 ## Eric Stansifer

I am a graduate student at MIT. I work with Dan Rothman in the EAPS Department on geometric properties of river networks and started in fall 2013. You can reach me via email at erst at mit.

## Publications, etc.

• Yi, R., Cohen, Y., Seybold, H., Stansifer, E., McDonald, R., Mineev-Weinstein, M., & Rothman, D. H. (2017, June). A free-boundary model of diffusive valley growth: theory and observation. In Proc. R. Soc. A (Vol. 473, No. 2202, p. 20170159). The Royal Society.
• Stansifer, E.M., O'Gorman, P.A. & Holt, J.I. (2016) Accurate computation of moist available potential energy with the Munkres algorithm. Quarterly Journal of the Royal Meteorological Society doi:10.1002/qj.2921 [pdf]
• Stansifer, E.M. (2012) Leopoldt's Conjecture for Abelian and non-Abelian cases. (Master's thesis) [pdf]

My resume.

## Personal projects and writings

### Cambridge forecast

The current forecast for Cambridge, updated hourly.

### Math diversions

The following are some math problems I enjoyed working on and wrote up my solutions for. I have aimed my writeups to be accessible to those familiar with complex numbers and basic calculus, including infinite sums and products, although some of the techniques are not typically seen at a high school level. In several places I assume the Fundamental Theorem of Arithmetic, that every integer can be uniquely factored into prime numbers.

• An infinite product for pi. I ran into an interesting expression giving pi as an infinite product that involves the prime numbers. In the end I found it more difficult to prove the correctness of the expression than I expected. Along the way I prove three other expressions for pi, and at the end I have a brief discussion of the Riemann zeta function.
• Asymptotic density of sums of squares. A classic result of elementary number theory, Fermat's theorem on sums of squares, shows that certain primes (in fact, half of them) are equal to the sum of some two perfect squares, for example 61 = 36 + 25. It turns out that this is special to prime numbers: most integers are not equal to the sum of some two perfect squares. In this writeup I briefly discuss this connection between prime numbers and sums of squares and prove my claim that the asymptotic density of sums of squares is zero. Along the way I give an elementary proof that the sum of 1/p over all primes p that are congruent to 3 mod 4 is infinite. The result requires Fermat's theorem to classify which numbers are sums of squares, so to avoid depending on results beyond calculus, I have also written up a proof of Fermat's theorem on sums of squares. This second proof is rather more technical and also requires me to define some terminology from ring theory so is only included for completeness' sake.
• Sum of 1/p for primes 1 mod 4. In the previous writeup we saw that the sum of 1/p over all primes p that are congruent to 3 mod 4 is infinite. It is natural to ask about the sum of 1/p over all primes p that are congruent to 1 mod 4, but the above proof does not work for this problem despite the superficial similarity. Here we give a proof of this result; like the previous writeup, this uses Fermat's theorem on sums of squares.

### Puzzles

I have been collecting a list of logic or math puzzles that I have enjoyed. For some of these puzzles I have included an estimate on how long it took me to solve, but these times should be considered very approximate.