# 60th Anniversary Reunion-Conference

*1957 - 2017*

Some alumni and friends of the Ross Program joined us for:

**Reunion & Conference, in Columbus, Ohio**, June 16 - 18, 2017

**Alumni and friends are invited to contact us!**

- Did your Ross Program experiences influence your life or career?
- What have you been doing in recent years?

Email us at ross@math.osu.edu and we will post responses at News and Comments, (if you give permission). Statements from all types of alumni will help document our Program's influence, providing a useful step in our continual search for scholarship funding. (And it's fun to read them.)

You are also invited to contribute a Ross Program story. Our alumni page has additional information you might find interesting.

### History:

In 1957 at the University of Notre Dame, Professor Arnold Ross launched his innovative summer program dedicated to involving young people with mathematical explorations. In 1964 Dr. Ross moved to Ohio State, where the Ross Program is still going strong. Dr. Ross led the number theory course every summer until he stepped down in 2000 at the age of 94. He passed away in 2002.

Since 2000 Daniel Shapiro has been leading the Ross Program, assisted in recent years by Jim Fowler. You might enjoy reading the short or long histories of Arnold Ross and his program.

### Funding, and your contributions

This Sixtieth Anniversary is an appropriate moment to ask for your support. Your financial contribution will help ensure that the Ross Program not only exists, but continues to be excellent. **CLICK HERE** for information about ways you can contribute to the Ross Endowment Fund.

## Reunion-Conference Schedule

Several people lectured on topics of broad interest to Ross Program alumni and students. Here is a schedule of events.

Reunion Headquarters = MW 724.

Note: MW = Math Tower at OSU: 231 W 18th Ave, Columbus, OH.

We also displayed some Ross memorabilia, and sold some QR T-shirts, and Z-hats.

Reunion lectures were in room EA 160, at 209 W 18th Ave.

**SCHEDULE OF EVENTS**.

**Friday, June 16**:

Arrivals. Gather in MW 724 after lunch.

1:30: Daniel Shapiro, *Arnold Ross and his program*.

2:45: Tom Weston, *The Banach-Tarski paradox.*

4:00: Glenn Stevens, *Connecting the dots: The art of extrapolation.*

6:00: Ross Alumni Party at Bernie Baltz's home.

**Saturday, June 17**:

9:15: Brian Conrad, *The ABC conjecture*.

10:30: Lola Thompson, *(Mind the) gaps between primes.*

11:30: Group photo.

12:00: Lunch time.

2:00: David Pollack, *The arithmetic of elliptic curves.*

3:15: Keith Conrad, *Codes and congruences.*

4:30: David Saltman, *Breaking German codes during World War Two*.

Audience members saw an original German enigma machine.

6:00 - 9:00: Banquet at the *Ohio Union*.

**Sunday, June 18**:

9:00: Paul Pollack, *Summing divisors: a status report on the first 2000 years.*

10:15: Jeffrey Meyer, *Hyperbolic geometry, Pell’s equation, and beyond.*

11:30: William Dunham, *The infinitude of primes: Euclid, Euler, Erdös.*

12:45: Goodbyes and Departures.

## Abstracts for Ross Reunion 2017

Brian Conrad (Stanford):** The ABC conjecture**

The ABC Conjecture, formulated in the mid-1980’s by Oesterle and Masser, is one of the most important conjectures in number theory. It has many deep consequences, but its basic formulation can be given in entirely elementary terms. In September 2012, a 500-page solution was announced by Shinichi Mochizuki (building on several thousand pages of work he has carried out over the last 20 years).

I will explain what the conjecture asserts, illustrate why it is important, and discuss the surreal and unprecedented circumstances that have engulfed the process of dissemination of the ideas to the wider mathematical community.

Keith Conrad (UConn): **Codes and congruences.**

Buying something online, sending a text, making a phone call, and listening to a digital music file all involve the transfer of information over a channel that may have noise. If the signal that is received differs from the signal that was sent then errors in the received signal can be detected and often corrected if the signal was encoded in a suitable mathematical form. We will discuss some of the algorithms that play a role in this everyday application of number theory, involving congruences, continued fractions, and the analogy between integers and polynomials.

William Dunham (Bryn Mawr): *The infinitude of primes: Euclid, Euler, Erdös.*

The foundation of the Ross Program lies in number theory, and the foundation of number theory lies among the primes. It thus seems fitting to examine three different proofs of the infinitude of the prime numbers.

We first look at Euclid’s argument from 300 BCE, which appears as Proposition 20 of Book IX of the Elements. Although “Euclid’s proof of the infinitude of primes” is a standard in every number theory textbook, some people might be surprised to see his argument in its original form.

Next we consider Euler’s analytic proof from 1737. Like so much of his work, this features a blizzard of formulas, manipulated with a maximum of agility and a minimum of rigor. But the outcome is spectacular.

Finally we examine Erdös’s combinatorial proof from the 20th century. This is an elementary argument, but it reminds us once again that “elementary” does not mean “trivial.” Taken together, these proofs suggest that, to establish the infinitude of primes, it helps to have a two-syllable last name starting with “E.” More to the point, they show mathematics as a subject whose creative variety knows no bounds.

David Pollack (Wesleyan): *The arithmetic of elliptic curves.*

Paul Pollack (UGA): **Summing divisors: a status report on the first 2000 years.**

For each natural number n, let s(n) be the sum of the proper divisors of n. For example, s(10) = 1+2+5 = 8. Despite having such a simple definition, the function s is the centerpiece of many of the oldest unsolved problems in mathematics. I will survey these problems and discuss what we’ve learned in the past 2000 years.

Glenn Stevens (BU): *Connecting the dots: the art of extrapolation.*

We will discuss Jon Wallis's discovery (~ 1655) of his famous product formula for π/2 in the light of reflections on Arnold Ross and what we have learned from him.

Lola Thompson (Oberlin): ** (Mind the) gaps between primes.**

In this talk, we will discuss some of what is currently known about gaps between primes through the lens of a Ross Program student: we will look for patterns in numerical examples, formulate conjectures, and discuss how one might approach the task of turning our conjectures into theorems.

Tom Weston (UMass Amherst): *The Banach-Tarski paradox.*

Abstract: The Banach-Tarski paradox is often described as saying that a pea can be divided into a finite number of pieces which can be reassembled to form an exact copy of the sun. In this talk we will explain the proof of this result, which is based on some very simple mathematics (including a tiny bit of elementary number theory).