A large crowd had gathered in Harvard Square, and I was curious what all the cheering and gasping was about. Working my way through the crowd, I saw a street performer who (according to the handwritten red letters on his cardboard sign) went by the name “Zorn the Magnificent”. He displayed a large orange, borrowed an extremely sharp knife from his assistant, and proceeded to chop the orange into five exotic-looking pieces while standing on one hand. Working with almost unfathomably deft precision, he rearranged the pieces into two oranges, each the same size as the original one. The oranges were given out for inspection and the crowd cheered wildly. I clapped as well — even though I was familiar with the old Banach-Tarski paradox — since it was nevertheless an impressive display of skill and I had never seen it done one-handed before. I heard a man with a long white beard whisper to the woman next to him “He hides it well, but I know that he’s secretly using the Axiom of Choice.” Continue reading
Like many fellow mathematicians, I was very sad to hear the news that Alexander Grothendieck passed away yesterday. The word “genius” is overused; or rather, does not possess sufficiently fine gradations. I know quite a few mathematical geniuses, but Grothendieck was a singularity. His ideas were so original, so profound, and so revolutionary – and he had so many of them! – that I will not even attempt to summarize his contributions to mathematics here. Rather, I thought that I would share some of my favorite passages from the fascinating Grothendieck-Serre Correspondence, published in a bilingual edition by the AMS and SMF. They illuminate in brief flashes what made Grothendieck so extraordinary — but also human. They also illustrate how influential Serre was on Grothendieck’s mathematical development. Before I begin, here is a quote from another wonderful book, Alexander Grothendieck: A Mathematical Portrait, edited by Leila Schneps:
…the features which constitute in some sense his personal mathematical signature… are very familiar to those who know Grothendieck’s work: the search for maximum generality, the focus on the harmonious aspects of structure, the lack of interest in special cases, the transfer of attention from objects themselves to morphisms between them, and—perhaps most appealingly—Grothendieck’s unique approach to difficulties that consisted in turning them, somehow, upside down, and making them into the actual central point and object of study, an attitude which has the power to subtly change them from annoying obstacles into valuable tools that actually help solve problems and prove theorems… Of course, Grothendieck also possessed tremendous technical prowess, not even to mention a capacity for work that led him to concentrate on mathematics for upward of sixteen hours a day in his prime, but those are not the elements that characterize the magic in his style. Rather, it was the absolute simplicity (in his own words, “nobody before me had stooped so low”) and the total freshness and fearlessness of his vision, seemingly unaffected by long-established views and vantage points, that made Grothendieck who he was.
After somewhat of a hiatus, I’m back to blogging again. The purpose of this post is to advertise the conference “p-adic Methods in Number Theory”, which will be held in Berkeley, CA from May 26-30, 2015. The conference, which I am helping to organize, is in honor of the mathematical legacy of Robert Coleman. Please spread the word! Here is the current version of the conference poster, which will be mailed out soon to a math department near you:
Many thanks to Janet Ziebell of the Georgia Tech College of Sciences for her help creating this poster, and to Ken McMurdy for designing the conference website.
Here is a memorial article about Robert which I co-authored with Barry Mazur and Ken Ribet. I encourage you to read it! It will be published in the new open access journal Research in the Mathematical Sciences, in a special volume dedicated to Robert.
In my last post, I wrote about two infinite games whose analysis leads to interesting questions about subsets of the real numbers. In this post, I will talk about two more infinite games, one related to the Baire Category Theorem and one to Diophantine approximation. I’ll then talk about the role which such Diophantine approximation questions play in the theory of dynamical systems.
The Choquet game and the Baire Category Theorem
The Cantor game from Part I of this post can be used to prove that every perfect subset of is uncountable. There is a similar game which can be used to prove the Baire Category Theorem. Let be a metric space. In Choquet’s game, Alice moves first by choosing a non-empty open set in . Then Bob moves by choosing a non-empty open set . Alice then chooses a non-empty open set , and so on, yielding two decreasing sequences and of non-empty open sets with for all . Note that ; we denote this set by . Alice wins if is empty, and Bob wins if is non-empty. Continue reading
In this post I’d like to illustrate how one can use infinite games to prove theorems about the real numbers. I’ll begin with a game-theoretic proof that the set of real numbers is uncountable, following the exposition in this paper of mine. This will lead us somewhat unexpectedly into the realm of descriptive set theory, where we will discuss how games are used in cutting-edge explorations of the Axiom of Choice, the Continuum Hypothesis, and the foundations of second-order arithmetic. In a sequel post I will discuss how infinite games can be used to study Diophantine approximation, with applications to complex dynamics.
Countable versus uncountable infinities
When my daughter was 5 years old, she asked me if there is just one infinity. I proudly kissed her on the forehead and told her what an excellent question that was. I told her no, infinity comes in many different flavors. I pretty much left it at that, but since she’s 10 now, here are some more details for her. (The reader familiar with the basics of set theory can move on to the next section.) Continue reading
It’s been a while since my last blog post — one reason being that I recently got married. In honor of that occasion, and my return to math blogging, here is a post on Hall’s Marriage Theorem.
Consider the following game of solitaire: you deal a deck of cards into 13 piles of 4 cards each, and your goal is to select one card from each pile so that no value (Ace through King) is repeated. It is a beautiful mathematical fact that this can always been done, no matter how the cards were originally dealt!
We will deduce this from a more general result due to Philip Hall commonly known as Hall’s Marriage Theorem. Suppose you are given finite sets and you wish to find distinct elements . (In our solitaire example, take to be the values of the cards in the pile.) Such a collection is called a transversal or SDR (system of distinct representatives). Under what conditions is this possible? Well, for a transversal to exist it is necessary that for each subset , the set contains at least elements. Hall’s theorem asserts that these conditions are also sufficient. Continue reading
A famous result of David Hilbert asserts that there exist irreducible polynomials of every degree over having the largest possible Galois group . However, Hilbert’s proof, based on his famous irreducibility theorem, is non-constructive. Issai Schur proved a constructive (and explicit) version of this result: the Laguerre polynomial is irreducible and has Galois group over .
In this post, we give a simple proof of Schur’s result using the theory of Newton polygons. The ideas behind this proof are due to Robert Coleman and are taken from his elegant paper On the Galois Groups of the Exponential Taylor Polynomials. (Thanks to Farshid Hajir for pointing out to me that Coleman’s method applies equally well to the Laguerre polynomials.) Before we begin, here is a quote from Ken Ribet taken from the comments section of this post:
One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if is an algebraic curve of genus at least 2, then the set of rational points on is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set . The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on , but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of provided that the Mordell-Weil rank of the Jacobian of is not too large. (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)
I am very sad to report that my Ph.D. advisor, Robert Coleman, died last night in his sleep at the age of 59. His loving wife Tessa called me this afternoon with the heartbreaking news. Robert was a startlingly original and creative mathematician who has had a profound influence on modern number theory and arithmetic geometry. He was an inspiration to me and many others and will be dearly missed.
Robert was born on November 22, 1954 and earned a mathematics degree from Harvard University. He subsequently completed Part III of the mathematical tripos at Cambridge, where he worked with John Coates and made important contributions to local class field theory. By the time he entered graduate school at Princeton, Robert had essentially already written his doctoral dissertation, but his formal thesis advisor was Kenkichi Iwasawa. He began teaching at UC Berkeley in 1983 and was a recipient of a MacArthur “Genius” Fellowship in 1987. Robert published 63 papers, including 8 papers in the prestigious journal Inventiones Mathematicae and 5 in the Duke Mathematical Journal. He had an amazing intuition for everything p-adic. Long before the invention of Berkovich spaces, Robert could somehow visualize paths and structures in p-adic geometry which no one else in the world saw as keenly or as profoundly. I rarely saw him reading papers or books. He seemed to figure out whatever he needed to know almost from scratch, which often made his papers quite difficult to read but this went hand in hand with his brilliance and originality.
When I was a graduate student at Berkeley, Robert hosted an invitation-only wine and cheese gathering in his office every Friday afternoon code-named “Potatoes”. Among the regular attendees were Loïc Merel and Kevin Buzzard, who were postdocs at the time. It was a wonderful tradition. In the summer of 1997, while I was still a graduate student, Robert invited me to accompany him for three weeks in Paris to a workshop on p-adic Cohomology at the Institut Henri Poincare. That was the first time I met luminaries like Faltings, Fontaine, and Mazur. Since the workshop was (a) totally in French and (b) on a topic I knew almost nothing about, I was in completely over my head. But I fell in love with Paris (which I’ve since returned to many times) and my best memories from that trip are of dining with Robert and seeing the city with him.
The trip also taught me to appreciate the significant challenges which Robert, who had Multiple Sclerosis, bravely faced every day. I remember helping Robert check into his hotel room near the Luxembourg Gardens, only to find out that his wheelchair did not fit in the elevator. We had to find another hotel room for him, which was not so easy given the level of our French! Curbs were a constant challenge for Robert, as finding on- and off- ramps for wheelchairs in Paris was like trying to get a vegan meal in rural Arkansas.
This past weekend I had the privilege to speak at the Southern California Number Theory Day along with Manjul Bhargava, Elena Fuchs, and Chris Skinner. Manjul and Chris spoke about a series of remarkable results which, when combined, prove that at least 66.48% of elliptic curves over satisfy the (rank part of the) Birch and Swinnerton-Dyer (BSD) Conjecture (and have finite Shafarevich-Tate group). Bhargava’s work with Arul Shankar also proves that at least 20.6% of elliptic curves over have rank 0, at least 83.75% have rank at most 1, and the average rank is at most 0.885. Conjecturally, 50% of elliptic curves have rank 0, 50% have rank 1, and 0% have rank bigger than 1, and thus the average rank should be 0.5. (And conjecturally, 100% of elliptic curves satisfy the BSD conjecture. :)) Before the work of Bhargava-Shankar and Bhargava-Skinner (which makes use of recent results of Skinner-Urban. Wei Zhang, and the Dokchitser brothers among others), the best known unconditional results in this direction were that at least 0% of elliptic curves have rank 0, at least 0% have rank 1, the average rank is at most infinity, and at least 0% of curves satisfy the BSD conjecture.
I will attempt to briefly summarize some of the main ideas from their talks; see these papers by Bhargava-Skinner and Bhargava-Shankar for more details and references. (The paper of Bhargava, Skinner, and Wei Zhang showing 66.48% is forthcoming. [Note added 7/8/14: that paper has now appeared at http://arxiv.org/abs/1407.1826.]) Continue reading
In this post I’ll talk about another favorite recreational math puzzle, the (in)famous “Pentagon Problem”. First, though, I wanted to provide a solution to the Ghost Bugs problem from my last blog post. The puzzle is the following:
You are given four lines in a plane in general position (no two parallel, no three intersecting in a common point). On each line a ghost bug crawls at some constant velocity (possibly different for each bug). Being ghosts, if two bugs happen to cross paths they just continue crawling through each other uninterrupted. Suppose that five of the possible six meetings actually happen. Prove that the sixth does as well.
Here is the promised solution. The idea (like in Einstein’s theory of general relativity) is to add an extra dimension corresponding to time. We thus lift the problem out of the page and replace the four lines by the graph of the bugs’ positions as a function of time. Since each bug travels at a constant speed, each of the four resulting graphs is a straight line. By construction, two lines and intersect if and only if the corresponding bugs cross paths.
Suppose that every pair of bugs cross paths except possibly for bugs 3 and 4. Then the lines each intersect one another (in distinct points) and therefore they lie in a common plane. Since line intersects lines and in distinct points, it must lie in the same plane. The line cannot be parallel to , since their projections to the page (corresponding to forgetting the time dimension) intersect. Thus and must intersect, which means that bugs 3 and 4 do indeed cross paths.
Cool, huh? As I mentioned in my last post, I can still vividly remember how I felt in the AHA! moment when I discovered this solution more than 15 years ago.
I just moved into a new house and haven’t had time to blog much lately. But I did want to advertise my friend Manya Raman-Sundström’s upcoming Workshop on Beauty and Explanation in Mathematics at Umeå University in Sweden: http://mathbeauty.wordpress.com/wbem/
The list of invited speakers includes Hendrik Lenstra, one of my graduate school teachers. (If you haven’t see it before, you should check out Lenstra’s lovely short article Profinite Fibonacci Numbers.) Continue reading
First, a bit of background. In a 2007 paper, Emeric Gioan introduced the cycle-cocycle reversal system on a (finite, connected, unoriented) graph G, which is a certain natural equivalence relation on the set of orientations of G. Recall that an orientation of G is the choice of a direction for each edge. A cycle flip on an orientation consists of reversing all the edges in a directed cycle in . Similarly, a cocycle flip consists of reversing all the edges in a directed cocycle in , where a directed cocycle (also called a directed cut) is the collection of all oriented edges connecting a subset of vertices of G to its complement. The cycle-cocycle reversal system is the equivalence relation on the set of orientations of G generated by all cycle and cocycle flips. In his paper, Gioan proves (via a deletion-contraction recursion) the surprising fact that the number of equivalence classes equals the number of spanning trees in G. A bijective proof of this result was subsequently obtained by Bernardi. Continue reading
In an earlier post, I described a graph-theoretic analogue of the Riemann-Roch theorem and some of its applications. In this post, I’d like to discuss a proof of that theorem which is a bit more streamlined than the one which Norine and I gave in our original paper [BN]. Like our original proof, the one we’ll give here is based on the concept of reduced divisors. Continue reading
This is a sequel to last week’s blog post “Two applications of finite fields to computational number theory“. The main reason I decided to write a follow-up is that I’ve learned a lot about Concluding Observations #1 and #6 from that post during the last week. In Observation #1, I mentioned without further comment a recursive procedure for computing square roots modulo a prime; it turns out that this procedure is essentially equivalent to Cipolla’s algorithm, but was discovered independently by Lehmer (who it seems did not know about Cipolla’s work). I learned this from the wonderful book “Édouard Lucas and Primality Testing” by Hugh C. Williams, which I highly recommend to anyone interested in the history of mathematics. In explaining the connection between the algorithms of Cipolla and Lehmer, I’ll make a digression into the general theory of Lucas sequences, which I had some vague knowledge of before but which I’ve learned a lot about in the last week from reading Williams’ book. In Observation #6, I asked if there was a conceptual explanation for the fact that the Chebyshev polynomial shows up in the Lucas-Lehmer test; Greg Kuperberg sent me just such an explanation and I will expand on his comments below. Continue reading