Mathematics

And so it came to pass, that an almost millenial quest found a safe resting place… Like all analytic number theorists, I’ve been amazed to learn that Yitang Zhang has proved that there exist infinitely many pairs of prime numbers with bounded by an absolute constant . So, how did he do it? Well, since the paper just became available, I don’t have anything intelligent to say yet on the new ideas that he introduced (but I certainly hope to come back to this!). However, one can easily list those previously-known tools that he uses, which involve some of the deepest and most clever results in analytic number theory of the last 30 to 35 years. (1) At the core, the proof is based on the method discovered about ten years ago by Goldston, Pintz and Y?ld?r?m to show that As I discussed a while back, this remarkable result — besides its intrinsic interest — was notable for being the first to bring the problem of bounded gaps between primes within a circle of well-studied and widely believed conjectures on primes in arithmetic progressions to large moduli. Precisely, Goldston, Pintz and Y?ld?r?m had derived the statement above, after many ingenious steps, by applying the Bombieri-Vinogradov Theorem, and they showed that any progress beyond it towards the so-called Elliott-Halberstam Conjecture would imply the bounded gap property. However, in my former blog post, I discussed why it seemed extremely difficult to go in that direction… (2) … despite the existence of some results going beyond the Bombieri-Vinogradov theorem, due first to Fouvry-Iwaniec and later improved by Bombieri-Friedlander-Iwaniec; but Zhang uses indeed these results… (3) … results which themselves depend crucially on two big ideas: the well-factorable weights of the linear sieve, due to Iwaniec, and the development and applications of the Kuznetsov formula and other results concerning the spectral theory of automorphic forms and estimates for sums of Kloosterman sums, the outcome of the work of Deshouillers and Iwaniec; (4) but furthermore, Zhang uses also an estimate for a certain character sum over finite fields which had appeared in the work of Friedlander and Iwaniec on the exponent of distribution for the ternary divisor function; this sum is a three-variable additive character sum, and its estimation (with square-root cancellation), proved by Bombieri and Birch in an Appendix to the paper of Friedlander and Iwaniec, depends crucially on the Riemann Hypothesis over finite fields of Deligne. Here are some references to surveys or explanations of some of these tools. Amusingly, I have written something on most of them… There have been many surveys of the work of Goldston, Pintz and Y?ld?r?m, and in particular I wrote a Bourbaki report on it, which may be interesting to those who read French; Concerning the automorphic Kloostermania that comes into the Fouvry-Iwaniec and Bombieri-Friedlander-Iwaniec circle of ideas, I happened to write a few years ago, for a book on Poincaré’s mathematical work, an account of the applications of Poincaré series to analytic number theory, which are used to prove the Kuznetsov formula; Fouvry has written a survey Cinquante ans de théorie analytique des nombres from the point of view of sieve methods, which discusses the philosophy of extending the ranges of exponents of distribution for important sequences, as well as the well-factorable weights of Iwaniec; Fans of trace functions may remember that I noticed in a previous post (see the very end) that the exponential sum of Friedlander-Iwaniec, estimated by Birch and Bombieri, is (for prime moduli) just a special case of the general “correlation sums” that appeared in my recent work with Fouvry and Ph. Michel — in particular, our arguments (based on the sheaf-theoretic Fourier transform of Deligne, Laumon, Katz and others) gives a conceptually simple proof of that estimate; And although it doesnR

Enlightenment at a red traffic light Wolf Prize laureate Prof. George Daniel Mostow made his greatest scientific breakthrough while driving. Haaretz tells the story of how Dan Mostow reached his breakthrough known as Mostow’s rigidity theorem. Congratulations, Dan! French-Isreali Meeting and Günterfest More updates: If you are in Paris On Wednesday and Thursday this week there will be a lovely French-Isreali interacademic meeting on mathematics.  The problem is very interesting, and I will give a talk quite similar to my recent MIT talk on quantum computers. In the  weekend  we will celebrate Günter Ziegler’s 50th birthday in Berlin. Günter started very very young so we had to wait long for this.

In this day and age of being green, some folks are looking at ways to save paper. Math worksheets are a great teaching resource since math is a practice subject. After much thought about expanding my worksheet library, I decided to work on a way to create online worksheets. I am proud to announce our new Math Worksheet Generator.Now visitors can create their own worksheets in seconds! Each worksheet is interactive, with a timer and instant scoring. These resources are intended for on-screen use, just like we also provided the option of printing the worksheet and the answer key. Many thanks to Noetic Learning for partnering with us on this venture.

Math Goodies Blog: Math Worksheet Generator

The academic year finished off with two rather different events: my LMS–Gresham lecture about the Mathematical Structures course, and marking the approximately 270 scripts. The LMS–Gresham Lecture Last week I gave the annual LMS–Gresham lecture. When they asked me to do it, a year ago, I was writing the course material for Mathematical Structures, and couldn’t think of anything else, so I said I would talk about that. In fact, I believe that a cohort of talented and passionate mathematicians is important – why else would I be doing this job? Anyway, the lecture went well. You can see the slides here; the slides don’t have all the ad libs, or the questions afterwards. After the lecture, we went for a pleasant meal in the nearby Indian restaurant. Incidentally, I went to the LMS–Gresham lecture last year to see how someone else (in this case Bernard Silverman) did it. But this year I have no idea who my successor is. Marking the exam The good news here is that the students seem to have done rather well in the exam. Of course there is lots of monitoring and standardization to be done yet, but it looks as if both the median and the average mark are just above the B/C borderline. The students have done me proud. (I did wish, though, that I had a little stamp saying “An example is not a proof!”) However, I noticed a couple of curious tendencies: Having given a counterexample to one statement, people tend to hunt for a different counterexample to the next, even if the same one works (for example, if the second statement is the contrapositive of the first). It’s as if the counterexample has exhausted its potency on one statement. When asked whether several properties hold (e.g. is a relation reflexive, symmetric, transitive?), there is a tendency to mention only the ones that do hold. We all have a mindset that inclines towards the positive, but to test a hypothesis you have to look at possible negatives. Here is a selection of things written by students which caught my eye. These are not in any sense a representative selection. Some of them show a lack of understanding which you might find worrying; I would excuse a lot of this on the grounds that in the stress of an exam you will almost certainly write things that with calm consideration you wouldn’t. Some contain good sense hidden under poor expression. Some show original and creative thought. Some of them show that the students have picked up my passion for mathematics! I have sometimes lightly edited these. √2 must be irrational since it falls between √1 and √3. “Clearly” cannot be used in a proof since nothing is clear without a proof of it. [This was a common reaction. One candidate said "You cannot be lazy and write "Clearly P(0) is true", you must prove it. It may be clear to you, not always clear to the reader". Bravo!] A set with no elements contains a single element which is the empty set. [After working out the cases n = 1,2,3] So it seems the formula is correct. However the above is not a proof. m2 is also even, since anything squared becomes even. If “less than” is related to ℕ, then ℕ is related to “less than”. Hence it is shown that P(n) is true for P(n+1). The proof does not include a conclusion box ☐ to indicate the end of proof [and is therefore invalid]. Therefore A is infinitely countable. [This quite common.] The relation “less than” on ℕ is not reflexive since a is not related to a for most natural numbers a. We need to prove by rejection. Suppose √2 is a rational number. Then it can be written in the form m/n. But we cannot write √2 in this form, so contradiction. The statement is not true as this cannot be proven. The contrapositive is not true as the statement is often true. Previous

The issue of random sampling in compressive sensing was made more clear, in my view, by the work of Ben Adcock and Anders Hansen ( see A Q&A with Ben Adcock and Anders Hansen: Infinite Dimensional Compressive Sensing, Generalized Sampling, Wavelet Crimes, Safe Zones and the Incoherence Barrier. ). In short, the whole random sampling story has some problems at low frequencies. In MRI, the field at the leading edge of compressive sensing, several sampling techniques have been evaluated to corner this issue down. What is most interesting in this whole story is the connection with the hardware: We are here seeing a direct connection between actual hardware constraints (sampling authorized by the hardware) and the mathematics of sampling. One needs to realize that this connection between those two fields is rare. Here is another example of the connection between mathematics of sampling and hardware constraints: Travelling salesman-based compressive sampling by Nicolas Chauffert, Philippe CIUCIU, Jonas Kahn, Pierre Weiss. The abstract reads: Compressed sensing theory indicates that selecting a few measurements independently at random is a near optimal strategy to sense sparse or compressible signals. This is infeasible in practice for many acquisition devices that acquire samples along \textit{continuous} trajectories. Examples include magnetic resonance imaging (MRI), radio-interferometry, mobile-robot sampling, ... In this paper, we propose to generate continuous sampling trajectories by drawing a small set of measurements independently and joining them using a travelling salesman problem solver. Our contribution lies in the theoretical derivation of the appropriate probability density of the initial drawings. Preliminary simulation results show that this strategy is as efficient as independent drawings while being implementable on real acquisition systems. If somebody were to explain to me why they have pi^2 as opposed to pi^(1/2), I'd really appreciate it. Join the CompressiveSensing subreddit or the Google+ Community and post there ! Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.

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Whoever ... proves his pointand demonstrates the prime truth geometricallyshould be believed by all the world,for there we are captured~Albrecht DurerThe 141st day of the year; 141 is the first non-trivial palindrome appearing in the decimal expansion of Pi, appearing immediately after the decimal point, 3.14159. Tanya Khovanova, Number Gossip EVENTS1819 the first bicycle in the U.S. was seen in New York City. Such bicycle velocipedes or "swift walkers" had been imported that same year. Shortly thereafter, on 19 Aug 1819, the city's Common Council passed a law to "prevent the use of velocipedes in the public places and on the sidewalks of the city of New York."**TIS (Skateborders take note, you are not the first to be banned from the sidewalks)1908 Glenn (Hammond) Curtiss was a pioneer in the development of U.S. aviation whose aircraft were widely used during World War I. That the Wrights made the first powered flights has generally been accepted, but the achievements of Curtiss spanned several decades and took the airplane from its wood, fabric and wire beginnings to the forerunners of modern transport aircraft. Curtiss made his first flight on his 30th birthday, 21 May 1908, in White Wing, a design of the Aerial Experiment Association, a group led by Alexander Graham Bell. White Wing was the first plane in America to be controlled by ailerons instead of the wing-warping used by the Wrights. It was also the first plane on wheels in the U.S. *TIS (See 1878 Birth below)1901 the first U.S. State motor car legislation was an act to regulate the speed of motor vehicle, passed in Connecticut. A limit was established of 12 mph within city limits and 15 mph outside, which were higher than the 8 mph city and 12mph country speeds in the bill as originally presented. Also, the car driver was required to reduce speed upon meeting or passing a horse-drawn vehicle, and if necessary, to stop to avoid frightening the horse.*TISThis last part about meeting (or passing) a horse, with or without cart, is still essentially the law in England and Ireland.In 1916, Daylight Saving Time was introduced in Britain as a war-time measure to save fuel. The idea began when a London builder, William Willett, presented a scheme of shifting the clock to better use the hours of daylight in summer. He campaigned and published a brochure on the subject in 1907 (in which his proposal was to adjust the clocks in four weekly adjustments of 10-mins). When Parliament did consider a Daylight Saving Bill, to implement a seasonal one-hour change, it failed for lack of support. However, a little more than a year after his death after his death, the idea was finally adopted during WW I for wartime fuel savings. Now most of the countries in the northern hemisphere use a form of daylight saving time. *TIS 1932 Amelia Earhart ?ew alone across the Atlantic, being the ?rst woman to do so. *VFR1952 IBM Announces Model 701, "Defense Calculator.": IBM announced its 701 machine and by doing so emphasized its commitment to innovation in electronic computing. The company's first computer designed for scientific computations. The IBM 701 had an electrostatic storage tube memory and kept information on magnetic tape. The company eventually sold 19 of the machines -- more than expected -- to the government and large companies and universities for complex research.*CHM BIRTHS429 B.C. Plato born in Athens. He died on the same date in 348 B.C. [Muller] [Should it be 427 B.C.?] *VFR 1471 Albrecht Durer, (21 May 1471 – 6 April 1528) German painter and engraver. Mathematicians are fond of his etching Melancholia for it contains the magic square. Oldstyle numerals are used in the two center squares to emphacize the year that this etching was done by Durer. There is still debate about the shape of the solid in the foreground of the picture. *TIS He also published a book on geometric constructions (1535) using a straight-edge and compass. Although designed to enable artists better represent a

Jo Boaler has kindly asked me to spread the word about her free, upcoming course How to Learn Math. It sounds intriguing; in fact, I signed up and hope to be able to attend (to have the time).Here's her description of it:The course is a short intervention designed to change students' relationships with math. I have taught this intervention successfully in the past (in classrooms); it caused students to re-engage successfully with math, taking a new approach to the subject and their learning.In the 2013-2014 school year the course will be offered to learners of math but in July of 2013 I will release a version of the course designed for teachers and other helpers of math learners, such as parents. In the teacher/parent version I will share the ideas I will present to students and hold a conversation with teachers and parents about the ideas. There will also be sessions giving teachers/parents particular strategies for achieving changes in students and opportunities for participants to work together on ideas through the forum pages. Concepts1. Knocking down the myths about math.Math is not about speed, memorization or learning lots of rules. There is no such thing as “math people” and non-math people. Girls are equally capable of the highest achievement. This session will include interviews with students.2. Math and Mindset.Participants will be encouraged to develop a growth mindset, they will see evidence of how mindset changes students’ learning trajectories, and learn how it can be developed.3. Teaching Math for a Growth Mindset.This session will give strategies to teachers and parents for helping students develop a growth mindset and will include an interview with Carol Dweck.4. Mistakes, Challenges & Persistence.What is math persistence? Why are mistakes so important? How is math linked to creativity? This session will focus on the importance of mistakes, struggles and persistence.5. Conceptual Learning. Part I. Number Sense.Math is a conceptual subject– we will see evidence of the importance of conceptual thinking and participants will be given number problems that can be solved in many ways and represented visually.6. Conceptual Learning. Part II. Connections, Representations, Questions.In this session we will look at and solve math problems at many different grade levels and see the difference in approaching them procedurally and conceptually. Interviews with successful users of math in different, interesting jobs (film maker, inventor of self-driving cars etc) will show the importance of conceptual math.7. Appreciating Algebra.Participants will be asked to engage in problems illustrating the beautiful simplicity of a subject with which they may have had terrible experiences.8. Going From This Course to a New Mathematical Future.This session will review where you are, what you can do and the strategies you can use to be really successful.

(Click on the cartoon to see the full image.) (C)Copyright 2013, C. Burke.A big hit for the Ram-Ones about 34 years ago.