Right now I’m driving from Chicago to Salt Lake City to start a job at the University of Utah. My route has taken me past a lot of silos and water towers, which naturally make me think of related rates. (Sadly, none of them seemed to be leaking into polyhedral receptacles.)
A cylindrical silo in South Dakota. Image: flickr user Lars Plougmann.
A group of seasoned mathematicians including Terence Tao are working to improve on Yitang Zhang’s exciting recent proof that there are infinitely many pairs of consecutive primes separated by no more than 70 million. A glance at the polymath8 wiki shows that the gap has now been shrunk to a measly 250,000. Although this is still nowhere near the gap of two required by the twin primes conjecture, one fact stands out to me as a junior faculty. As Mario Livio recently stated in his new book “The road to triumph [is] paved with blunders.” And on the path to furthering their (and our) knowledge, those contributing to polymath8 are recording all progress, including mistakes! It’s exciting to see research happening in real time, and even as I wrote this, I noticed new World Records being established for the smallest bound on the gap. The current record-holder is MIT Computational Number Theorist Andrew Sutherland.
Terrence Tao’s June 3rd post gives a great introduction to recent events as he clearly lays out the definition of H admissible k-tuples of integers and how they relate to the twin primes conjecture. A k-tuple of integers (a list of k distinct integers) is admissible if it avoids at least one residue class modp for every prime p. These are the k-tuples that have a fighting chance at possessing the desirable characteristic that infinitely many translates consist exclusively of primes. For instance, an example of a NON-admissible 3-tuple is {-2, 0, 2} since each residue class mod 3 is represented. Notice that every translate of {-2,0,2} must contain a multiple of 3, and therefore there are only finitely many translates containing only primes. Filtering all k-tuples to find these special ones is accomplished by creating a “sieve”, thus a series of posts on sieve theory can be found on Tao’s blog and a quick encapsulation of many sieves can be found on another polymath wiki.
The famous Hardy-Littlewood conjecture asserts that all H-admissible k-tuples have infinitely many translates consisting exclusively of primes. But this over-arching theorem is extremely difficult (the twin primes conjecture is a corollary where k=2). Tao proceeds to explain various dilutions and progress made including the 2005 breakthrough by Goldston-Pintz-Yildirim. Apparently, finding narrow H-admissible k-tuples leads to a reduction in the gap between consecutive primes, thus improving on Zhang’s result. This is just one approach to improving the result.
As a non-number theorist, it’s great to be able to understand at least the introductory notions used to explore these questions and to see some of the relationships between older and newer results. It seems that Zhang used many common techniques, but applied them in novel ways, and presented his work so clearly that it was quickly confirmed as valid. While many articles have dwelled on Zhang’s relative obscurity, it’s worth noting that Zhang clearly has a taste for problems that are easy to state and hard to solve as he tackled The Jacobian Conjecture for his dissertation work. Zhang’s dissertation was in the field of Algebraic Geometry (not number theory), and that his advisor remembers him as “a free spirit” who did not even request letters of recommendation for jobs! From now on it seems that Zhang need only his own accomplishments to recommend him.
Astrophysicist Carol Jo Crannell, mathematician Annalisa Crannell, and Iolanthe Good, three generations of women who love STEM. Image: Ximena Catepillan.
It’s not strictly mathematical, but Grandma Got STEM is one of my favorite blogs. It’s a collection of stories about grandmothers and other older women who have or had careers in science, technology, engineering, and mathematics. Continue reading
A Physicist, two Mathematicians and a Philosopher walk into a room and start to discuss infinity. This sounds like the set up to a great joke. But it’s actually one of the events that will occur tonight at 8:00PM EST (momentarily) as part of the World Science Festival. Mathematician Keith Devlin will moderate, and you can watch it live here.
On Saturday, Devlin will moderate a salon aimed at graduate students, post-docs and well-informed members of the general public. This salon will focus on progress made on the Continuum Hypothesis.
The World Science Festival was co-founded by Emmy Award winning Journalist Tracy Day and Columbia Physics professor Brian Greene. This is the 5th year of the festival, which started in NY on the 29th and ends this Sunday. As part of the Ultimate Science Street Fair (to be held in Washington Square), the Museum of Mathematics will present Math Midway, giving passers by the opportunity to help create a “gigantic fractal tetrahedron”. National Public Radio’s “Ask Me Another” will air from the festival with mathematical guest Steven Strogatz. As part of a free event, Strogatz, Astrophysicist Mario Livio (author of “Innumeracy”), and many other best-selling science authors will be available to discuss, give readings, and sign their books.
For a sweeter take on mathematics and science, the microstructures and crystal networks of ice cream will be discussed during an event called the “Scientific Kitchen” , which will be hosted by several chefs and a microbiologist.
Are you in NY now? Send us some pictures of the Fair on Sunday! Comment and tell us how the “mathy” events were.
Four pregnancy tests. Image: Zwager, via Wikimedia Commons.
I have never been pregnant, but from what I understand, it is full of bizarre cravings, frequent bathroom breaks, and a smorgasbord of medical scans and tests. This last part is what concerns Kate Owens. She is a visiting assistant professor in the math department at the College of Charleston, and she is also pregnant with her second child. On her blog, she recently started writing about being a “pregnant mathematician.” Owens’s two posts on the subject have focused on the value of screening tests. “It’s tough having to juggle my emotions as a mother, my knowledge of statistics as a mathematician, and my interest in minimizing unnecessary financial expenses.”
A visualization of the twin primes using an Ulam spiral. Created by Silveira Neto and shared under a Creative Commons-attribution-share alike license.
By now you’ve probably heard the announcements of two big results in number theory: Yitang Zhang of the University of New Hampshire proved that there are infinitely many pairs of primes whose differences are under 70 million, and Harald Helfgott of the École Normale Supériure in Paris posted his completion of the proof of the ternary, or weak, Goldbach conjecture to the arxiv. That conjecture states that every odd integer greater than 5 can be written as the sum of three primes.
Last year over 60 new mathematical articles were appearing on arXiv every day! What if every mathematics research article were freely available online? What format would you want this World Digital Mathematical Library (WDML) to take on other than a gigantic arXiv? The debate concerning open access to academic research continues almost 7 years since the entire editorial board of Topology resigned and 16 months since the boycott of Elsevier led by mathematician Tim Gowers began. But Ingrid Daubechies, the current president of the International Mathematical Union (IMU), wants to know what young mathematicians think about how articles already accessible online could be used.
All kinds of questions have arisen: How should online articles be annotated by readers? How should errata be identified and discussed? Might there be a way for non-traditional contributions to research mathematics to be officially recognized within this context? How can it be made easier to update and create accurate bibliographies? There are 29 comments and counting posted on Terry Tao’s blog concerning Dr. Daubechies’ guest post on this topic last Wednesday. For example, graduate student Jason Rute suggests online reading groups for specific papers and the ability to create reading lists as one does on Amazon. The answers and ideas you might have may very well become a reality as the project is being backed by the Sloan Foundation, which has also supported the Sloan Digital Sky Survey, a 14-year ongoing project that has greatly impacted astronomy research.
Are you having deja vu? Parallel to the the NIH making its research public in 2004, there was a push to retro-digitize hundreds of mathematics journals. So there is currently a website for the WDML that is inactive but still has some interesting links. You can read an AMS Notices article about this project from 2003. For an update, Peter Olver, chair of the IMU’s Committee on Electronic Information and Communication (CEIC) which is working on the current project, has a slideshow “What’s Happening with the World Digital Mathematics Library?”
The white house still finds “open science” to be a hot topic, as the Office of Science and Technology is accepting nominations up until today for “Open Science” Champions of Change who provide free access to data or publications generated from scientific research or who lead research that uses publicly available scientific data.
A “binary bonsai” created using an algorithmic knitting process. Image copyright Madeleine Shepherd. Used with permission.
Many of us have seen Fibonacci numbers in sunflowers and hyperbolic curvature in kale leaves. Botanica Mathematica, “a textile taxonomy of mathematical plant forms,” takes mathematical-botanical correspondences like these and throws in a little fiber art. “The Botanica Mathematica project is about using simple mathematical rules to generate pieces of knitting or crochet,” says one post. The blog is a companion to an art exhibit in progress. Later this year, Botanica Mathematica creators Julia Collins and Madeleine Shepherd will put together an art installation of the pieces they make and receive from knitters and crocheters around the world. Shepherd says that pieces have been donated from England, Scotland, Germany, the Netherlands, and the US, and anyone is welcome to contribute.
As the end of the semester draws near, I find myself looking for quick ways to procrastinate. And playing a little game won’t take up that much time, right? Like, say, we could play a friendly game of Tic-Tac-Toe…
(Courtesy of xkcd)
Rogue waves in art and nature. Image: J.M. Dudley, V. Sarano, F. Dias.
Happy Earth Day! It seems appropriate today to highlight the Mathematics of Planet Earth blog. In fact, it’s triply appropriate: today is Earth Day, April is Mathematics Awareness Month (with a theme this year of the mathematics of sustainability), and 2013 is the year for Mathematics of Planet Earth.
