide and water, the outer edge of our Sun’s habitable zone extends to a distance of about 1.7 AU – just outside the orbit of Mars. Beyond this, the condensation and scattering of CO² molecules make a greenhouse effect negligible.
However, if one factors in the outgassing of sufficient levels of H², that habitable zone can extend that outer edge to about 2.4 AUs. At this distance, planets that are the same distance from the Sun as the Asteroid Belt would theoretically be able to sustain life – provided enough volcanic activity was present. This is certainly exciting news, especially in light of the recent announcement of seven exoplanets orbiting the nearby TRAPPIST-1 star.
Of these planets, three are believed to orbit within the star’s habitable zone. But as Lisa Kaltenegger – also a member of the Carl Sagan Institute and the co-author on the paper – indicated, their research could add another planet to this
“Finding multiple planets in the habitable zone of their host star is a great discovery because it means that there can be even more potentially habitable planets per star than we thought. Finding more rocky planets in the habitable zone – per star – increases our odds of finding life… Although uncertainties with the orbit of the outermost Trappist-1 planet ‘h’ means that we’ll have to wait and see on that one.”
Another upside of this study is that the presence of volcanically-produced hydrogen gas would be easy to detect by both ground-based and space-based telescopes (which routinely conduct spectroscopic surveys on distant exoplanets). So not only would volcanic activity increase the likelihood of there being life on a planet, it would also be relatively easy to confirm.
“We just increased the width of the habitable zone by about half, adding a lot more planets to our ‘search here’ target list,” said Ramirez. “Adding hydrogen to the air of an exoplanet is a good thing if you’re an astronomer trying to observe potential life from a telescope or a space mission. It increases your signal, making it easier to spot the makeup of the atmosphere as compared to planets without hydrogen.”
Already, missions like Spitzer and the Hubble Space Telescope are used to study exoplanets for signs of hydrogen and helium – mainly to determine if they are gas giants or rocky planets. But by looking for hydrogen gas along with other biosignatures (i.e. methane and ozone), next-generation instruments like the James Webb Space Telescope or the European Extremely Large Telescope, could narrow the search for life.
It is, of course, far too soon to say if this study will help in our search for extra-solar life. But in the coming years, we may find ourselves one step closer to resolving that troublesome Fermi Paradox!
Further Reading: Astrophysical Journal Letters