Invited Addresses
EARLE RAYMOND HEDRICK LECTURE SERIES
WHAT IS ARITHMETIC COMBINATORICS?
W.T. Gowers, Cambridge University
Some mathematical problems and results belong to areas of mathematics with well-established names such as functional analysis or algebraic geometry. Others are harder to categorize. In recent years increasing numbers of mathematicians have found themselves using techniques from number theory, analysis, and combinatorics, which leaves them unable to answer what for most mathematicians is a simple question: “What area of mathematics do you work in?” However, this social disadvantage may soon be a thing of the past because some fascinating common themes are emerging, enough for the area to deserve its own name. And there is even a name that people seem to be happy with: arithmetic combinatorics. In these talks, I shall introduce some of the main results and open problems of arithmetic combinatorics, give some ideas of how they are proved, and explain some of the surprising connections between them.
LECTURE 1: THE BACKGROUND: SOME THEOREMS AND OPEN PROBLEMS IN NUMBER THEORY, ANALYSIS, AND COMBINATORICS
Thursday, August 10, 9:30 am - 10:20 am
This lecture will set the scene for additive combinatorics by discussing several famous theorems and problems that predate it. Amongst them are van der Waerden’s theorem on arithmetic progressions, Goldbach’s conjecture, the Kakeya problem, some deceptively simple problems of Paul Erdos and a recent breakthrough of Ben Green and Terrance Tao. They have in common that they are easy to state, but hard to solve. Deeper similarities between them will be explored in subsequent lectures.
LECTURE 2: DISCRETE FOURIER ANALYSIS: ITS POWER AND ITS LIMITATIONS
Friday, August 11, 9:30 am – 10:20 am
Fourier analysis is concerned with the splitting up of a signal, such as a sound wave, into simple parts, such as pure sine waves. Its first cousin, discrete Fourier analysis, is an extremely useful tool for thinking about a certain kind of problem in number theory. I shall explain why this might be and then give examples of how Fourier analysis can be used to solve, or at least begin to attack, some of the problems mentioned in the first lecture. Oddly enough, the failure of Fourier analysis to solve some of those problems is almost more interesting than its successes, because it raises the possibility that arithmetic combinatorics can repay some of the debt it owes to analysis. The difficulties one encounters when one tries to improve the best known results about arithmetic progressions are genuine ones, and it looks likely that, in order for them to be overcome, it will be necessary to develop a new, more powerful version of Fourier analysis, which would almost certainly have applications beyond arithmetic combinatorics.
LECTURE 3: FREIMAN’S THEOREM AND ARITHMETIC PROGRESSIONS OF LENGTH 4
Saturday, August 12, 9:30 am – 10:20 am
Some of the most interesting problems in arithmetic combinatorics concern arithmetic progressions, that is sequences like 5,11,17,23,29, where each number is obtained from the previous one by adding some fixed amount (in this case 6). Fourier analysis is very useful for analyzing progressions of length 3, but runs into difficulties for longer progressions. However, ways have been found for getting around some of these difficulties, and these are leading to a new and more powerful theory of “quadratic” Fourier analysis. I shall explain in very broad terms what this means and why it is still by no means fully understood. A central role in this development is played by a fascinating result known as Freiman’s theorem. Once again, the theorem is much easier to state than it is to prove, but it is possible to describe some of the ideas from a beautiful proof due to Imre Ruzsa. There are some questions that Ruzsa’s techniques are not strong enough to answer, and a central problem in arithmetic combinatorics is to strengthen them. This could have profound consequences: the solution of several problems, the development of a new form of “linear” Fourier analysis, and a significant increase in the cohesion and maturity of arithmetic combinatorics.
MAA INVITED ADDRESS
Dorothy Buck, Imperial College London
THE CIRCLE (AND KNOT AND LINK) OF LIFE: HOW TOPOLOGY UNTANGLES KNOTTY DNA QUESTIONS
Thursday, August 10, 8:30 am – 9:20 am
DNA is often referred to as ‘the staff of life’, as it is the blueprint for all hereditary traits and diseases, as well as the template for all proteins. But the structure of DNA structure is often more complicated than a straight ‘staff’. DNA molecules can have a circular (e.g. bacterial), or topologically constrained (e.g. human), central axis. The axis can even be knotted or linked! We’ll discuss how the topology of this axis affects important biological processes – both local (e.g. which proteins attach to DNA) and global (e.g. how a cell divides). We’ll conclude with some examples of how topology (knot theory) has helped our understanding of these processes.
MAA INVITED ADDRESS
SOME OPEN QUESTIONS ABOUT CONVEX POLYHEDRA:
Jesus A. De Loera, University of California Davis
Thursday, August 10, 10:30 am – 11:20 am
Convex polyhedra are familiar objects since our childhood. Indeed, cubes, pyramids, and triangles are common staples in all kindergartens! Unknown to most people polyhedra, in their high-dimensional version, are also widely used in applied mathematics (e.g. operations research, finances, computer networks, and more). Their beauty and simplicity appeal to all, but very few people know of the many easy-to-state difficult unsolved mathematical problems that hide behind their beauty. The purpose of this lecture is to introduce an audience without prior background to some of these open questions.
MAA STUDENT LECTURE
Promoting Students' Appreciation for Math through Applications to Very Cool Activities
Richard Tapia, Rice University
Thursday, August 10, 1:00 pm – 1:50 pm
For many years the speaker was involved in BMX bicycle racing as a supportive father for his son Richard and also for many years he has been involved in car show activity. In the first part of this talk the speaker uses several lively videos to identify and illustrate what he calls the Curse of Lane 8 or The Fair Lane Assignment Problem in BMX bicycle racing.
He then uses his mathematical training to formulate the problem as a mathematical problem and with the aid of a Rice student he solves this problem using a computer and mathematics. This solution technique will be described. In the second part of the talk the speaker will show and describe the making of an exciting video that was made with the assistance of a Rice undergraduate art-math major to accompany the showing of his 1970 Chevelle Malibu SS at various car shows across the country. Both the car and video are entitled "Heavy Metal". The video attempts to depict the time period of the late1960's and early 1970's in terms of muscle cars, heavy metal music, and unrestlessness and rebellion. The psychedelic video images are constructed entirely using mathematics. Numerical simulations of fluid flow in and around the car are obtained using numerical methods to solve the Navier-Stokes partial differential equations that govern fluid flow. By being creative with the mathematical parameters and solution techniques some very interesting images and patterns are obtained. In this way the video demonstrates that mathematics can take us places where physics can't. The video sound track consists of Heavy Metal music and adds to the excitement of the video.
NAM DAVID BLACKWELL LECTURE
PUBLIC HEALTH AND MATHEMATICS: SOME EMERGING CHALLENGES AND PARADIGMS AT THE INTERFACE
Dominic P. Clemence, North Carolina A&T State University
Friday, August 11, 8:30 am – 9:20 am
Public health issues concern us all: just a few include emerging and re-emerging diseases, the shrinking global neighborhood, health disparities, deliberately released infectious agents, and natural disasters. While in the past mathematics has played a significant role in addressing some public health concerns, one cannot but wonder, 'can more be done?' and in particular, 'in what ways can mathematicians contribute more?' when one looks at the status of public health world-wide. We share a mathematician's perspective on a few of these issues, and highlight some paradigms and challenges emerging at the public health-mathematics interface.
JAMES R. LEITZEL LECTURE
TEACHING RESEARCH: ENCOURAGING DISCOVERIES
Francis Edward Su, Harvey Mudd College
Friday, August 11, 10:30 am – 11:20 am
What does it take to turn a learner into a discoverer? Or to turn a teacher into a coadventurer? I will describe a handful of experiences, from teaching a middle-school math class to doing research with undergraduates, that have changed the way that I would answer these questions. Some of the lessons I've learned have surprised me.
PME J. SUTHERLAND FRAME LECTURE
ELLIPSES AND CIRCLES? TO UNDERSTAND VOTING PROBLEMS??!
Donald Saari, UC Irvine
Friday, August 11, 8:00 pm – 8:50 pm
Why is it that whenever we put forth a carefully considered proposal, somebody can put forth an "improvement?" Sure. Yet, attend any meeting, even the MAA business meetings, and it happens on a regular basis. Why? Insight is possible by using just the geometry of circles. And then, to introduce a new game theoretic solution concept, I will use the geometry of ellipses.
AWM-MAA ETTA Z. FALCONER LECTURE
CANCER MODELING: FROM THE CLASSICAL TO THE CONTEMPORARY
Trachette Jackson, University of Michigan
Saturday, August 12, 8:30 am – 9:20 am
Cancer is one of the leading causes of death in the world today, and an abundance of research is being conducted in order to better understand tumor development, to evolve existing cancer therapies, and to discover new approaches to combat the disease at the cellular and molecular levels. Mathematical modeling, aided by computational tools and combined with the experimental data, have the potential to facilitate a deeper and broader understanding of the cellular and molecular interactions associated with tumor initiation, progression, and treatment, and can guide experimental design and interpretation. Many of the challenges cancer researchers are facing lie at the intersection of the mathematical and biomedical sciences and in this talk I will review the progress that has been made in modeling the various aspects of avascular and vascular tumor growth.
MAA INVITED ADDRESS
STORIES FROM THE HISTORY OF MATHEMATICS
David Bressoud, Macalester College
Saturday, August 12, 10:30 am – 11:20 am
This is a collection of some of my favorite stories from the history of mathematics, stories that I use in my classes to illustrate what it really means to "do" mathematics.
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