Talks

Abstract: Elementary number theory – the study of the positive integers and the primes – seems to many students to be a world in which there is no ambiguity. After all, the number 7653 is either a prime or it is not; no chance is involved. Nevertheless, probabilistic and statistical thinking have permeated many parts of the study of number theory. This talk will examine several of these, with a focus on projects with which the speakers has worked on with his students.

(This is a good talk for math majors who have completed a calculus sequence, but is not very accessible to a general audience)

The Life, Legacy, and Lost Library books of Leonhard Euler

Abstract This talk will examine the life of Euler, and will discuss some of his major accomplishments, in fields ranging from number theory to geometry. We will also tell stories of the speaker’s role in creating the online “Euler Archive”, and of the fascinating old books by Euler he discovered in several libraries.

(This talk is accessible to anyone)

Zeta Functions and the Mathematics of Juggling

Abstract: We will give an introduction to the classical Riemann Zeta function, and also discuss some zeta functions which have been developed in other areas of mathematics. After this, we shall give a primer on the mathematics of juggling, with a special focus on “siteswap” notation. Finally, we shall combine these to introduce the "Juggling Zeta Function", and note a few of its properties.

(This talk is aimed at junior or senior math majors)

Interdisciplinary Talks

I’m sometimes asked to give an interdisciplinary talk, which can be jointly sponsored by mathematics and another department. This is a really fun chance to explore connections which we miss if we only stay within our own departments.

Math and Astronomy

Rethinking the universe: the gravity crisis of the 1740’s.

Abstract: In November of 1747, a young man stood before the prestigious Paris Academy of Science. The man, Alexis Clairaut, was well known to the academy, and was considered by many of its members to be the greatest physicist in the world. On this day, however, he seemed unusually nervous. Taking a deep breath, he announced to those assembled that Isaac Newton’s inverse-square law of gravity, the most certain of all physical laws, was wrong.

Clairaut’s desperate move came at the end of a decade in which all of the leading scientists of the day found reason to question Newton’s theories. This talk will tell the story of the fascinating people involved, the reasons they came to doubt, and their many attempts at solutions. In the process, we will consider how we know anything in science, and the possibility that we are still mistaken about some fundamental ideas.

(This talk is accessible to a general audience; about 10% of the talk requires some knowledge of calculus, but those parts can be ignored without losing the story.)

Math and Linguistics

Euler and Phonetics: The Untold Story of the Mathematics of Language

Abstract: It is well known that Euler made seminal contributions to a wide range of fields. Recent scholarship demonstrates that he contributed to as least one other which has not been described in the literature. In this paper we will describe two fascinating contributions to the field of articulatory phonetics. First, it was Euler who convinced the St. Petersburg Academy to make the nature of the vowels an annual prize question, leading directly to one of the most influential works in the history of phonetics, the Tentamen of Kratzenstein. Second, Euler himself wrote a short work, the Meditatio de formatione vocum, which, as an article on articulatory phonetics, strikingly presages 20th-century work in vowel classification. Euler’s rather mathematical treatment of vowels as existing in two-dimensional “vocal space” was two centuries ahead of its time. This talk will survey Euler’s work in this area.

(This talk is more historical and linguistically-based than it is mathematical, but can make a nice interdisciplinary seminar, if only because mathematicians can enjoy seeing Euler apply mathematical-style thinking to another field. The work has since been published in Historiographia Linguistica).

Math and Biology

Darwin, Malthus, Süssmilch, and Euler: The Ultimate Origin of the Motivation for the Theory of Natural Selection

Abstract: It is fairly well known that Darwin was inspired to formulate his theory of natural selection by reading Thomas Malthus’s Essay on the Principle of Population. In fact, by reading Darwin’s notebooks, we can even locate one particular sentence which started Darwin thinking about population and selection. What has not been done before is to explain exactly where this sentence – essentially Malthus’s ideas about geometric population growth – came from. In this essay we show that eighteenth century mathematician Leonhard Euler is responsible for this sentence, and in fact forms the beginning of the logical chain which leads to the creation of the theory of natural selection. We shall examine the fascinating path taken by a mathematical calculation, the many different lenses through which it was viewed, and the path through which it eventually influenced Darwin.

(This talk has some math, but is accessible to a scientifically-literate audience. It is based on work which has been published in the Journal of the History of Biology)

Math and Religion

Euler’s Rettung: An anonymous work on the limits of mathematics, science, and faith

Abstract: In 1747, Euler wrote (and anonymously published) his Rettung der gottlichen Offenbahrung (E92), defending the validity of divine revelation as a valid source of knowledge, while considering perceived inconsistencies in mathematics and science. Drawing on a new translation of the Rettung, we shall attempt to elucidate some of Euler’s religious views, and draw connections between this work and the work he was doing at the time in astronomy and mathematics. We claim that a close familiarity with Euler’s mathematical works of the 1740’s is necessary to understand this important document, and we will demonstrate some of the mathematical theorems and problems besetting Euler at this time.

Workshops

I am also available to give workshops on the incorporation of primary sources into the curriculum. The description below comes from a workshop I gave in Puebla, Mexico last November.

Primary Sources in Every Classroom: an interactive introduction

This workshop will introduce participants to curricular modules, known as Primary Source Projects, based entirely on primary historical source material, developed by a team of mathematicians at nine universities in the United States. The team is beginning a five-grant funded by the National Science Foundation through which modules will be developed, implemented, tested and published. The expansion will further support classroom testing of new modules by faculty at forty other institutions, and provide training in their use to graduate students and faculty.

Designed to capture the spark of discovery and motivate subsequent lines of inquiry, each module is built around primary source material close to or representing the discovery of a key concept. Through guided reading and activities, students explore the mathematics of the original discovery and develop their own understanding of the subject. To place the source in context, a module also provides biographical information about its author, and historical background about the problems with which the author was concerned. For example, motivated by the problems of computing odds in a game of chance and of finding the summation of powers (with the eventual goal of computing the area under certain curves), Pascal arranged figurate numbers into columns of a table, today called Pascal’s Triangle. Having noticed certain patterns in the table which he wished to justify, he formulated verbally what has become mathematical induction. After reading Pascal’s original writings in his 1653 Treatise on the Arithmetical Triangle, students are asked to explore the validity of his claims with concrete numerical values, and then grapple with the logic behind induction techniques.

During this workshop, the presenter will share some of the research-based findings concerning the benefits of using primary sources in the classroom, and will outline the research structure planned by the grant team. After introducing a Primary Source Project and discussing its classroom implementation, participants will have the opportunity to work in groups on some of these Projects, developed under an earlier grant, which focused exclusively on Discrete Mathematics and Computer Science. Workshop activities will be facilitated by project team members from the lead institution, who developed and tested modules under the pilot grant. After a brief overview of how the materials can be used in undergraduate courses, the workshop format will allow small groups of participants to engage in detailed study and discussion of two different historical modules of their choosing, and will end with a summary workshop discussion.