The twin prime conjecture states that there are infinitely many pairs of primes that differ by two. A new paper about to published in the Annals proves that there are infinitely many pairs of primes differing by no more than 70,000. There is a nice write-up about this in Nature. For a few more details, see the n-Category Cafe.
Algorithmic, yet readable, proofs.
Over the past couple of weeks, Tim Gowers has been conducting an experiment where three proofs each of five well-known facts are presented, one written by an undergrad, one written by a grad student, and one written by a computer program developed over the past few years by Gowers and Mohan Ganesalingam. The goal of the program is to attempt to copy how human mathematicians actually approach producing such proofs (unlike, say, chess programs, which typically play by analyzing much farther into the future than any human could). The goal of the poll is to tell which is which.
It’s too late to participate in the poll, but if you’d like to test your judgment, you can see the original experiment here (be aware that polling results are now shown at the bottom).
In either case, you should check out the results of the poll and Gowers’ write-up. How well do his assumptions about our thought-processes ring true to you?
Abel Prize
The Abel Prize was awarded this morning to Pierre Deligne, “for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields.” This award is well-deserved, and I encourage you to read the writings of Tim Gowers and Arne Sletsjøe at the link above that discuss his work.
An interesting note about axiomatics
Andrew Wiles’ proof of Fermat’s Last Theorem relied on some very heavy machinery in algebraic geometry. What I had not realized (this is not at all my field) was that this machinery (built by Grothendieck in the middle of the 20th century) involved an additional axiom beyond those of ordinary set theory. Thus Wiles’ proof FLT relied on an axiom well beyond those needed to state the theorem. This is disconcerting.
Colin McLarty of Case Western Reserve University has just shown that Grothendieck’s work (or maybe just that part needed for FLT, I’m not sure) could in fact be built on a more familiar, standard set of axioms, meaning that the proof of FLT is a little more concerting (which should be the opposite of disconcerting, n’est-ce pas?).
Meeting Summary
We had a great meeting last night! Many thanks to our new officers Julia (treasurer) and Alice (secretary). Here’s a list of some upcoming events we talked about:
- Feb 23: The Math/Stat department at South is hosting a high school math competition, and we need volunteers for proctoring tests. This involves sitting in a room in ILB for approximately two hours doing whatever you want that is quiet. The level of effort involved is approximately ε. On the other hand, I believe there will be food. Please contact Sonna Farmer if you are interested.
- Mar 23: USA also hosts GEMS (Girls Exploring Math and Science). This is an event where approximately 300-400 middle-school girls who show an interest in math and science visit USA and, among other things, attend two workshops. We are interested in anyone who would like to run a workshop. These are typically led by two people for a group of about 20 girls. The level of effort here is significantly greater than ε, but it is very rewarding (and looks great on a CV). There are several people in the department who have led workshops before and would be happy to answer any questions you might have or even possibly assist you in leading a workshop. If you are interested, contact Kalyn Hode, Trey Trampel, or Dr. Barnard, and we can put you in touch with the right people.
- Apr 6: Troy MathFest 2013. This is a conference intended primarily for undergraduates where short talks are given by undergrads, grad students, and professors, all aimed at undergraduate mathematicians. It is free and should be a lot of fun (and there’s food). If you are interested in giving a 15 minute talk, you need to supply your information by March 15, and if you register by March 1 you can win prizes. There’s also a Calculus contest with cash prizes. We expect several people to go, so we can probably carpool. There are some travel funds available from Troy and possibly also from the USA Math/Stat department. Contact Kalyn, Trey, or Dr. Barnard if you are interested in going.
- REUs! It’s not too late to register for an REU (research experience for undergraduates), but deadlines are fast approaching. They can be a lot of fun, look great on grad school applications, and can give you a much better idea of what it is mathematicians actually do. The AMS has a wonderful page on REUs. In particular, there is a wonderful REU called SMILE at LSU that Trey attended last year and absolutely loved. Look into it!
Math Videos
I was surprised to discover (via the UCSB Math Dept facebook page) that there is a collection of over 1300 math lecture videos on pinterest.
3d-printing shadows of 4d
Scientific American has a blog post about a puzzle created by 3d-printing pieces of a projection of the 4d regular polytope known as the 120-cell.