Melting Permafrost (Part 3)

Melting permafrost is in the news! Check out this great slide show and article:

• Josh Hane, Hunting for clues to global warming, New York Times, 16 December 2011.

• Justin Gillis, As permafrost thaws, scientists study the risks, New York Times, 16 December 2011.

They track Katey M. Walter Anthony, an assistant professor at the Water and Environmental Research Center at the University of Alaska Fairbanks, as she studies methane bubbling up from lakes—as shown above.

These lakes form in an interesting way. Permafrost is permanently frozen soil lying beneath a layer 0.6 to 4 meters thick of soil that thaws in the summer and refreezes in the winter: the active layer. The permafrost itself can be much thicker—up to 1500 meters in parts of Siberia!

As far as I can tell, talik is permanently unfrozen soil on top of, amid or beneath the permafrost.

Permafrost is rock-hard and solid. Liquid water does not pass through it, so permafrost environments tend to be poorly drained and boggy. But when permafrost starts to melt, it becomes soft. Soil sinks down into marshy hollows separated by small hills, forming a kind of terrain called thermokarst.

Trees in this terrain can lean crazily as their roots sink, creating drunken forests.

On flat ground, melted water can pool into a thermokarst lake. On slopes, water pours downhill and the land can rip open in a thermokarst failure. Here are Breck Bowden and Michael Gooseff exploring a thermokarst failure in Alaska:

For more on this, see:

• Emily Stone, When the ground collapses like a soufflé: Studying the effect of thermokarst on the Arctic, Field Notes: the Polar Field Services Newsletter

All these are natural processes that are widespread at the end of each glacial period. Here’s a surprisingly delightful book which discusses this in detail:

• Evelyn C. Pielou, After the Ice Age: the Return of Life to Glaciated North America, U. Chicago Press, Chicago, 1991.

So, please don’t misunderstand: I’m not trying to say that thermokarst lakes, drunken forests and the like are signs of disaster. However, as the Earth warms, new regions of permafrost are melting, and we’ll see these phenomena in new regions. We need to understand how they work, and the positive and negative feedbacks. For example, thermokarst lakes are darker than their surroundings, so they absorb more sunlight and warm the area.

Most importantly, as permafrost thaws, it releases trapped carbon in the form of carbon dioxide and methane, which are both greenhouse gases. Since there are roughly 1.7 trillion tons of carbon in northern soils, with about 90% locked in permafrost, that’s a big deal.

At least once so far, the tundra has even caught fire:

One day in 2007, on the plain in northern Alaska, a lightning strike set the tundra on fire.

Historically, tundra, a landscape of lichens, mosses and delicate plants, was too damp to burn. But the climate in the area is warming and drying, and fires in both the tundra and forest regions of Alaska are increasing.

The Anaktuvuk River fire burned about 400 square miles of tundra, and work on lake sediments showed that no fire of that scale had occurred in the region in at least 5,000 years.

Scientists have calculated that the fire and its aftermath sent a huge pulse of carbon into the air — as much as would be emitted in two years by a city the size of Miami. Scientists say the fire thawed the upper layer of permafrost and set off what they fear will be permanent shifts in the landscape.

Up to now, the Arctic has been absorbing carbon, on balance, and was once expected to keep doing so throughout this century. But recent analyses suggest that the permafrost thaw could turn the Arctic into a net source of carbon, possibly within a decade or two, and those studies did not account for fire.

“I maintain that the fastest way you’re going to lose permafrost and release permafrost carbon to the atmosphere is increasing fire frequency,” said Michelle C. Mack, a University of Florida scientist who is studying the Anaktuvuk fire. “It’s a rapid and catastrophic way you could completely change everything.”

By the way, if you click on these scientists’ portraits, you’ll see where they work. If you’re a student looking for an interesting career, consider these options! For example, Michelle C. Mack—shown above—runs a lab, and you can see her postdocs and grad students, and what they do.

For previous posts in this series, see:

• Melting Permafrost (Part 1).

• Melting Permafrost (Part 2).

Melting Permafrost (Part 2)

This summer a Russian research ship found hundreds of plumes of methane, “of a fantastic scale”, bubbling up from the sea floor off the East Siberian coast:

• Steve Connor, Shock as retreat of Arctic sea ice releases deadly greenhouse gas, 13 December 2011.

Here are the quotes with actual new information:

In late summer, the Russian research vessel Academician Lavrentiev conducted an extensive survey of about 10,000 square miles of sea off the East Siberian coast. Scientists deployed four highly sensitive instruments, both seismic and acoustic, to monitor the “fountains” or plumes of methane bubbles rising to the sea surface from beneath the seabed.

“In a very small area, less than 10,000 square miles, we have counted more than 100 fountains, or torch-like structures, bubbling through the water column and injected directly into the atmosphere from the seabed,” Dr Semiletov said. “We carried out checks at about 115 stationary points and discovered methane fields of a fantastic scale – I think on a scale not seen before. Some plumes were a kilometre or more wide and the emissions went directly into the atmosphere – the concentration was a hundred times higher than normal.”

and

“This is the first time that we’ve found continuous, powerful and impressive seeping structures, more than 1,000 metres in diameter. It’s amazing,” Dr Semiletov said. “I was most impressed by the sheer scale and high density of the plumes. Over a relatively small area we found more than 100, but over a wider area there should be thousands of them.”

Scientists estimate that there are hundreds of millions of tonnes of methane gas locked away beneath the Arctic permafrost, which extends from the mainland into the seabed of the relatively shallow sea of the East Siberian Arctic Shelf. One of the greatest fears is that with the disappearance of the Arctic sea-ice in summer, and rapidly rising temperatures across the entire region, which are already melting the Siberian permafrost, the trapped methane could be suddenly released into the atmosphere leading to rapid and severe climate change.

Dr Semiletov’s team published a study in 2010 estimating that the methane emissions from this region were about eight million tonnes a year, but the latest expedition suggests this is a significant underestimate of the phenomenon.

I’d like to know more about Igor Semiletov’s work and what he’s just found. He was mentioned in this earlier very good article:

• Amanda Leigh Mascarelli, A sleeping giant?, Nature Reports Climate Change, 5 March 2009.

Namely:

The Siberian Shelf alone harbours an estimated 1,400 billion tonnes of methane in gas hydrates, about twice as much carbon as is contained in all the trees, grasses and flowers on the planet. If just one per cent of this escaped into the atmosphere within a few decades, it would be enough to cause abrupt climate change, says Shakhova. “When hydrates are destabilized, gas is released under very high pressure,” she says. “So emissions could be massive and non-gradual.” Shakhova and her colleague Igor Semiletov of the University of Alaska, Fairbanks, believe the plumes they’ve observed confirm previous reports that the permafrost cap is beginning to destabilize, allowing methane to escape from the frozen hydrates below. “Subsea permafrost is like a rock,” explains Semiletov. “It works like a lid to prevent escape of any gas. We believe that the subsea permafrost is failing to seal the ancient carbon pool.”

But Carolyn Ruppel, a geophysicist with the US Geological Survey in Woods Hole, Massachusetts, isn’t yet ready to attribute the methane plumes to a breakdown in methane hydrates in the subsea permafrost. “We have proof from studies that have been carried out in the past few years that there’s a lot of methane in certain shallow marine environments offshore in the Arctic,” says Ruppel. “But can we prove that the methane comes from methane hydrates? That is a critical question.”

Why is it critical? Because people are worried about global warming melting permafrost and gas hydrates on the ocean floor. Suppose these release large amounts of methane, a greenhouse gas vastly more potent than carbon dioxide. This will then makes the Earth even warmer, and so on: we have a feedback loop. In a real nightmare scenario, we could imagine that this feedback actually leads to a ‘tipping point’, where the climate flips over to a much warmer state. And in the worst nightmare of all, we can imagine something like Paleocene-Eocene Thermal Maximum, a spike of heat that lasted about 20,000 years, causing significant extinctions.

Are any of these nightmares really possible? I wrote about this question before, assembling what facts I could easily find:

• Melting permafrost (Part 1).

How much new light does Semiletov’s work shed on this question?

Luckily, a team of scientists is gearing up to answer it:

• Permafrost Carbon Network (RCN).

Here’s a paper by this team:

• Edward A. G. Schuur, Benjamin Abbott and the Permafrost Carbon Network, High risk of permafrost thaw, Nature 480 (1 December 2011), 32-33.

To get the ball rolling, they surveyed themselves. That may seem like a lazy way to write a paper, but I don’t mind it as a quick way to get a sense of the conventional wisdom… and they probably wanted to do it just to find out what they all thought! Here are the results—emphasis mine:

Our survey asks what percentage of the surface permafrost is likely to thaw, how much carbon will be released, and how much of that carbon will be CH₄, for three time periods and under four warming scenarios that will be part of the Intergovernmental Panel on Climate Change Fifth Assessment Report. The lowest warming scenario projects 1.5 °C Arctic warming over the 1985–2004 average by the year 2040, ramping up to 2 °C by 2100; the highest warming scenario considers 2.5 °C by 2040, and 7.5 °C by 2100. In all cases, we posited that the temperature would remain steady from 2100 to 2300 so that we could assess opinions about the time lag in the response of permafrost carbon to temperature change.

The survey was filled out this year by 41 international scientists, listed as authors here, who publish on various aspects of permafrost. The results are striking. Collectively, we hypothesize that the high warming scenario will degrade 9–15% of the top 3 metres of permafrost by 2040, increasing to 47–61% by 2100 and 67–79% by 2300 (these ranges are the 95% confidence intervals around the group’s mean estimate). The estimated carbon release from this degradation is 30 billion to 63 billion tonnes of carbon by 2040, reaching 232 billion to 380 billion tonnes by 2100 and 549 billion to 865 billion tonnes by 2300. These values, expressed in CO₂ equivalents, combine the effect of carbon released as both CO₂ and as CH₄.

Our estimate for the amount of carbon released by 2100 is 1.7–5.2 times larger than those reported in several recent modelling studies, all of which used a similar warming scenario. This reflects, in part, our perceived importance of the abrupt thaw processes, as well as our heightened awareness of deep carbon pools. Active research is aimed at incorporating these main issues, along with others, into models.

Are our projected rapid changes to the permafrost soil carbon pool plausible? The survey predicts a 7–11% drop in the size of the permafrost carbon pool by 2100 under the high-warming scenario. That scale of carbon loss has happened before: a 7–14% decrease has been measured in soil carbon inventories across thousands of sites in the temperate-zone United Kingdom as a result of climate change. Also, data scaled up from a single permafrost field site point to a potential 5% loss over a century as a result of widespread permafrost thaw. These field results generally agree with the collective carbon-loss projection made by this survey, so it should indeed be plausible.

Across all the warming scenarios, we project that most of the released carbon will be in the form of CO₂ with only about 2.7% in the form of CH₄. However, because CH₄ has a higher global-warming potential, almost half the effect of future permafrost-zone carbon emissions on climate forcing is likely to be from CH₄. That is roughly consistent with the tens of billions of tonnes of CH₄ thought to have come from oxygen-limited environments in northern ecosystems after the end of the last glacial period.

All this points towards significant carbon releases from permafrost-zone soils over policy-relevant timescales. It also highlights important lags whereby permafrost degradation and carbon emissions are expected to continue for decades or centuries after global temperatures stabilize at new, higher levels. Of course, temperatures might not reach such high levels. Our group’s estimate for carbon release under the lowest warming scenario, although still quite sizeable, is about one-third of that predicted under the strongest warming scenario.

I found this sentence is a bit confusing:

These values, expressed in CO₂ equivalents, combine the effect of carbon released as both CO₂ and as CH₄.

But I guess that combined with a guess like “30 billion to 63 billion tonnes of carbon by 2040”, it means that they’re expecting a release of carbon dioxide and methane that’s equal, in its global warming potential, to what you’d get from burning 30 to 63 billion tonnes of carbon, turning it all into carbon dioxide, and releasing it into the atmosphere.

For comparison, in 2010 humanity burnt 8.3 billion tonnes of carbon. So, at least up to 2040, I guess they’re expecting the effect of melting permafrost to be roughly 1/8 to 1/4 of the direct effect of burning carbon.

What’s Up With Solar Power?

What’s going on with solar power? On the one hand, I read things like this:

• Paul Krugman, Here comes the sun, New York Times, 6 November 2011.

In fact, progress in solar panels has been so dramatic and sustained that, as a blog post at Scientific American put it, “there’s now frequent talk of a ‘Moore’s law’ in solar energy,” with prices adjusted for inflation falling around 7 percent a year.

This has already led to rapid growth in solar installations, but even more change may be just around the corner. If the downward trend continues–and if anything it seems to be accelerating—we’re just a few years from the point at which electricity from solar panels becomes cheaper than electricity generated by burning coal.

This would be a big deal! As you may have noticed, attempted political remedies for global warming aren’t working too well yet. Cheap solar power won’t be enough to solve the problem: even if we can build a grid that deals with the intermittency of solar power, the problem is that electric power only accounts for some of the fossil fuel burnt. But it could help.

On the other hand, I read things like this:

• Jackie Chang, Half of China solar firms halt production, says report, Digitimes, 9 December 2011.

About 50% of the firms in China’s solar industry have suspended production, according to the country’s Guangzhou Daily.

The daily cited the solar energy division of CSG Holding as claiming that half of the solar firms have stopped production, 30% have halved their output and 20% are trying to maintain certain levels of production.

Digitimes Research’s findings have indicated that only tier-one solar firms in China had capacity utilization rates over 80% in the first half of 2011 while tier-two and tier-three firms were already facing falling capacity utilization rates.

Guangzhou Daily stated that oversupply and significant price drops are the reasons for the firms to shut down production.

The report also indicated that China firms have been facing increasing production costs following news on September 2011 that one of the large-size solar players had a chemical leak at one of its plants that polluted a nearby river. This means the other solar firms now face increasing costs to prevent such pollution while suffering from sharp price drops and low demand.

And this:

• Yuliya Chernova, Chinese solar industry fueled by unsustainable debt, analysts say, Wall Street Journal, 8 December 2011.

Even now, as the U.S. reevaluates its federal loan and other subsidy programs for renewable energy, some lawmakers invoke the strong support the Chinese government offers to its own renewable energy industry as a call for the U.S. to match up with its own support.

Indeed, easy access to low-interest loans over the past three years helped Chinese solar makers build up capacity, and quickly take over market share from European and U.S. manufacturers. In 2010 alone, the China Development Bank made $35 billion in low-interest credit available to Chinese renewable energy companies, according to Bloomberg New Energy Finance, a figure cited by Energy Secretary Steven Chu in his testimony to the House Energy and Commerce Committee in mid-November.

But, perhaps an unintended consequence of this easy access to capital was that the cheap, plentiful production of solar panels resulted in a cutthroat pricing competition, which, in turn is now starting to suffocate the very same large, leading Chinese manufacturers.

“We remain concerned about debt levels across the solar manufacturing complex given the compression of profit margins,” wrote Think Equity analysts in a recent report. “With increasing net debt and reduced module prices, it is hard to imagine absolute gross margin dollars growing enough to offset existing OpEx and interest payments.”

It’s hard to know who to trust. Of course all three of these news reports could be true! Or none.

Do you know what’s really going on with solar power?

The Beauty of Roots

I feel like talking about some pure math, just for fun on a Sunday afternooon.

Back in 2006, Dan Christensen did something rather simple and got a surprisingly complex and interesting result. He took a whole bunch of polynomials with integer coefficients and drew their roots as points on the complex plane. The patterns were astounding!

Then Sam Derbyshire joined in the game. After experimenting a bit, he decided that his favorite were polynomials whose coefficients were all 1 or -1. So, he computed all the roots of all the polynomials of this sort having degree 24. That’s 2²⁴ polynomials, and about 24 × 2²⁴ roots—or about 400 million roots! It took Mathematica four days to generate the coordinates of all these roots, producing about 5 gigabytes of data.

He then plotted all the roots using some Java programs, and created this amazing image:

You really need to click on it and see a bigger version, to understand how nice it is. But Sam also zoomed in on specific locations, shown here:

Here’s a closeup of the hole at 1:

Note the line along the real axis! It’s oddly hard to see on my computer screen right now, but it’s there and it’s important. It exists because lots more of these polynomials have real roots than nearly real roots.

Next, here’s the hole at :

And here’s the hole at :

Note how the density of roots increases as we get closer to
this point, but then suddenly drops off right next to it. Note also the subtle patterns in the density of roots.

But the feathery structures as we move inside the unit circle are even more beautiful! Here is what they look near the real axis — this plot is centered at the point :

They have a very different character near the point :

But I think my favorite is the region near the point This image is almost a metaphor of how mathematical patterns emerge from confusion… like sharply defined figures looming from the mist as you drive by with your headlights on at night:

Now, you may remember that I said all this already in “week285” of This Week’s Finds. So why am I talking about it again?

Well, the patterns I’ve just showed you are tantalizing, and at first quite mysterious… but ‘some guy on the street’ and Greg Egan figured out how to understand some of them during the discussion of “week285”. The resulting story is quite beautiful! But this discussion was a bit hard to follow, since it involved smart people figuring out things as they went along. So, I doubt many people understood it—at least compared to the number of people who could understand it.

Let me just explain one pattern here. Why does this region near :

look so much like the fractal called a dragon?

You can create a dragon in various ways. In the animated image above, we’re starting with a single horizontal straight line segment (which is not shown for some idiotic reason) and repeatedly doing the same thing. Namely, at each step we’re replacing every segment by two shorter segments at right angles to each other:

At each step, we have a continuous curve. The dragon that appears in the limit of infinitely many steps is also a continuous curve. But it’s a space-filling curve: it has nonzero area!

Here’s another, more relevant way to create a dragon. Take these two functions from the complex plane to itself:

Pick a point in the plane and keep hitting it with either of these functions: you can randomly make up your mind which to use each time. No matter what choices you make, you’ll get a sequence of points that converges… and it converges to a point in the dragon! We can get all points in the dragon this way.

But where did these two functions come from? What’s so special about them?

To get the specific dragon I just showed you, we need these specific functions. They have the effect of taking the horizontal line segment from the point 0 to the point 1, and mapping it to the two segments that form the simple picture shown at the far left here:

As we repeatedly apply them, we get more and more segments, which form the ever more fancy curves in this sequence.

But if all we want is some sort of interesting set of points in the plane, we don’t need to use these specific functions. The most important thing is that our functions be contractions, meaning they reduce distances between points. Suppose we have two contractions and from the plane to itself. Then there is a unique closed and bounded set in the plane with

Moreover, suppose we start with some point in the plane and keep hitting it with and/or in any way we like. Then we’ll get a sequence that converges to a point in And even better, every point in show up as a limit of a sequence like this. We can even get them all starting from the same

All this follows from a famous theorem due to John Hutchinson.

“Cute,” you’re thinking. “But what does this have to do with roots of polynomials whose coefficients are all 1 or -1?”

Well, we can get polynomials of this type by starting with the number 0 and repeatedly applying these two functions, which depend on a parameter :

For example:

and so on. All these polynomials have constant term 1, never -1. But apart from that, we can get all polynomials with coefficients 1 or -1 using this trick. So, we get them all up to an overall sign—and that’s good enough for studying their roots.

Now, depending on what is, the functions and will give us different generalized dragon sets. We need for these functions to be contraction mappings. Given that, we get a generalized dragon set in the way I explained. Let’s call it to indicate that it depends on .

Greg Egan drew some of these sets . Here’s one that looks like a dragon:

Here’s one that looks more like a feather:

Now here’s the devastatingly cool fact:

Near the point z in the complex plane, the set Sam Derbyshire drew looks a bit like the generalized dragon set S_z!

The words ‘a bit like’ are weasel words, because I don’t know the precise theorem. If you look at Sam’s picture again:

you’ll see a lot of ‘haze’ near the unit circle, which is where and cease to be contraction mappings. Outside the unit circle—well, I don’t want to talk about that now! But inside the unit circle, you should be able to see that I’m at least roughly right. For example, if we zoom in near , we get dragons:

which look at least roughly like this:

In fact they should look very much like this, but I’m too lazy to find the point and zoom in very closely to that point in Sam’s picture, to check!

Similarly, near the point , we get feathers:

that look a lot like this:

Again, it would be more convincing if I could exactly locate the point and zoom in there. But I think I can persuade Dan Christensen to do that for me.

Now there are lots of questions left to answer, like “What about all the black regions in the middle of Sam’s picture?” and “what about those funny-looking holes near the unit circle?” But the most urgent question is this:

If you take the set Sam Derbyshire drew and zoom in near the point z, why should it look like the generalized dragon set S_z?

And the answer was discovered by ‘some guy on the street’—our pseudonymous, nearly anonymous friend. It’s related to something called the Julia–Mandelbrot correspondence. I wish I could explain it clearly, but I don’t understand it well enough to do a really convincing job. So, I’ll just muddle through by copying Greg Egan’s explanation.

First, let’s define a Littlewood polynomial to be one whose coefficients are all 1 or -1.

We have already seen that if we take any number , then we get the image of under all the Littlewood polynomials of degree by starting with the point and applying these functions over and over:

a total of times.

Moreover, we have seen that as we keep applying these functions over and over to , we get sequences that converge to points in the generalized dragon set .

So, is the set of limits of sequences that we get by taking the number and applying Littlewood polynomials of larger and larger degree.

Now, suppose 0 is in . Then there are Littlewood polynomials of large degree applied to that come very close to 0. We get a picture like this:

where the arrows represent different Littlewood polynomials being applied to . If we zoom in close enough that a linear approximation is good, we can see what the inverse image of 0 will look like under these polynomials:

It will look the same! But these inverse images are just the roots of the Littlewood polynomials. So the roots of the Littlewood polynomials near will look like the generalized dragon set .

As Greg put it:

But if we grab all these arrows:

and squeeze their tips together so that they all map precisely to 0—and if we’re working in a small enough neighbourhood of 0 that the arrows don’t really change much as we move them—the pattern that imposes on the tails of the arrows will look a lot like the original pattern:

There’s a lot more to say, but I think I’ll stop soon. I just want to emphasize that all this is modeled after the incredibly cool relationship between the Mandelbrot set and Julia sets. It goes like this;

Consider this function, which depends on a complex parameter :

If we fix this function defines a map from the complex plane to itself. We can start with any number and keep applying this map over and over. We get a sequence of numbers. Sometimes this sequence shoots off to infinity and sometimes it doesn’t. The boundary of the set where it doesn’t is called the Julia set for this number

On the other hand, we can start with , and draw the set of numbers for which the resulting sequence doesn’t shoot off to infinity. That’s called the Mandelbrot set.

Here’s the cool relationship: in the vicinity of the number the Mandelbrot set tends to look like the Julia set for that number This is especially true right at the boundary of the Mandelbrot set.

For example, the Julia set for

looks like this:

while this:

is a tiny patch of the Mandelbrot set centered at the
same value of They’re shockingly similar!

This is why the Mandelbrot set is so complicated. Julia sets are
already very complicated. But the Mandelbrot set looks like a
lot of Julia sets! It’s like a big picture of someone’s face made of little pictures of different people’s faces.

Here’s a great picture illustrating this fact. As with all the pictures here, you can click on it for a bigger view:

But this one you really must click on!

So, the Mandelbrot set is like an illustrated catalog of Julia sets. Similarly, it seems the set of roots of Littlewood polynomials (up to a given degree) resembles a catalog of generalized dragon sets. However, making this into a theorem would require me to make precise many things I’ve glossed over—mainly because I don’t understand them very well!

So, I’ll stop here.

For references to earlier work on this subject, try “week285”.

The Global Amphibian Crisis

There’s a fungus that infects many kinds of amphibians. Some get wiped out entirely—but it’s harbored harmlessly by others, so it’s impossible to eradicate. Over a hundred species have disappeared in the last 20 years!

You’ve got to read this:

• Joseph R. Mendelson III, Lessons of the lost, American Scientist 99 (November-December 2011), 438.

The fungus causes a disease called chytridiomycosis. The effects are gruesome: when spores land on a susceptible amphibian, they quickly sprout and form a vase-shaped structure that harvests energy from the animal’s skin. This produces more spores, which swim around using flagella and spread. The disease progresses as these reinfect the host. The victim may become lethargic, lose skin over its body, go into convulsions, and die.

Amphibian populations have been dropping rapidly worldwide since the 1980s. There were about 6500 species, but now 30% of these are endangered, about 130 are ‘missing’, and about 30 are extinct in the wild.

There were many theories about the cause of this decline, but now we know this disease is playing a big role. As Mendelson says:

Herpetologists and wildlife biologists began observing inexplicable disappearances of amphibians around the globe in the mid-1970s and especially by the mid-1980s but were at a complete loss to explain them. Finally, in the late 1990s, an insightful team of pathologists at the U.S. National Zoo, led by Don Nichols, collaborated with one of the few chytrid fungus scholars in the world, Joyce Longcore, and identified this quite unusual new genus and species.

Conservationists and disease ecologists were unprepared for the reality of a pathogen capable of directly and rapidly—mere months!—causing the elimination of a population or an entire species that was otherwise robust. Classical host-pathogen theory held that such dramatic consequences to the host population or species were only realized when the host population was already drastically reduced in size or otherwise compromised. The concept of a lightning extinction was foreign to researchers and conservationists, and we argued vehemently about it throughout the 1990s at symposia worldwide.

In retrospect, the scenario of a spreading pathogen is parsimonious and clear, but in the midst of the massacre we were entangled in logical quagmires along these lines: “The disappearances cannot be the result of disease; diseases are not capable of such.” Not to mention the fact that the smoking gun, the pathogen itself, was not described until 1999. While we were debating the issue, a terrible lesson was playing out for us around the world as an unknown disease decimated amphibian populations.

What are the ‘lessons’ that Mendelson is talking about? Here are some:

Our powerlessness in this terrible crisis must be balanced by increased efforts in realms that we can control, such as reducing carbon emissions to protect what habitat remains from chemical and physical disruption. We can go further and restore what has been wounded but can still be salvaged. We need to inspire and fund truly innovative research on pathogens in order to better predict and thwart emerging infectious diseases. The lessons we learn here will extend far beyond the amphibians. We must support funding for programs such as the Amphibian Ark and the Amphibian Survival Alliance. We must keep looking for species gone missing, and continue biodiversity surveys, despite the sometimes paralyzing depression that both activities can induce in this era. But especially, we need to pay close attention to the lessons that legions of dead amphibians are teaching us. I note with some satisfaction that our colleagues in bat research and conservation did not spend a decade arguing whether the fungus that causes white-nose syndrome could possibly eliminate entire colonies of bats in a single season. Our colleagues assumed that it was possible and reacted quickly. We can thank the amphibians for leaving us that lesson, but at such cost.

Yes, millions of bats in America have died from a new fungal disease called white-nose syndrome.

What role, if any, do people play in the spread of these new diseases? Why are they happening now?

In the case of amphibians, people helped spread American bullfrogs. These are resistant to the disease, but carry it. They’ve largely taken over here in Singapore.

Global warming seems not to be responsible, because the worst outbreaks happen at high elevations, where it’s cool: that’s where the fungus thrives.

As for the bats, the same fungus that’s killing bats in America is found in healthy bats in Europe, which suggests the disease spread from there. People might carry spores on their clothes from infected caves to not-yet-infected ones, so visitors to caves with bats are being asked to limit their activities, and disinfect clothing and equipment. It’s completely against the rules to visit some caves now.

There have been successful attempts to cure some amphibians of chytridiomycosis:

• Reid Harris of James Madison University has claims that coating frogs with Janthinobacterium lividum protects them from chytridiomycosis.

• A team of scientists published a paper claiming that Archey’s frog (Leiopelma archeyi), a critically endangered species in New Zealand, was successfully cured of chytridiomycosis by applying chloramphenicol topically.

• Don Nichols claims to have cured several species of frogs using a drug called itraconazole.

• Jay Redmond at WWT Slimbridge, Gloucestershire claims that raising poison dart frogs in water containing Rooibos tea (Aspalathus linearis) wards off chytridiomycosis.

The Amphibian Ark is trying to keep populations alive that have died in the wild. They have a list of suggestions on what you can do to help. For starters:

• Don’t ever release pet amphibians into the wild.

• Build a frog pond: here’s how. Even in arid places like Riverside California, our friends who built some ponds soon found them occupied by sweetly chirping frogs.

• Get involved in collaborations that promote sustainable breeding and management, like the Amphibian Steward Network.

• Figure things out. Zoos don’t even know how to breed common toads without using artificial hormone injections! If you could find a way, maybe the same technique could be used with threatened species.

• If you’re a student, go to James Madison University and work with Reid Harris:

or go to the University of Maine and work with Joyce Longcore:

or find a university closer to you with someone leading a group that studies chytridiomycosis!

(Click on the pictures for even more info.)

I thank Allen Knutson for pointing out the American Scientist article. This is the best popular science magazine in the English language, but I let my subscription lapse when I came to Singapore!

Mathematics 1001

Just in time for your holiday gift-giving, here’s my suggestion:

• Richard Elwes, Mathematics 1001, Quercus Books, 2010.

The idea: 1001 bite-sized, juicy explanations of math topics ranging from the elementary:

• how to do long division
• why multiplying two negative numbers gives a positive one
• why 0.9999… equals 1

to the slightly less elementary:

• what’s the equation of a straight line?
• what are the basic identities that trig functions obey?
• how do you prove the product rule in calculus?

to the fun and easy:

• What’s Goldbach’s conjecture and how much evidence do we have for it?
• What’s the golden ratio?
• What’s a Koch snowflake?

to the fun and harder:

• What’s the abc conjecture?
• What’s a Julia set?
• What’s Chaitin’s number Ω?
• What’s Tarski’s geometric decidability theorem?

to the stuff all mathematicians need to know, but ordinary folks don’t:

• What’s the Cauchy-Schwarz inequality?
• What’s a Fourier series?
• What are grad, div and curl?
• What’s the classification of closed surfaces?
• What’s a permutation group?

to the stuff all mathematicians want to understand, but find intimidating:

• What is a scheme?
• What’s the Langlands program?
• What’s forcing?
• What’s a derived category?

The more advanced topics are covered in a sketchy way, so you will not walk out knowing complete answers to all these questions. And that’s just as well, because otherwise the book would be a lot longer than 409 pages, and nobody would be able to read it all!

What you will get is some rough sense of all what all these things are… and you’ll have a lot of fun in the process. You can start reading the book anywhere: it’s not a treatise, it’s a bunch of tasty appetizers. So, you can set it by your bed or somewhere in the living room, and dip into it whenever you have some spare time.

I really, really wish I’d had a book like this when I was younger. Math is a huge enchanted kingdom with many highways and byways, dark forests and dragon-infested mountain ranges. It took me decades to wander across it, extricate myself from various tar pits, and get some sense of the overall lay of the land. This book would have sped that process immensely, in a very pleasant way.

So if you’d like to know more about math, get this book! And if you know a kid, or friend, who is interested in math but not yet an expert, give them this book!

As a professional showoff and know-it-all, I was chagrined (though also secretly pleased) that there are some things in this book I did not know about. They’re mainly open problems in number theory. I guess this subject holds a lot of the most easily stated and hard-to-prove conjectures, so Elwes included a bunch, including these I’d never heard of:

• Legendre’s conjecture.
• de Polignac’s conjecture.
• Hypothesis H.
• Andrica’s conjecture.
• Beals’ conjecture.

By the way, if you saw the recent post Richard Elwes and I wrote on Babylonian mathematics, you may think it’s some sort of conflict of interest for me to be touting his book now. However, that would be putting Descartes before the horse! In fact what happened is this. He sent me a copy of his book, I liked it, I wanted to post an article about it, I saw an interesting post of his about a Babylonian clay tablet on Google+, I decided to expand it with him here on this blog, and finally I’m talking about his book!

Probabilities Versus Amplitudes

Here are the slides of the talk I’m giving at the CQT Annual Symposium on Wednesday afternoon, which is Tuesday morning for a lot of you. If you catch mistakes, I’d love to hear about them before then!

• Probabilities versus amplitudes.

Abstract: Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of “chemical reaction networks”, which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas—but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?