## Earle Raymond Hedrick Lecture Series

**Hendrik Lenstra,** *Universiteit Leiden*

*The trick is not to waive the rules,*

but stick to groups, rings, fields as tools.

(Algebraist's motto)

The main message of the 2016 Hedrick Lectures is that “abstract algebra” is a misnomer: for those with eyes to see - groups, rings, and fields are everywhere. The script that *concrete algebra* is written in may not be readily decipherable, but mastering it is a sure step on the road to insight and wisdom. The three lectures sketch several of the sights to be seen.

### Hedrick Lecture 1. The Group Law on Elliptic Curves

*Thursday, August 4, 10:30 a.m. - 11:20 a.m., Regency Ballroom*

Click here to view the slides from this lecture

The theory of elliptic curves is a showpiece of modern mathematics. Its implications are felt from the brightest parts of number theory, where it supplied the key to Fermat's Last Theorem, to the darkest corners of cyberspace, where it provides the workhorses of secret communication. How much of the theory can with lucidity and rigor be developed in an undergraduate algebra course? The lecture outlines an approach to at least establishing the group law, using no other tools than what such a course ordinarily already covers.

### Hedrick Lecture 2. The Combinatorial Nullstellensatz

*Friday, August 5, 9:30 a.m. - 10:20 a.m., Regency Ballroom*

Click here to view the slides from this lecture

Noga Alon's combinatorial Nullstellensatz (1999) is a quantitative sharpening of the familiar fact that a non-zero polynomial in several variables over an infinite field defines a function that does not vanish everywhere. It has an impressive number of consequences of a combinatorial nature, and forms an excellent example of what algebra can do for the non-algebraist. As the lecture shows, the combinatorial Nullstellensatz also belongs to the algebraist's own toolbox. A characteristic application is an elegant theorem about matrices that is just one step away from one of the loveliest theorems of Galois theory: the existence of a *normal basis* in any finite Galois extension.

### Hedrick Lecture 3. Profinite Number Theory

*Saturday, August 6, 9:30 a.m. - 10:20 a.m., Regency Ballroom*

Click here to view the slides from this lecture

What is a *number*? Surprisingly, one of the fundamental notions of mathematics is never given a rigorous definition. Usage determines the meaning of the word. When one refers to the typical element of some ring as a “number”, one expresses a certain familiarity with that ring. The algebraist who defines a new ring creates a new species of number, and may feel master of the universe. *Profinite numbers* are elements of the ring of profinite integers, an important instrument in both infinite Galois theory and arithmetic geometry. In the lecture, the world of profinite numbers with its wealth of wonders is explored for its own sake.

*Illustration by Willem Jan Palenstijn*

## AMS-MAA Joint Invited Address

### Understanding Symplectic Geometry and Topology through Polytopes and Lattice Points (NEW)

*Thursday, August 4, 9:30 a.m. - 10:20 a.m., Regency Ballroom*

**Tara Holm****, ***Cornell University*

Symplectic geometry, the natural geometry of classical mechanics, is floppier than Riemannian geometry but more rigid than topology. I will describe how the geometry and topology of group actions on symplectic manifolds relate to properties of polytopes, motivated by many pictures and examples. I will conclude with how some of my recent work, joint with Daniel Cristofaro-Gardiner, Alessia Mandini and Ana Rita Pires, comes to feature continued fractions, counting lattice points, and the Philadelphia subway system.

**Please Note: This talk is replacing Ravi Vakil's address listed in the program. Dr. Vakil is unable to speak at MAA MathFest 2016.**

## MAA Invited Address

### Mathematical Sense and Nonsense outside the Classroom: How Well Are We Preparing Our Students to Tell the Difference?

*Thursday, August 4, 8:30 a.m. - 9:20 a.m., Regency Ballroom*

**Robert Megginson, ***University of Michigan*

"Mathematics will always be a key element of liberal education, since it promotes logical reasoning." You have likely heard this claim, or perhaps made it yourself. And we generally do a decent job of teaching our mathematics and statistics students how to avoid certain types of errors in their own deductive and inductive reasoning. But it is not so clear that we have done as good a job of preparing our students to examine critically the reasoning, mathematical and otherwise, of others who are trying to convince us to buy their product or adopt their position on an issue. In this presentation, which expands upon an invited 20 minute talk given at JMM 2013 in San Diego, the speaker will propose a three-category classification of types of fallacious mathematical arguments that have been used to try to convince the public of the wisdom of a policy decision or the safety of a new product, in the hope of starting a conversation about where and how in our school and college mathematics curricula we could better prepare students to be suspicious when presented with arguments in each category, and help them think about the questions they should ask when their suspicions are aroused. Examples will be given, several with roots in applications of mathematics to climate science, one of the speaker’s interests. For those involved in teacher preparation or who are K-12 teachers themselves, some connections to the Common Core standards will be given.

## MAA Invited Address

### Magical Mathematics

*Friday, August 5, 10:30 a.m. - 11:20 a.m., Regency Ballroom*

**Arthur Benjamin,** *Harvey Mudd College*

We will explore mathematical magic tricks using cards, calculations, and combinatorics! Sometimes the underlying mathematical secret is just as interesting as the magic trick itself.

## MAA Invited Address

### Immersion in Mathematics via Digital Art

*Saturday, August 6, 10:30 a.m. - 11:20 a.m., Regency Ballroom*

**Judy Holdener,** *Kenyon College*

The relationship between mathematics and art has a long and rich history. Artists past and present have used mathematics in significant ways to carry out their artistic vision, and mathematicians have used formulas, algorithms and computers to produce art. In my own case, I find art to be a good medium for conveying the nature of mathematics to a wide audience. In this lecture I examine my recent venture into digital art with the creation of a mathematical artwork I title “Immersion”. The surface patterns in the piece reflect my own day-to-day immersion in mathematics, depicting patterns that relate to the content of courses I teach as well as research I have conducted with undergraduates in the area of dynamical systems. I will describe how patterns in the piece reflect the connection between two well-known mathematical objects: the Thue-Morse sequence and the von Koch curve. Additionally, I will describe how the formal mathematical meaning of “immersion” plays a role in the composition of my piece. In particular, my work highlights “Boy’s Surface”, which is an immersion of the real projective plane into three-dimensional Euclidean space.

## MAA James R.C. Leitzel Lecture

### Inquiry, Encouragement, Home Cooking (And Other Boundary Value Problems)

*Saturday, August 6, 8:30 a.m. - 9:20 a.m., Regency Ballroom*

**Annalisa Crannell, ***Franklin & Marshall College*

When you teach an abstract algebra class in a bagel shop, there's this problem: you have no chalkboards. Likewise, no chalk. Worse yet, the acoustics are lousy for lecturing, especially if you are trying to keep your voice down so you don't annoy the non-algebra bagel customers.

But crossing the threshold into new physical spaces can lead to crossing metaphorical boundaries into strange and wonderful new pedagogies. In this talk we'll explore a personal approach to stumbling into inquiry-based learning. Along the way, we'll meander into developing pragmatic strategies that help us in cheerleading for our students, and of course, we'll celebrate food.

## AWM-MAA Etta Z. Falconer Lecture

### Harmonic Analysis and Additive Combinatorics on Fractals

*Friday, August 5, 8:30 a.m. - 9:20 a.m., Regency Ballroom*

**Izabella Laba, ***University of British Columbia*

A plane is flat; a sphere is curved. Both are smooth, well behaved surfaces on which one can define measure and integration. If a harmonic analyst only knows the behaviour of analytic objects associated with a given surface, for example singular or oscillatory integrals, can she tell whether the surface is curved or flat? It turns out that, yes, the geometry of the surface is indeed reflected in such analytic estimates.

It might be somewhat surprising that similar phenomena have also been observed for fractals, including Cantor-type sets on the line. Some fractals behave, from the analytic point of view, as if they were flat; others display features typical of the sphere, and we have also seen additional types of behaviours that are never observed for smooth surfaces. The recent work investigating such phenomena highlights the connection to arithmetic properties of fractals, expressed in terms of "randomness" and "structure."

## MAA Chan Stanek Lecture for Students

### Zombies & Calculus: A Survival Guide

*Thursday, August 4, 1:00 p.m. - 1:50 p.m., Regency Ballroom*

**Colin Adams**, *Williams College*

If you are reading this, then you have managed to survive the zombie apocalypse so far. Congratulations! But as the world sinks further into ruin, what additional strategies can you apply to endure the onslaught? Learn how calculus can help you to defeat the zombie hordes. The lecture room will be certified a safe haven for the duration of the talk.

## Pi Mu Epsilon J. Sutherland Frame Lecture

### Combinatorics - The Mathematics That Counts

*Friday, August 5, 8:00 p.m. - 8:50 p.m., Regency Ballroom*

**Robin Wilson****,** *Open University*

How many Sudoku puzzles are there? Are there 33 Londoners with the same number of hairs on their head? Can a knight visit all the squares of a chessboard just once? And can we tile a floor with squares and regular hexagons? These are all problems in combinatorics, the branch of mathematics concerned with selecting, arranging, counting and listing things. In this talk I shall illustrate the nature and uses of combinatorics by means of a number of entertaining problems.

## NAM David Harold Blackwell Lecture

### Urban Analytics: The Case for Smart Parking

*Friday, August 5, 1:00 p.m. - 1:50 p.m., Regency Ballroom*

**Robert Hampshire****,** *University of Michigan*

Parking management has been a vexing problem for cities since the invention of the automobile. One concern is excess travel,congestion, air pollution and greenhouse gas (GHG) emissions that are caused by drivers searching for available parking – an activity colloquially known as cruising.

A recent study by UCLA economist and urban planner, Donald Shoup, found that in a 15-block area of Westwood, cruising for parking generates 950,000 excess vehicle-miles of travel, wastes 47,000 gallons of gas, 100,000 hours and produces 730 tons of greenhouse gas carbon dioxide per year.

I present the results of an investigation of this problem using queueing theory, stochastic processes, statistics including the Rao-Blackwell theorem, optimization and machine learning. The analysis is enabled by a large dataset of sensor data, cell phone data, parking payment data, and connected vehicle data.