mm-4799 === Subject: Re: Algebraic Geometry in Europe I would appreciate if any could help me with locating good universities > in Europe (especially France) which have a good research team in > Algebraic Geometry. I know about ENS and maybe Cambridge, but that's it. Galois Perhaps the MPI Bonn (http://www.mpim-bonn.mpg.de/) is interesting for you, too. Zagier and Faltings are there. === Subject: Re: Algebraic Geometry in Europe > Hi there, > I would appreciate if any could help me with locating good universities > in Europe (especially France) which have a good research team in > Algebraic Geometry. I know about ENS and maybe Cambridge, but that's > it. > Galois Perhaps the MPI Bonn (http://www.mpim-bonn.mpg.de/) is interesting for > you, too. Zagier and Faltings are there. Hi there, Sorry, I see I forgot to mention that this is supposed to be a PhD application. Unfortunately, I think MPIM does not have a graduate programme. Galois === Subject: Re: MatheRealism posting-account=lwNmEAoAAAB9yKqyOQ9ijwax7bRvEMO6 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) And that's the problem. I want to know whether there > can be a theory in which Cantor's theorem is false > and 10^100 is the largest number. If you mean by Cantor's theorem you mean N < R, then I think the answer is no, so long as the theory is natural and allows you to express the theorem. It obviously depends on how you define reals, but the most natural definition in an ultrafinitist setting (where L is the largest number) would be something like: a real is a natural number + a binary expansion of length <= L. In this case there are obviously more reals than naturals. You should note that, however, that in such a setting, an important part of the Cantor hierarchy changes (just not the part everyone worries about). Namely, in an ultrafinitist setting, you also have N < Q, where Q is the set of rationals. That is, the importance of an infinitary outlook in Cantor's theory is that it collapses the low- lying sets into the same infinity, which an ultrafinitist perspective would keep them separate. === Subject: Re: MatheRealism <8763ljx4ia.fsf@alatheia.dsl.inet.fi> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2008102920 Firefox/3.0.4,gzip(gfe),gzip(gfe) > So I wish to discuss ultrafinitism, without trying to > prove ZFC inconsistent. But unfortunately, as Tribble > points out, it's impossible to separate this concepts > in the eyes of WM. If you're interested in ultra-finitism, why not acquaint yourself with > the literature instead of obsessing over M.9fckenheim's blather? He's already said he doesn't want to have to buy books (and, presumably, he's not close enough to a university library).I want to relax each night on a plush leather sofa, but I don't want to buy or rent one, and I don't want people telling me that is what I need to do to relax each night on a plush leather sofa! I want to learn archery, but I don't want to buy or rent bow and arrows, and I don't want people telling me that is what I need to do to learn archery! I want to learn about advanced topics in mathematics, but I don't want to buy or borrow the books to learn about it, and I don't want people telling me that is what I need to do! > See Randall Holmes's Elementary Set Theory with a Universal > Set , Section 8.3 for some details. invitation to read it. MoeBlee === Subject: This Week's Finds in Mathematical Physics (Week 273) Also available at http://math.ucr.edu/home/baez/week273.html December 14, 2008 This Week's Finds in Mathematical Physics (Week 273) John Baez Today I'd like to talk about the history of the Earth, and then say a bit about locally compact abelian groups. But first, a few more words about Enceladus. Last week we visited the geysers of Saturn's moon Enceladus. subject by Carolyn Porco, leader of the imaging team for the Cassini-Huygens mission - the team that's been taking the intriguing theories about what powers these geysers: 1) Carolyn Porco, Enceladus: secrets of Saturn's strangest moon, Scientific American, November 2008, available at And it's free online! - at least for now. I've criticized the Scientific American before here, but if they keep coming out with well-written: There is obviously a tale writ on the countenance of this little moon that tells of dramatic events in its past, but its present, we were about to find out, is more stunning by far. In its excursion over the outskirts of the south polar terrain, coming from the region of the tiger stripes. Two other instruments detected water vapor, and one of them delivered the signature of carbon dioxide, nitrogen and methane. Cas[CapitalEth]si[CapitalEth]ni had passed through a tenuous cloud. What is more, the thermal infrared imager sensed elevated temperatures along the fractures - possibly as high as 180 kelvins, well above the 70 kelvins that would be expected from simple heating by sunlight. These locales pump out an extraordinary 60 watts per square meter, many times more than the 2.5 watts per square meter of heat arising from Yellowstone's geothermal area. And smaller patches of surface, beyond the resolving power of the infrared instrument, could be even hotter. For another, it tackles a fascinating mystery. Where does all this power come from? The geysers near the south pole of Enceladus emit about 6 gigawatts of heat. Enceladus is too small to have that much radioactive heating at its core - only about 0.3 gigawatts, probably. The rest must come from tidal heating. This happens when stuff sloshes back and forth in a changing gravitional field: friction converts this motion to heat. So, what causes tides on Enceladus? It may be important that Enceladus has a 2:1 resonance with Dione: it orbits Saturn twice for each orbit of that larger moon. This sort of resonance is known to cause tidal heating. For example, in week269, I showed you how Jupiter's moon Io is locked in resonances with Europa and Ganymede. The resulting tidal heat powers its mighty volcanos. Unfortunately, the resonance with Dione doesn't seem powerful enough to produce the heat we see on Enceladus. Unless something funny is going on, there should only be 0.1 gigawatts of tidal heating - not nearly enough! At least that's what Porco estimated in 2006: 2) Carolyn Porco et al, Cassini observes the active south pole of Enceladus, Science 311 (2006), 1393-1401. So, we need to dream up a more complicated story. Here's one: there could be a kind of slow cycle where the orbit of Enceladus gets more eccentric, tidal heating increases, ice beneath its surface melts, more sloshing water causes more tidal heating, and then the release of heat energy damps its eccentric orbit, until it freezes solid and the whole cycle starts over. We could be near the end of such a cycle right now. Here's another: maybe Enceladus has an sea of liquid water under the frozen surface of its south pole. With enough water sloshing around, there could be a lot more tidal heating than you'd naively guess... and this heating, in turn, could keep the water liquid. The fun thing about this second scenario is that a permanent liquid ocean on Enceladus raises the possibility of life! Nobody knows for sure what's going on - but Carolyn Porco examines the options in a clear and engaging way. If you like celestial mechanics, also try this paper: 3) Jennifer Meyer, Jack Wisdom, Tidal heating in Enceladus, Icarus 188 (2007), 535-539. Also available at http://groups.csail.mit.edu/mac/users/wisdom/meyerwisdom1.pdf stuff. He knows a lot about resonances. For related work on the Jupiter-Saturn resonance, the Neptune-Pluto resonance, and the math of continued fractions, also try the addenda to week222. Next I'd like to give you a quick trip through the Earth's history. In week196 we looked back into the deep past, all the way to the electroweak phase transition 10 picoseconds after the Big Bang. On the other hand, here: 4) John Baez, The end of the universe, http://math.ucr.edu/home/baez/end.html you can zip forwards into the deep future - for example, 10^{19} years from now, when the galaxies boil off, shooting dead stars into the the vast night. But now I'd like to zoom in closer to home and quickly tell the history of the Earth, focusing on an aspect you may never have thought about. You see, Kevin Kelly recently pointed me to this fascinating paper on mineral evolution: 5) Robert M. Hazen, Dominic Papineau, Wouter Bleeker, Robert T. Downs, John M. Ferry, Timothy J. McCoy, Dmitri A. Sverjensky and Henxiong Yang, Mineral evolution, American Mineralogist 91 (2008), 1693-1720. Ever since the Earth was formed, the number of different minerals has kept increasing - and ever since *life* ran wild, it's soared! Some examples are obvious: seashells become limestone, which gets squashed into marble. But others are less so. Here's a timeline loosely taken from this paper: THE ERA OF PLANETARY FORMATION 1. Primary chondrite minerals (over 4.56 billion years ago): 60 species of mineral. Chondrites are stony meteorites that formed early in the history of the solar system. They're made of chondrules - millimeter-sized spheres of olivine, pyroxene and other minerals - together with nuggets called CAIs (calcium-aluminum rich inclusions) and other stuff. These chondrules began life as molten droplets back when the Sun was a T Tauri star, heated only by gravitational collapse. 2. Aqueous alteration, thermal alteration, and shocks form achondrites and iron-nickel meteorites (4.56 to 4.55 billion years ago): 250 species of mineral. This is the era when the disk of dust circling the early Sun started forming lumps. As these lumps collided, they got bigger and bigger, eventually forming the asteroids and planets we see today. Some of these proto-planets melted, letting heavier metals sink to their core while lighter material stayed on top. But then some crashed into each other, shattering and forming new kinds of meteorites: iron-nickel radioactive dating of these, scientists claim a shockingly precise knowledge of when all this happened: sometime between 4.56 and 4.55 billion years ago. THE ERA OF CRUST AND MANTLE REWORKING 3. Igneous rock evolution (4.55 to 4 billion years ago): 350 species of mineral. The Earth's history is divided into four eons: Hadean, Archean, Proterozoic and Phanerozoic. Back when I was a kid, the Cambrian era seemed really old - but that's just the start of the current eon, when multicellular life emerged: the Phanerozoic. We're digging much deeper now: the Phanerozoic will be *end* of today's story. The Hadean eon began with a bang: the event that formed the Moon around 4.55 billion years ago! What made the moon? The current most popular explanation is the giant impact theory - sometimes called the Big Splat Theory. Dana Mackenzie spends a lot of time writing about math, but he's also written a book about this: 6) Dana Mackenzie, The Big Splat, or How Our Moon Came To Be, The idea is that another planet formed in one of the Lagrange points of Earth's orbit - a stable spot 60 degrees ahead or behind the Earth: 7) John Baez, Lagrange points, http://math.ucr.edu/home/baez/lagrange.html But when this planet reached about the mass of Mars, it would no longer be stable in this location. So, it gradually drifted toward Earth, and eventually smacked right into us! The impact was incredibly energetic, melting the Earth's entire crust and outer mantle. The iron core of this other planet sank into Earth's core, while about 2% of the outer part formed an orbiting ring of debris. Within a century, about half of this ring formed the Moon we know and love. It's an amazing story, but most of the evidence seems to support it. The early Moon is known to have been much closer to Earth than it is now - it's been receding ever since. For this and many other reasons, the giant impact theory is sufficiently solid that the hypothetical doomed planet that hit Earth has a name: Theia! You can even watch a simulation of it hitting Earth, produced by Robin Canup: 8) Dana Mackenzie, The Big Splat (animation), http://www.danamackenzie.com/big_splat_animation.htm Let me quote Mackenzie on this: The simulation shows the first twenty-four hours after the giant impact. It begins with Theia about to strike the Earth. After the impact, one hemisphere of the Earth is sheared off and flung into space. The remaining part of Earth is very lopsided, and sets up a gravitational torque on the debris. This boosts some of the debris into orbit. (Without such a boost, it would all simply fall back down again.) Within a few hours, the debris has formed an arm that smashes spectacularly back into the Earth. This crash is nearly as explosive as the original impact! (The second explosion can be seen much more vividly in the video than in the still frames published in my book.) Notice how the temperature of the Earth has risen, from the blues and greens of the early frames to yellows and reds, indicating more than 2000 degrees Kelvin. Earth has literally become a blast furnace. As the fateful day continues, the debris gets more uniformly distributed in a disk around the Earth. Notice, though, that this disk is not stable like the rings of Saturn. It develops shock waves that whirl around the Earth, collecting material into spiral arms. According to Alastair Cameron, another modeler the giant impact, these spiral arms also play an important role in the development of the Moon, by siphoning debris up from lower orbits into higher ones. Scientists have estimated that a mass at least twice the present mass of the Moon had to be lifted beyond the Roche limit, roughly twelve thousand miles or three Earth radii above the surface. Any debris that does not make it past the Roche limit will be torn apart by tidal forces, and cannot form a permanent moon. This simulation stops after 24 hours, a long time before the disk of debris condenses into our Moon. The Moon was not formed in a day! However, it did form much more rapidly than you might expect; current estimates range from 1 to 100 years. This is astounding, compared to ordinary geological time scales. An entire new planet was born within the life span of a single human. No rocks on Earth are known to survive from before 4.03 billion years ago, so the details of this time period are hotly debated. However, many igneous rocks, especially basalt, must have been formed at this time. Even after the surface cooled enough to form a crust, volcanoes continued to release steam, carbon dioxide, and ammonia. This led to what is called the Earth's second atmosphere. The first atmosphere was mainly hydrogen and helium; the second was mainly carbon dioxide and water vapor, with some nitrogen but almost no oxygen. This second atmosphere had about 100 times as much gas as today's third atmosphere! As the Earth cooled, oceans formed, and much of the carbon dioxide dissolved into the seawater and later precipitated out as carbonates. 4. Granitoid formation and the first cratons (4 to 3.2 billion years ago): 1000 species of mineral. Between 4 and 3.8 billion years ago there was another scary time: the Late Heavy Bombardment. A lot of large craters on the Moon date to this period, so probably the Earth, Venus and Mars got hit too. Why so many impacts after a period of relative calm? One theory is that Jupiter and Saturn locked into their current 2:1 resonance at this time, causing a big shakedown in the asteroid belt and Kuiper belt. Wikipedia has a nice quick review of this and other theories: 9) Wikipedia, Late heavy bombardment, http://en.wikipedia.org/wiki/Late_heavy_bombardment This time also marked the rise of cratons. Cratons are a bit like small early plates in the sense of plate tectonics: they're ancient tightly-knit pieces of the earth's crust and mantle, many of which survive today. While most cratons only finished forming 2.7 billion years ago, nearly all started growing earlier, in the Eoarchean era. Cratons are made largely of granitoids. Granitoids are more sophisticated igneous rocks than basalt. Modern granite is one of these. Granite is made in a variety of ways, for example by the remelting of sedimentary rock. Early granitoids were probably simpler. 5. Emergence of plate tectonics (3.2 to 2.8 billion years ago): 1500 species of mineral. In the Paleoarchean and Mesoarchean eras, plate tectonics as we know it began. A key aspect of this process is the recycling of the Earth's crust through subduction: oceanic plates slide under continental plates and get pushed down into the mantle. Another feature is underwater volcanism and hydrothermal activity. 6. Anoxic biology leading up to photosynthesis (3.9 to 2.5 billion years ago): 1500 species of mineral. The earliest hints of life include some rocks called banded iron formations that date back 3.85 billion years. The real fun starts with the rise of photosynthesis leading up to the Great Oxidation Event about 2.5 billion years ago - more on that later. But organisms from the domain Archaea can do well in a wide variety of extreme environments without oxygen, and as their name suggests, many of these organisms are very ancient. These organisms gave rise to an active sulfur cycle and deposits of sulfate ores starting in the Paleoarchean era. They also made the atmosphere increasingly rich in methane throughout the Mesoarchean and Neoarchean. So, life was already beginning to affect mineral evolution. THE ERA OF BIO-MEDIATED MINERAL FORMATION 7. The Great Oxidation Event (2.5 to 1.9 billion years ago): over 4000 species of mineral. The Archean eon ended and the Proterozoic began with the Great Oxidation Event 2.5 billion years ago. In this event, also known as the Oxygen Catastrophe, photosynthesis put enough oxygen into the atmosphere to make it lethal to most organisms of the time! Luckily evolution found a way out of this impasee. The oyxgen-rich atmosphere in turn led to a wide variety of new minerals. 8. The intermediate ocean (1.9 to 1 billion years ago): over 4000 species of mineral. In the Mesoproterozoic era, increased oxygen levels in the ocean put an end to many anoxic life forms. For example, around 1.85 billion years ago, banded iron formations suddenly ceased. The next gigayear was rather static and dull - if you're mainly interested in new minerals, that is. The term intermediate ocean means that during this period, the seawater contained a lot more oxygen than before, but still much less than today. 9. Snowball Earth and the Neoproterozoic oxygenation events (1 to 0.54 billion years ago): over 4000 species of mineral. The Neoproterozoic era probably saw several Snowball Earth events: episodes of runaway glaciation during which most or all the Earth was covered with ice. Since ice reflects sunlight, making the Earth even colder, it's easy to guess how this runaway feedback might happen. The opposite sort of feedback is happening now, as melting ice makes the Earth darker and thus even warmer. The interesting questions are why this instability doesn't keep driving the Earth to extreme temperatures one way or another - and what stopped the Snowball Earth events back then! Here's a currently popular answer to the second question. Ice sheets slow down the weathering of rock. Weathering of rock is one of the main long-term processes that use up atmospheric carbon dioxide, by converting it into various carbonate minerals. On the other hand, even on an ice-covered Earth, volcanic activity would keep putting carbon dioxide into the atmosphere. So, eventually carbon dioxide would build up, and the greenhouse effect would warm things up again. This process might be very dramatic, with perhaps as much as 13% of the atmosphere being carbon dioxide (350 times what we see today), and temperatures soaring to 50 Celsius! But the details are still the subject of much controversy. At the end of these glacial cycles, it's believed that oxygen increased from 2% of the atmosphere to 15%. (Now it's 21%.) This may be why multi-celled oxygen-breathing organisms date back to this time. Others argue that the freeze-fry cycle imposed tremendous evolutionary pressure on life and led to the rise of multicellular organisms. Both these theories could be true. During the glacial cycles, few new minerals were formed - unless you count ice. Afterwards, surface rocks were weathered in new ways involving oxidation. 10. Phanerozoic biomineralization (0.54 billion years ago to now): over 4300 species of mineral. The Phanerozoic eon, beginning with the Cambrian 540 million years ago, marks the rise of life as we know it. During this time, sea life has given rise to extensive deposits of biominerals such as calcite, aragonite, dolomite, hydroxylapatite, and opal. There has also been increased production of clay and many different types of soil. This is the end of our story - but of course the story isn't over. We're now in the Anthropocene epoch of the Cenozoic era of the Phanerozoic eon. New things are happening. Humans are boosting atmosphieric carbon dioxide levels. If the temperature rises one more degree, the Earth's temperature will be the hottest it's been in 1.35 million years, before the Ice Ages began. There's no telling when this trend will stop. We're filling the oceans and land with plastic and other debris. In millions of years, these may form new species of minerals. Regardless, there will probably still be rocks - but we'll either be gone or drastically changed. Next: Pontryagin duality! Like last week's math topic, I needed to learn more about this for my work on infinite-dimensional representations of 2-groups. And like last week's math topic, it involves a lot of analysis. But it also involves a lot of algebra and category theory. You may know about Fourier series, which lets you take a sufficiently nice complex-valued function on the circle and write it like this: f(x) = sum_k g_k exp(ikx) Here k ranges over all integers, so what you're really doing here is taking a function on the circle: f: S^1 -> C and expressing it in terms of a function on the integers: g: Z -> C More precisely, any L^2 function on the circle can be expressed this way for some L^2 function on the integers - and conversely. In fact, if we normalize things right, the Fourier series gives a unitary isomorphism between the Hilbert spaces L^2(S^1) and L^2(Z). You may also know about the Fourier transform, which lets you take a sufficiently nice complex-valued function on the real line and write it like this: f(x) = integral g(k) exp(ink) dk Here k also ranges over the real line, so what you're really doing is taking a function on the line: f: R -> C and expressing it in terms of another function on the line: g: R -> C In fact, any L^2 function f: R -> C can be expressed this way for some L^2 function g: R -> C. And if we normalize things right, the Fourier transform is a unitary isomorphism from L^2(R) to itself. Pontryagin duality is the grand generalization of these two examples! Any locally compact Hausdorff abelian group A has a dual A* consisting of all continuous homomorphisms from A to S^1. The dual is again a locally compact Hausdorff abelian group - or LCA group, for short. When you take duals twice, you get back where you started. And the Fourier transform gives a unitary isomorphism between the Hilbert spaces L^2(A) and L^2(A*). It's fun to take the Pontryagin duals of specific groups, or specific classes of groups, and see what we get. We've already seen that the dual of S^1 is Z, the dual of Z is S^1, and the dual of R is R. More generally the dual of the n-dimensional torus is Z^n, and vice versa, while the dual of R^n is isomorphic to R^n. What can we glean from these examples? Well, any discrete abelian group is an LCA group - a good example is Z^n. So is any compact Hausdorff abelian group - a good example is the n-dimensional torus. And there's a nice general theorem saying that the dual of any group of the first kind is a group of the second kind, and vice versa! In particular, if we have an abelian group that's both compact and discrete, its dual must be too. But the only abelian groups like this are the *finite* abelian groups - products of finite cyclic groups Z/n. So, this collection of groups is closed under Pontryagin duality! In fact, it's easy to see that for any finite abelian group, A* is isomorphic to A. But not canonically! To get a canonical isomorphism we need to take duals twice: for any LCA group, we get a canonical isomorphism between A and A**. This should remind you of duality for finite-dimensional vector spaces - another famous collection of LCA groups that's closed under Pontryagin duality. You can take any collection of LCA groups, stare at it through the looking-glass of Pontryagin duality, and see what it looks like. I've mentioned a few examples so far: A is compact iff A* is discrete. A is finite iff A* is finite. A is a finite-dimensional vector space iff A* is a finite-dimensional vector space. Here are some fancier ones: A is torsion-free and discrete iff A* is connected and compact. A is compact and metrizable iff A* is countable. A is a Lie group iff A* has finite rank. A is metrizable iff A* is sigma-compact. A is second countable iff A* is second countable. If you know more snappy results like this, tell me! I'm collecting them - they're sort of addictive. Because Pontryagin duality turns compact LCA groups into discrete ones - and vice versa - we can use it to turn some topology questions into algebra questions, and vice versa. After all, a discrete abelian group has no more structure than an abstract abelian group - one without a topology! Sometimes this change of viewpoint helps, but sometimes it merely reveals how hard a problem really is. For example, here's an innocent-sounding question: what are the compact path-connected LCA groups? The obvious example is the circle. More generally, we could take any product of circles - even an *infinite* product. Are there any others? It turns out that this question cannot be settled by Zermelo-Fraenkel set theory together with the axiom of choice! Here's why. An LCA group is compact and path-connected iff its dual is a Whitehead group. What's that? It's an abelian group A such any short exact sequence of abelian groups like this splits: 0 -> Z -> B -> A -> 0 where Z is the integers and B is any abelian group. We call this sort of short exact sequence an extension of A by Z. So, if you want to show off your sophistication, you can say that A is a Whitehead group if Ext(A,Z) = 0. The obvious examples of Whitehead groups are free abelian groups. Indeed, these are precisely the guys whose Pontryagin dual is a product of circles! So the question is: are there any others? Or is every Whitehead group a free abelian group? This is a famous old problem, called the Whitehead problem: 10) Wikipedia, Whitehead problem, http://en.wikipedia.org/wiki/Whitehead_problem In 1971, the logician Saharon Shelah showed the answer to this problem was undecidable using the axioms of ZFC! This was one of the first problems mathematicians really cared about that turned out to be undecidable. If you want an easy introduction to Pontryagin duality and the structure of LCA groups, you can't beat this: 10) Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge U. Press, Cambridge, 1977. This classic treatment is still great, too: 14) Lev S. Pontrjagin, Topological Groups, Princeton University Press, Princeton, 1939. To dig deeper, you need to read this - it's a real mine of information: 15) E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1979. This book has a lot of interesting newer results: 16) David L. Armacost, The Structure of Locally Compact Abelian Groups, Dekker, New York, 1981. In particular, this is where I learned about path-connected LCA groups and the Whitehead problem. I'd like to dedicate this issue of This Week's Finds to my father, Peter Baez, who died yesterday around midnight at the age of 87. His health had been failing for a long time, so this did not come as a shock. It's a curious coincidence that I was already writing an issue about minerals, since my dad majored in chemistry and returned to school for a master's in soil science after serving in the Army in World War II. After that he worked in the Blackfeet Nation in Browning Montana, riding around in a jeep, digging up soil samples, and testing them back at the lab for the Army Corps of Engineers. When he found medicine wheels - stone circles laid down by the native Americans for ritual reasons - he would report them to his friend the archeologist Tom Kehoe. Later he moved to California, became an editor for the Forest Service, and met my mother. He got me interested in science at an early age because he was always taking me to museums, bringing me books from the public library, and so on. As a little kid, when I spilled something, he'd say So you don't believe in the law of gravity? He liked to joke around. Whenever I said an ungrammatical sentence, he'd tease me for it. I'm not that hungry. What do you mean? You're not *how* hungry? I learned a lot of math, physics and chemistry from his 1947 edition of the CRC Handbook of Chemistry and Physics - an edition so old that it listed mesothorium among the radioactive isotopes. He brought logic - because it was in the math section of the library and he misread Goedel as Googol. He knew I liked large numbers! I didn't understand much of it, but it had a big effect on me. I owe a lot to him. ----------------------------------------------------------------------- Quote of the Week: People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion. - Albert Einstein ----------------------------------------------------------------------- Addenda: Mike Stay pointed out an interesting book on how humans may affect the future of mineral evolution: in the Rocks?, Oxford University Press, Oxford, 2009. It's not 2009 yet, but the best books about the future are actually published there! Here's a quote: The surface of the Earth is no place to preserve deep history. This is in spite of - and in large part because of - the many events that have taken place on it. The surface of the future Earth, one hundred million years now, will not have preserved evidence of contemporary human activity. One can be quite categorical about this. Whatever arrangement of oceans and continents, or whatever state of cool or warmth will exist then, the Earth's surface will have been wiped clean of human traces. Thus, one hundred million years from now, nothing will be left of our contemporary human empire at the Earth's surface. Our planet is too active, its surface too energetic, too abrasive, too corrosive, to allow even (say) the Egyptian Pyramids to exist for even a hundredth of that time. Leave a building carved out of solid diamond - were it even to be as big as the Ritz - exposed to the elements for that long and it would be worn away quite inexorably. So there will be no corroded cities amid the jungle that will, then, cover most of the land surface, no skyscraper remains akin to some future Angkor Wat for future archaeologists to pore over. Structures such as those might survive at the surface for thousands of years, but not for many millions. For more discussion, visit the n-Category Cafe, at: http://golem.ph.utexas.edu/category/2008/12/this_weeks_finds_in_mathematic_3 4.html ----------------------------------------------------------------------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === === Subject: Re: JSH: Breaking Galois Theory posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081029 Firefox/2.0.0.18,gzip(gfe),gzip(gfe) > [...] No, that's wrong. Arturo Magidin co-authored a paper with >Douglas Mackinnon, David McKinnon. -- > It's not denial. I'm just very selective about > what I accept as reality. > --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin > magidin-at-member-ams-org Arturo, Well, I spelled the last name right in at least one of two references to it - It is interesting that in this latest Harristotelian episode, he promoted an argument based on splitting 7 up into two factors, much as we have done in our many counterarguments, rather than insisting that the only way to factor 7 out of the product of two terms was to factor it /in toto/ out of one or the other term. That is, when it is convenient, he makes use of an argument that he has systematically ignored about a million times in the hope of arriving at a contradiction of Galois theory, or at least the way it is usually 'interpreted'. And it looks now like he has abandoned these threads with no explanation. This is typical Harristotelian behavior too - he argues one view ad nauseam, rejecting all kinds of perfectly sound rebuttals, then drops the argument for reasons entirely of his own, which presumably don't make any more sense than his original argument did and which have nothing to do with the rebuttals. Marcus. === Subject: Non-existence directional derivative (why?) posting-account=gc2kDQoAAADMsLO9kJjQL9hCJkI0D8qJ .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) Why there is no real-valued function f such that the directional derivative of f (f '(x;u)) is strictly greater than zero, for a fixed point x in R^n and for every nonzero vector u in R^n ?? === Subject: Re: Non-existence directional derivative (why?) posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20081201 Firefox/2.0.0.19,gzip(gfe),gzip(gfe) > Why there is no real-valued function f such that the directional > derivative of f (f '(x;u)) is strictly greater than zero, > for a fixed point x in R^n and for every nonzero vector u in R^n ?? Maybe I am missing something, but isn't the function f(x,y) = sqrt(x^2 + y^2) at (0,0) a counterexample? The directional derivative of f at (0,0) is positive in any direction. (Of course, the Gateaux differential of f at (0,0) is not linear, and f is not Frechet- differentiable at (0,0), but it still looks like a counterexample.) R.G. Vickson === Subject: Re: Non-existence directional derivative (why?) > Why there is no real-valued function f such that the directional > derivative of f (f '(x;u)) is strictly greater than zero, > for a fixed point x in R^n and for every nonzero vector u in R^n ?? Maybe I am missing something, but isn't the function f(x,y) = sqrt(x^2 > + y^2) at (0,0) a counterexample? The directional derivative of f at > (0,0) is positive in any direction. (Of course, the Gateaux > differential of f at (0,0) is not linear, and f is not Frechet- > differentiable at (0,0), but it still looks like a counterexample.) R.G. Vickson IIRC the usual definition of Df_u(x) is lim [f(x + tu) - f(x)]/t as t -> 0; you seem to be thinking of t -> 0+. === Subject: Re: Non-existence directional derivative (why?) > Why there is no real-valued function f such that the directional > derivative of f (f '(x;u)) is strictly greater than zero, > for a fixed point x in R^n and for every nonzero vector u in R^n ?? Hint: Take n = 1. What is the directional derivative of f(x) = x in the direction -1? === Subject: Re: Non-existence directional derivative (why?) posting-account=gc2kDQoAAADMsLO9kJjQL9hCJkI0D8qJ .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) Why there is no real-valued function f such that the directional > derivative of f (f '(x;u)) is strictly greater than zero, > for a fixed point x in R^n and for every nonzero vector u in R^n ?? Hint: Take n = 1. 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