2013 Section Meeting

Invited talks

Katherine Socha, Math for America

Sea battles, Benjamin Franklin's oil lamp, and jellybellies

Abstract: During our passage to Madeira, the weather being warm, and the cabbin windows constantly open for the benefit of the air, the candles at night flared and run very much, which was an inconvenience. At Madeira we got oil to burn, and with a common glass tumbler or beaker, slung in wire, and suspended to the ceiling of the cabbin, and a little wire hoop for the wick, furnish'd with corks to float on the oil, I made an Italian lamp, that gave us very good light...." (Benjamin Franklin, December 1, 1762 letter to John Pringle)

Observations of real phenomena have led to mathematical modeling of surface water waves, interfacial waves, and Lagrangian coherent structures among other examples. This expository talk will provide a quick tour of the (mostly advanced undergraduate level) mathematics needed to describe idealized versions of the rings formed by striking a surface of water with a large object (like a bomb), the oil-water waves observed by Founding Father Benjamin Franklin on his voyage to Madeira, and the motion of nutrient laden water being swept into the underbelly of a swimming jellyfish.

Ravi Vakil, Stanford University

The Mathematics of Doodling

Abstract: Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today.

William Dunham, Muhlenberg College

Two (More) Morsels from Euler

Abstract: Leonhard Euler (1707-1783) ranks among history's greatest mathematicians, and he is justly acclaimed for any number of famous results. This talk will focus on two of his lesser-known proofs - crumbs, as it were, from the great Eulerian feast.

The first is a 1781 argument in which Euler found four different whole numbers, the sum of any two of which is a perfect square. Remarkably, his answer was the fearsome foursome of: 18530 , 38114 , 45986 , 65570 .

The second is one of his methods for summing the infinite series of the reciprocals of the squares, the so-called "Basel Problem." He did this in various ways over his career, and I'll consider the 1755 argument in which he evaluated the series via three applications of l'Hospital's rule!

Both proofs, one from number theory and the other from analysis, will be given in complete mathematical detail. And, in both proofs, Euler's genius is on full display.