This edition of the Carnival of Mathematics is dedicated to the memory of Paul J. Cohen (April 2, 1934 - March 23, 2007).
Ironically, he died just 2 weeks ago, the date of the previous edition of this Carnival. Here's the New York Times obituary. There have already been a number of comments about and tributes to Paul in math blogs. One of the shortest, but most significant is from fellow Field's Medalist Terence Tao, who points out that Paul excelled not only in the area of set theory (in which his work best known to the public was done), but also in harmonic analysis.
In fact, Paul was one of the most universal mathematicians (a lot like Tao himself) of the last 50 years. I can personally vouch for that, because he was a Professor in the math department where I did my graduate work. He taught the first year graduate course in complex analysis, and made the subject absolutely inspiring, even though it was new to me at the time, and I was more into algebra than analysis. A year or two later, he conducted a seminar on class field theory (a form of advanced algebraic number theory), and was equally inspiring there. I had the chance to see him in action also in courses or seminars on such subjects as analytic number theory and quadratic forms.
But one thing that impressed me just as much was what a decent, friendly, approachable person he was. Unlike a number of other high-powered mathematicians on the faculty at that time, who were pompous and overbearing. (I won't name names.)
Here's a tribute from another logician, Barkley Rosser, who gives a semi-technical account of the work for which Paul received the Fields Medal. And Jason Rosenhouse at Evolutionblog gives another semi-technical explanation of Cohen's work on the independence from ZFC set theory of the Axiom of Choice and the Continuum Hypothesis.
Paul's set theory work actually settled one of the famous Hilbert Problems, enuciated by David Hilbert in 1900 – the very first, in fact, concerning Georg Cantor's question about the cardinality of the continuum. Was it ℵ1, or indeed any of the ℵs? Kurt Gödel had earlier shown that the answer was "maybe". What Cohen showed was that the answer was "maybe not". It is a measure of how bright this guy was that set theory and logic were not even specialties of his at the time he took on the problem. According to Ben Yandell's book on Hilbert's problems (The Honors Class), Cohen had no training as a logician, but in 1959 he was looking for a challenging new problem to work on, asked logician Solomon Fefferman for some suggestions, and was told about the continuum problem. By 1963 Cohen had cracked the problem in an entirely original way.
Moving on. There has been a lot more news about mathematicians and mathematics that has appeared in the public media recently – somewhat of an anomaly. One item that received relatively light (but non-null) coverage, because of its very technical nature, is the solution of a problem concerning "mock thera functions", originating in some cryptic notes of another mathematical prodigy, Srinivasa Ramanujan. You can read accounts of this work here, here, here, here, and here. Although I haven't come across any blog articles yet that go into much more detail on this, there is a good, detailed post about the fascinating Ramanujan himself from M. Balamurugan's blog.
Another story that got a bit more attention was the discovery that some ornamental art found on Islamic architecture of the 13th century (CE) and later has striking affinities to "quasicrystals" and "Penrose tilings". Two of the better articles were done by Philip Ball and Julie Rehmeyer. More stories about this can be found here, here, here, here, here, here, and here. But the most fascinating aspect of this to me is not the Penrose connection, but instead the connection with "noncommutative geometry", as explained by Masoud Khalkhali at (what else?) the Noncommutative Geometry blog.
Of course, the other big piece of mathematical news recently was the determination of the structure of the exceptional (a technical term, not an encomium) Lie group E8. There have been a number of news reports and blog reports about the work, but this one is interesting, as it comes from a physicist, Clifford Johnson at Asymptotia. That's appropriate, as Clifford is a string theorist, and E8 plays a big role in string theory, with heterotic strings in particular. Interestingly, Clifford notes that E8 is also connected with Penrose tilings.
On the other hand, Peter Woit (a mathematician) at Not Even Wrong is not only skeptical of string theory (to put it mildly), but considers all the publicity surrounding E8 to be a little excessive.
Nevertheless, if you're interested in learning a bit about the mathematics behind E8, you should certainly take a look at two postings from John Armstrong at The Unapologetic Mathematician – here and here. And if that still isn't enough to satisfy your curiosity, try this wonkier post on Lie groups, Lie algebras, and representations.
By the way, the reason E8 is called an "exceptional" Lie group is not because of some particularly noteworth properties it possesses. The actual reason is that all Lie groups can be understood in terms of a subclass known as "simple" Lie groups. And most of these, in turn, can be classified in terms of several different infinite subclasses. But there are five other simple Lie groups that defy classification – the exceptional Lie groups. And of these, E8 is the largest.
Likewise, we have several additional items to mention here that do not fit into a classification scheme.
If you're a fan of hard problems, there are few problems harder than understanding the Navier-Stokes equations. But we can thank Fields medalist Terry Tao for a lucid explanation of
Why global regularity for Navier-Stokes is hard.
If you're more into logical puzzles and computer science, you know all about the "halting problem". Alexandre Borovik at Mathematics under the Microscope uses that topic to illustrate the difference between formal and informal proofs. And incidentally, if you are fond of multi-disciplinary investigations, Alexandre has a book you can download from his site (which is named after the book) that looks at mathematics from the standpoint of cognitive psychology and neuroscience.
My own contribution to this carnival offers a quick (well, compared to a whole book) overview of field theory and Galois theory. It's one step on the way to examining some of the deeper mysteries of algebraic number theory.
Perhaps you've never quite understood why some folks make such a big deal about set theory and the continuum hypotheses, which Paul Cohen did so much to clarify. One illustration of how set theory can actually be applied is to a whole new and elegant way of "constructing" the real numbers. This construction, known as "surreal numbers", was invented by John Horton Conway, another "universal" mathematician. He's also known for, among other things, inventing the "game of life" (based on cellular automata), his role in classifying finite simple groups (only distantly related to Lie groups), and his work (with logician Simon Kochen) on the "free will theorem" in quantum mechanics. Mark Chu-Carroll at Good Math, Bad Math gives us a nice overview of surreal numbers.
Or perhaps you were somehow involved in the "New Math" debacle back in the 70s, and now just don't really care that much for even attempting to explain set theory to junior high students, or practically anyone else for that matter. That's too bad, but math teacher JD2718 has some thoughts on the subject.
In spite of all that, there are rewards to trying to teach math. Dave Marain at MathNotations gives a quick review of the recently published book Coincidence, Chaos, and All That Math Jazz given to him in gratitude by one of his top students. He liked the book.
Thanks, everyone, for reading. Come back again in 2 weeks for the next Carnival of Mathematics, to be hosted by Graeme Taylor at Modulo Errors on April 20.
Note: I had a problem with one of my mailboxes. If you tried to submit an article for this edition of the carnival by sending email to cgd at scienceandreason.net, and the article isn't included here, it may have been affected by the problem. Please resend to carnival at scienceandreason.net, or else submit it for the next edition of the carnival. My apologies for any inconvenience.
Note 2: In the comments Mikael Johanssons observes that his contribution to the carnival got lost in my email problem, so please be sure to check it out. It's about modular representation theory. I'm very sorry about the glitch, Mikael.
Ironically, he died just 2 weeks ago, the date of the previous edition of this Carnival. Here's the New York Times obituary. There have already been a number of comments about and tributes to Paul in math blogs. One of the shortest, but most significant is from fellow Field's Medalist Terence Tao, who points out that Paul excelled not only in the area of set theory (in which his work best known to the public was done), but also in harmonic analysis.
In fact, Paul was one of the most universal mathematicians (a lot like Tao himself) of the last 50 years. I can personally vouch for that, because he was a Professor in the math department where I did my graduate work. He taught the first year graduate course in complex analysis, and made the subject absolutely inspiring, even though it was new to me at the time, and I was more into algebra than analysis. A year or two later, he conducted a seminar on class field theory (a form of advanced algebraic number theory), and was equally inspiring there. I had the chance to see him in action also in courses or seminars on such subjects as analytic number theory and quadratic forms.
But one thing that impressed me just as much was what a decent, friendly, approachable person he was. Unlike a number of other high-powered mathematicians on the faculty at that time, who were pompous and overbearing. (I won't name names.)
Here's a tribute from another logician, Barkley Rosser, who gives a semi-technical account of the work for which Paul received the Fields Medal. And Jason Rosenhouse at Evolutionblog gives another semi-technical explanation of Cohen's work on the independence from ZFC set theory of the Axiom of Choice and the Continuum Hypothesis.
Paul's set theory work actually settled one of the famous Hilbert Problems, enuciated by David Hilbert in 1900 – the very first, in fact, concerning Georg Cantor's question about the cardinality of the continuum. Was it ℵ1, or indeed any of the ℵs? Kurt Gödel had earlier shown that the answer was "maybe". What Cohen showed was that the answer was "maybe not". It is a measure of how bright this guy was that set theory and logic were not even specialties of his at the time he took on the problem. According to Ben Yandell's book on Hilbert's problems (The Honors Class), Cohen had no training as a logician, but in 1959 he was looking for a challenging new problem to work on, asked logician Solomon Fefferman for some suggestions, and was told about the continuum problem. By 1963 Cohen had cracked the problem in an entirely original way.
Moving on. There has been a lot more news about mathematicians and mathematics that has appeared in the public media recently – somewhat of an anomaly. One item that received relatively light (but non-null) coverage, because of its very technical nature, is the solution of a problem concerning "mock thera functions", originating in some cryptic notes of another mathematical prodigy, Srinivasa Ramanujan. You can read accounts of this work here, here, here, here, and here. Although I haven't come across any blog articles yet that go into much more detail on this, there is a good, detailed post about the fascinating Ramanujan himself from M. Balamurugan's blog.
Another story that got a bit more attention was the discovery that some ornamental art found on Islamic architecture of the 13th century (CE) and later has striking affinities to "quasicrystals" and "Penrose tilings". Two of the better articles were done by Philip Ball and Julie Rehmeyer. More stories about this can be found here, here, here, here, here, here, and here. But the most fascinating aspect of this to me is not the Penrose connection, but instead the connection with "noncommutative geometry", as explained by Masoud Khalkhali at (what else?) the Noncommutative Geometry blog.
Of course, the other big piece of mathematical news recently was the determination of the structure of the exceptional (a technical term, not an encomium) Lie group E8. There have been a number of news reports and blog reports about the work, but this one is interesting, as it comes from a physicist, Clifford Johnson at Asymptotia. That's appropriate, as Clifford is a string theorist, and E8 plays a big role in string theory, with heterotic strings in particular. Interestingly, Clifford notes that E8 is also connected with Penrose tilings.
On the other hand, Peter Woit (a mathematician) at Not Even Wrong is not only skeptical of string theory (to put it mildly), but considers all the publicity surrounding E8 to be a little excessive.
Nevertheless, if you're interested in learning a bit about the mathematics behind E8, you should certainly take a look at two postings from John Armstrong at The Unapologetic Mathematician – here and here. And if that still isn't enough to satisfy your curiosity, try this wonkier post on Lie groups, Lie algebras, and representations.
By the way, the reason E8 is called an "exceptional" Lie group is not because of some particularly noteworth properties it possesses. The actual reason is that all Lie groups can be understood in terms of a subclass known as "simple" Lie groups. And most of these, in turn, can be classified in terms of several different infinite subclasses. But there are five other simple Lie groups that defy classification – the exceptional Lie groups. And of these, E8 is the largest.
Likewise, we have several additional items to mention here that do not fit into a classification scheme.
If you're a fan of hard problems, there are few problems harder than understanding the Navier-Stokes equations. But we can thank Fields medalist Terry Tao for a lucid explanation of
Why global regularity for Navier-Stokes is hard.
If you're more into logical puzzles and computer science, you know all about the "halting problem". Alexandre Borovik at Mathematics under the Microscope uses that topic to illustrate the difference between formal and informal proofs. And incidentally, if you are fond of multi-disciplinary investigations, Alexandre has a book you can download from his site (which is named after the book) that looks at mathematics from the standpoint of cognitive psychology and neuroscience.
My own contribution to this carnival offers a quick (well, compared to a whole book) overview of field theory and Galois theory. It's one step on the way to examining some of the deeper mysteries of algebraic number theory.
Perhaps you've never quite understood why some folks make such a big deal about set theory and the continuum hypotheses, which Paul Cohen did so much to clarify. One illustration of how set theory can actually be applied is to a whole new and elegant way of "constructing" the real numbers. This construction, known as "surreal numbers", was invented by John Horton Conway, another "universal" mathematician. He's also known for, among other things, inventing the "game of life" (based on cellular automata), his role in classifying finite simple groups (only distantly related to Lie groups), and his work (with logician Simon Kochen) on the "free will theorem" in quantum mechanics. Mark Chu-Carroll at Good Math, Bad Math gives us a nice overview of surreal numbers.
Or perhaps you were somehow involved in the "New Math" debacle back in the 70s, and now just don't really care that much for even attempting to explain set theory to junior high students, or practically anyone else for that matter. That's too bad, but math teacher JD2718 has some thoughts on the subject.
In spite of all that, there are rewards to trying to teach math. Dave Marain at MathNotations gives a quick review of the recently published book Coincidence, Chaos, and All That Math Jazz given to him in gratitude by one of his top students. He liked the book.
Thanks, everyone, for reading. Come back again in 2 weeks for the next Carnival of Mathematics, to be hosted by Graeme Taylor at Modulo Errors on April 20.
Note: I had a problem with one of my mailboxes. If you tried to submit an article for this edition of the carnival by sending email to cgd at scienceandreason.net, and the article isn't included here, it may have been affected by the problem. Please resend to carnival at scienceandreason.net, or else submit it for the next edition of the carnival. My apologies for any inconvenience.
Note 2: In the comments Mikael Johanssons observes that his contribution to the carnival got lost in my email problem, so please be sure to check it out. It's about modular representation theory. I'm very sorry about the glitch, Mikael.