Vertically Integrated Summer Program in Computational Mathematics

The Ohio State University, Columbus, OH - Summer 2004

CLICK HERE FOR HANDOUTS ON VARIOUS TOPICS

CLICK HERE FOR MATHSCINET (ARTICLE SEARCHING)

CLICK HERE FOR MATHWORLD (MATH REFERENCE SITE)

PURPOSE:

Undergraduates, graduate students, post-docs and faculty will work for 10 to 12 weeks investigating interesting unsolved conjectures theoretically and experimentally. The goal is to help the undergraduates, graduate students and post-docs in making the transition from one level of their mathematical careers to another.

• Undergraduates: Many interesting conjectures are accessible to undergraduates; further, often there is very little numerical support for the sweeping generalizations that are claimed (the computations required are often non-trivial, and interesting in their own right). Undergraduates will learn the necessary theory and quickly become involved in cutting edge research. They will learn what types of problems mathematicians study, and experience what it is like to be a graduate student by doing original
  (guided) research.
  
  

• Graduate Students / Postdocs: while most graduate students / post-docs have taught a standard class before graduating, few have mentored students or helped design research programs. Both of these skills are essential to becoming a good professor. Under supervision of the faculty, these participants will choose problems to investigate and mentor the undergraduates.

The vertically integrated nature of the program, in which faculty work with post-docs and graduate students, and all three work with undergraduates, will provide valuable training and exposure which is not seen in a typical REU program.

HISTORY:

This program is based on guided undergraduate research classes taught at Princeton, NYU and Ohio State over the last four years, as well as a pilot summer program run last year at the American Institute of Mathematics. Previous year's investigations are available online at

• Princeton:    http://www.math.princeton.edu/mathlab/index.html

• NYU:          http://www.math.nyu.edu/Courses/V63.0393/

• AIM:           http://www.math.ohio-state.edu/~sjmiller/AIM/  (not all write-ups available yet)

• Ohio State:  http://www.math.ohio-state.edu/~sjmiller/ntwg/  (student write-ups not yet available)
   

DETAILS:

After the summer program at AIM, one of the biggest requests by students was for background material. As the students are working on the forefront of mathematics (with mixed backgrounds), the background material is scattered through a variety of sources. A variety of introductory notes (many assuming just calculus) will be made available for students interested in getting a head start.

For the summer program, in addition to introductory lectures, students will immediately begin working on projects on arrival. The projects we plan all have an element which can be investigated on a computer with essentially no preparation. This allows the student to accomplish something tangible on the first day. Throughout the summer we will maintain a mix of computer programming, analyzing data, background reading, and theoretical work.

We expect students to have wildly different backgrounds. Both at the summer program at AIM and the Princeton/NYU/Ohio State classes, we have never had a problem finding problems appropriate to student background. While the techniques which (we hope) will someday prove many modern conjectures are quite sophisticated, often the statement, simple cases and numerical investigations require little more than simple algebra and calculus. Graduate students and post-docs will run series of lectures on the needed background material (linear algebra, probability theory, basic number theory, and so on); the faculty (and probably the post-docs and graduate students) will lecture on more advanced topics and the problems to be investigated.

In addition to mentoring students, graduate students and post-docs will conduct original research, under the supervision of the faculty; these projects will be related to the undergraduate investigations.

It is important that the participants stay focused. Much to our surprise, we found last summer that the students appreciated mandatory weekly presentations. For 5 to 10 minutes, students would lecture to the group on what they accomplished the previous week. The students said this kept them focused and working; we will continue this tradition this year.

The problems will be chosen from Number Theory and related disciplines. We have found it is very useful to have large numbers of students working on related problems, as they are then able to discuss and work together.

The undergraduates have told us that they love doing original computations -- it makes them feel like they own a problem. Thus, while we will suggest projects and avenues (and make sure the students are making good progress), we feel it is important that  the students are co-investigators, helping to choose what to investigate.

At the end of the summer, participants will LaTeX their investigations, which will be published on the web; in addition to announcing their results, these reports will also help future investigators. Also, all students will give a 40 minute talk, as well as introducing another student. Giving talks at conferences / schools is a useful skill, but often undergraduates and graduate students have no preparation before their first presentation. Several days will be spent giving advice; undergraduates and graduate students will also give several practice talks to the post-docs and faculty before giving their final presentation. Students from Princeton, NYU and AIM, while often worrying about their presentations beforehand, have uniformly said it was a very useful experience.

The program is similar in spirit to regular research classes; however, as participants' time will not be split with other classes, we envision significantly greater productivity and results. The goal is to prepare participants for the next level of their mathematical careers; we feel such a vertical environment in the summer is a natural way to do so.

Tentatively, the program will run for a little over two months, starting in late June.
 

POSSIBLE PROBLEMS:

Research problems will be drawn primarily from Number Theory and Random Matrix Theory; depending on participants' interest, we may also investigate some problems in Graph Theory and Probability Theory. For example, below are three types of problems which give a flavor for the types of topics to be investigated. By no means are these complete descriptions, or the only problems. More problems will be listed (with more details) later. If you are not familiar with some of the terms below (such as eigenvalues, for example), don't worry -- background lecture series will be run for the program.

• Random Matrix Theory and L-Functions: In classical mechanics, we cannot "solve" how three point masses move in time under gravity; imagine how complicated a heavy nucleus like Uranium would be to describe (with over 200 protons and neutrons). In the 1950s, physicists combined the techniques of Statistical Mechanics and Random Matrix Theory to give a good model for the energy levels of heavy nuclei. Roughly, consider a probability distribution on certain spaces of matrices. For each matrix, calculate properties of its eigenvalues, and then average over all matrices. It turns out this is a very good model for nuclear energy levels. Surprisingly, it is also a good model for many Number Theory questions. In Number Theory, the zeros of L-functions provide enormous information for many of problems. An L-function is of the form L(s) = ∑_n a(n) n^-s. The most famous is the Riemann Zeta Function ζ(s), which has a(n) = 1 for all n. In this case, we can write ζ(s) = Π_p (1 - p^-s)^-1 (this follows from unique factorization of integers). Thus, it is not unreasonable to expect that knowledge of ζ(s) gives us knowledge of primes. It turns out that this function, which initially converges only if Re(s) > 1, can be analytically continued to a function for all complex s. Moreover, the location and distribution of its zeros gives information about primes, and the spacing between zeros of this function look like the spacings between eigenvalues of Complex Hermitian Matrices! This is just the type of the iceberg; there are a lot of problems in this field (lots of interesting sets of matrices to look at, how the largest eigenvalue behaves, interesting L-functions to study, order of vanishing of L-functions at the central point, Birch and Swinnerton-Dyer conjecture, random graphs, connectivity of computer networks, ....). For more information, see the handouts. The following notes are a subset of An Invitation to Modern Number Theory, by Steven J. Miller and Ramin Takloo-Bighash (Princeton University Press, to appear). These notes have been expanded by the first author in a series of lectures in guided research classes, and are still in very rough form. Please do not distribute these notes further.Comments and suggestions would be appreciated -- please email sjmiller@math.ohio-state.edu. I cannot recommend too strongly the following: JUST SKIM THESE NOTES. Do not read these line by line, trying to understand everything at once. Merely glance at them at first to see the types of systems we can investigate and the types of questions we might ask
  
  

• 3x+1 and Benford's Law: The following problem is quite simple to state, but has resisted solution for over 70 years; it's been called a Soviet conspiracy to slow down American mathematics because of all people who have worked on it. Let a₀ be a positive integer, and define a sequence {a_n} by the following: let a_n+1 = 3a_n + 1 if a_n is odd, and a_n / 2 if a_n is even. For example, if we start with 13, we get the sequence 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, and so on. It is conjectured that if we start with any positive integer, we eventually get the repeating pattern 4, 2, 1. This sequence exhibits lots of interesting properties, for example, if one looks at the first digit of each term in the sequence, one sees more 1s than 2s than 3s and so on. Many processes are known to have a bias in their first digit (Benford's Law). However, for a typical starting value a₀, it takes about log(a₀) terms to hit the 4,2,1 cycle. Thus, if we want to study a sequence with 1000 terms before reaching 4,2,1, we have to look at numbers a₀ of size 10¹⁰⁰⁰. Investigations of such large numbers are currently underway in the Winter Working Group here at OSU (and, in fact, it is very easy for these questions to look at such large numbers!). For more on the 3x+1 problem, see http://mathworld.wolfram.com/CollatzProblem.html; for more on Benford's law, see http://mathworld.wolfram.com/BenfordsLaw.html.
  
  

• Another great problem almost certain to be investigated concerns the distribution of prime numbers (or primes in arithmetic progression: for example, fix integers a and b, and look for primes p of the form p = an + b; these primes are in a progression, all are congruent to b modulo a). The celebrated Prime Number Theorem says the following: let π(x) be the number of primes at most x; for example, π(10) = 4 (only primes less than 10 are 2, 3, 5, 7). Then up to lower order terms, as x tends to infinity, π(x) = x / ln(x). This means the percent of numbers less than x that are prime is like 1 / ln(x), which tends to zero. So, in this sense, there are very few primes. Let S(x) be the number of perfect squares at most x; for example, S(10) = 3 (only squares less than 10 are 1, 4, 9). Then for large x, S(x) is basically √x (the error in fact is at most 1). Thus, the percent of numbers less than x which are perfect squares is like 1 / √x; thus, while there are infinitely many primes and infinitely many perfect squares, in this sense there are a lot more primes than perfect squares. Consider a large number x. Then on average the distance to the next prime is like ln(x). Thus, a very natural model of primes is to consider a coin with probability 1 / ln(x) of heads and 1 -1/ ln(x) of tails. If one flips a head, make that number prime, else make it composite (this generates a random sequence, and we can investigate what properties such random sequences should have; we are of course ignoring some number theory constraints -- for example, even numbers can never be prime for x > 2, multiples of 3 cannot be prime for x > 3, and so on). For many questions in number theory, this model leads to good heuristics and predictions; recently, however, Montgomery and Soundararajan (Beyond Pair Correlation) have shown that this model is inconsistent with certain simple numerical investigations of primes (these simple counting experiments can be explained and the programs written in a few minutes!), and in fact the Random Matrix Theory model of the zeros of the Riemann Zeta function give a prediction which agrees beautifully with experiments. There are many additional interesting sequences of primes to investigate, and see which model is correct. Candidates include primes in arithmetic progression, twin primes (primes p such that p+2 is also prime; it was by studying twin primes a few years ago that the Intel processor bug was discovered!), generalized twin primes (fix an integer k, look for primes such that p and p+2k are prime), prime tuples (fix integers k₁ through k_r such that p, p+2k₁, ..., p+2k_r are all prime), Germain primes (if p > 3 is prime, then p-1 cannot be prime as it is divisible by 2; if 2 is the only factor of p-1, then we say p, (p-1)/2 are a Germain prime pair; these occur in cryptography applications), and so on. In order to understand the Random Matrix Model, we will describe one of the most beautiful ideas in mathematics, the Circle Method. This method has allowed us to prove amazing statements concerning primes. Primes are defined by multiplicative properties; thus, questions about addition of primes is quite difficult. The Circle Method allows us to show that every large odd number is the sum of three primes, and gives us a heuristic for the answer to the famous Goldbach question (is every even number greater than 2 the sum of two primes).

APPLICATIONS / CONTACT INFORMATION:

Undergraduates, Graduate Students, Postdocs and Faculty interested in participating in this program should contact Steven Miller at sjmiller@math.ohio-state.edu, who will provide more information and application materials. Financial support will be available for some participants.