Earle Raymond Hedrick Lecture Series
What is Symplectic Geometry? An Introduction to Some Concepts and Open Questions
Lecture 1: Thursday, July 27, 10:30 a.m. - 11:20 a.m., International Ballroom North
Lecture 2: Friday, July 28, 9:30 a.m. - 10:20 a.m., International Ballroom North
Lecture 3: Saturday, July 29, 9:30 a.m. - 10:20 a.m., International Ballroom North
Dusa McDuff, Barnard College, Columbia University
Symplectic geometry has many faces. It takes place in even dimensions, and can be considered a version of complex algebraic geometry that is not constrained by the requirement that functions be polynomial, or more generally complex analytic. It also gives a framework in which to describe energy-conserving flows, and so has many applications to questions in dynamics. Recently it has turned out that symplectic structures (and their odd dimensional analog contact structures) can be used to help understand purely topological questions such as the possible ways that a circle can be twisted up in three-dimensional space, i.e. knot theory.
This set of lectures will first describe what a symplectic structure is, and then explain why such structures are interesting. Most of our examples will be very concrete and will concern objects in two, three and four dimensions. We will assume knowledge of multivariable calculus and basic linear algebra, but not too much else.
AMS-MAA Joint Invited Address
Computational Math Meets Geometry
Thursday, July 27, 9:30 a.m. - 10:20 a.m., International Ballroom North
Douglas Arnold, University of Minnesota
One of the joys of mathematical research occurs when seemingly distant branches of math come together. A beautiful example occurred over the last decade with the development of the field of compatible, or structure-preserving, discretizations of differential equations, in which ideas from topology and geometry have come to play a key role in numerical analysis. Very roughly, instead of applying standard all-purpose algorithms, such as Runge-Kutta methods and linear multistep methods for ODEs and finite difference or finite element methods for PDEs, far better results can be obtained for various classes of problems by constructing discretization methods which exactly preserve key geometric structures underlying the equations under consideration. Such structures include, for ordinary differential equations, symplecticity, symmetry, invariants and constraints, and, for partial differential equations, de Rham and other cohomologies and associated Hodge theory. We will tour this burgeoning field, demonstrating some of the advances in numerical methods made possible by the new geometrical and topological approaches, and even present a case where the numerical point of view has enabled the resolution of a long open question in algebraic topology.
MAA Invited Address
Is There a Better Way to Elect a President?
Friday, July 28, 10:30 a.m. - 11:20 a.m., International Ballroom North
Steven Brams, New York University
I describe properties of approval voting—whereby voters can approve of as many candidates as they like in a multi-candidate election, and the candidate with the most approval wins—and compare them with properties of (1) plurality voting, in which voters can vote for only one candidate; (2) ranking systems, such as the Borda count and the Hare system of single transferable vote (also called instant runoff or ranked choice voting); and (3) grading systems that have been proposed by mathematicians Warren Smith (range or score voting) and Michel Balinski and Rida Laraki (majority judgment voting).
I argue that approval voting, which is used by both the MAA and AMS, among other professional societies, is a simpler and more practicable alternative and should be used in presidential and other public elections. Extending approval voting to multi-winner elections, such as to a committee or council, will also be discussed.
MAA Invited Address
An Introduction to Spatial Graph Theory
Thursday, July 27, 8:30 a.m. - 9:20 a.m., International Ballroom North
Erica Flapan, Pomona College
Spatial graph theory developed in the early 1980's when topologists began using the tools of knot theory to study graphs embedded in \(3\)-dimensional space. Later, this area came to be known as spatial graph theory to distinguish it from the study of abstract graphs. Much of the current work in spatial graph theory can trace its roots back either to the ground breaking results of John Conway and Cameron Gordon on intrinsic knotting and linking of graphs or to the topology of non-rigid molecules. This talk will present the history of spatial graph theory and survey some of the current trends in the field.
MAA Invited Address
How to Create Periodic Functions from Geometric Shapes
Saturday, July 29, 10:30 a.m. - 11:20 a.m., International Ballroom North
Ronald Mickens, Clark Atlanta University
The trigonometric sine and cosine functions are generated from the geometrical properties of the unit circle. We demonstrate that other periodic functions can be constructed by generalizing the methodology used to analyze the properties of the circular, i.e., trigonometric, functions. In particular, we investigate the elliptic, “square,” and “triangular” periodic functions, and derive a number of their critical mathematical features using only elementary trigonometry. At a somewhat more advanced level, we introduce the functional equation, \(f(t)^2 + g(t)^2 = 1\), and show it has an unbounded set of periodic functions as solutions. An algorithm is given to explicitly calculate those periodic solutions possessing a second derivative. Finally, the following interesting and important result is obtained: the considered periodic functions always occur as a triplet of functions, rather than a pair.
MAA James R.C. Leitzel Lecture
Math's Other Half
Saturday, July 29, 8:30 a.m. - 9:20 a.m., International Ballroom North
Dan Meyer, Desmos
Whatever your job title, you are also an ambassador from the world of those who love math to the world of those who fear math. Your ambassadorship will either produce more people who love math or more people who fear math. Your effect will be non-zero. But the math that people fear is often just one half of math. Let's discuss methods for helping fearful people encounter math's other half.
AWM-MAA Etta Z. Falconer Lecture
Not So Hidden Figures: Unveiling Mathematical Talent
Friday, July 28, 8:30 a.m. - 9:20 a.m., International Ballroom North
Talithia Williams, Harvey Mudd College
In the past few months, the movie “Hidden Figures” has brought visibility to the lives of African American women who served as NASA “human computers” in the 1960s. During that same time, Dr. Etta Falconer, the 11th African American woman to receive a Ph.D. in mathematics, began her tenure at Spelman College, motivating young women of color to be and do more than they dreamed possible in a field where their presence was lacking. I was fortunate to take her classes, engage her mathematical mind and dream of following in her footsteps. At Harvey Mudd College, I now find myself replicating those “Falconer moments” with my own students. I'll share several of these strategies that you can use in and out of the classroom to encourage all students, particularly underrepresented students, to develop their mathematical talent and pursue mathematical sciences.
MAA Chan Stanek Lecture for Students
Four Tales of Impossibility
Thursday, July 27, 1:00 p.m. - 1:50 p.m., International Ballroom South
David Richeson, Dickinson College
"Nothing is impossible!" It is comforting to believe this greeting card sentiment; it is the American dream. Yet there are impossible things, and it is possible to prove that they are so. In this talk we will look at some of the most famous impossibility theorems—the so-called "problems of antiquity." The ancient Greek geometers and future generations of mathematicians tried and failed to square circles, trisect angles, double cubes, and construct regular polygons using only a compass and straightedge. It took two thousand years to prove conclusively that all four of these are mathematically impossible.
Pi Mu Epsilon J. Sutherland Frame Lecture
Wednesday, July 26, 8:00 p.m. - 8:50 p.m., International Ballroom North
Ingrid Daubechies, Duke University
NAM David Harold Blackwell Lecture
Hidden Figures: My Role as a Math Consultant for this Film
Friday, July 28, 1:00 p.m. - 1:50 p.m., International Ballroom South
Rudy L. Horne, Morehouse College
In January 2017, the movie Hidden Figures was released by 20th Century Fox studios. This movie tells the story of three African-American women mathematicians and engineers (Katherine Johnson, Mary Jackson and Dorothy Vaughan) who would play a pivotal role towards the successful mission of John Glenn’s spacecraft orbit around the Earth and the NASA missions to the moon.
For this talk, we give a brief review of the space race going on at the time between the United States of America and the former Soviet Union. We will discuss the lives and contributions that NASA mathematician Katherine Johnson and the NASA engineers Mary Jackson and Dorothy Vaughan made to the space race, particularly their work as it concerns John Glenn’s orbit around the Earth in 1962 and to the moon missions. Also, we will talk about the experiences of being a mathematical consultant for this film.