MASS Colloquia 2012

Thursday, September 6 Tanya Khovanova, Massachusetts Institute of Technology
2:30pm Manhole Covers and Convex Geometry
ABSTRACT Why are manhole covers round? Bring your answer to this famous interview question. We will use manhole covers as a starting point to discuss some modern research in convex geometry.
Thursday, September 13 Alexander Veselov, Loughborough University, UK
2:30pm Markov equation and the most irrational numbers
ABSTRACT You probably know that the golden ratio is "the most irrational number", although a precise meaning
of this is not so obvious. You might be able to guess "the second most irrational number", but you certainly will be very surprised to see the bronze medallist. It turns out that the hierarchy of the most irrational numbers is governed by a remarkable Diophantine equation called Markov equation. I will tell you about this story going back to XIX century with a surprising modern twist linking it with enumerative geometry and theory of Painleve equations.
Thursday, September 20 Mark Meckes, Case Western Reserve University
2:30pm 5-dimensional geometry is not like 2-dimensional geometry (and 3 and 4 are somewhere in between)
ABSTRACT There are many different ways to measure how "big" a geometric shape is, and their relationships are sometimes surprising. I will discuss three different problems (Shephard's problem, the Busemann-Petty 
problem, and one that I proposed several years ago) asking about monotonicity relationships among volumes. For example, Shephard's problem asks, if K and L are origin-symmetric convex bodies and every shadow cast by K has bigger area than the shadow cast by L in the same direction, is the volume of K bigger than that of L? In all three problems, the answer turns out to be yes in two dimensions, but 
surprisingly, no in five or more dimensions. In three and four dimensions the answer is different depending on the problem.
Thursday, September 27 Valentin Ovsienko, University of Lyon
2:30pm Donald Coxeter: tales of mystery and imagination
ABSTRACT Harold Scott MacDonald "Donald" Coxeter was one of the brightest stars of classical geometery and there is no mystery at all in his clearly written books and articles. Yet, his math is full of magic. Among other things, I will talk about a relatively little known notion of "friezes". Invented by Coxeter in 1970, it had to wait intill 21st century to show its real power.
Thursday, October 4 Donald Newhart, NSA
2:30pm A new approach to CRC processing
ABSTRACT High-speed digital communications must anticipate the possibility of channel errors in transmission. Nontrivial mathematical approaches to this date back to 1948, and constitute the subject of Algebraic Coding Theory. The principle of using the coefficients of polynomial remainders as a checksum to detect errors goes back to at least 1961; this idea blended well with the technology of shift registers, and is used in everything from the GPS system to the internet. Advances in modern technology such as Field Programmable Gate Arrays (FPGA), allows for very efficient vector addition of long binary inputs, and open new possibilities. 

This talk will explain CRC checksums from first principles, show how abstract algebra plays a pivotal role, and finally, present a new mathematical algorithm (developed at NSA) to process them. Although the approach will be explained with some simple linear algebra, it implicitly takes advantage of an underlying quotient-ring context that is usually ignored. Potential advantages for FPGA use will be discussed.

Thursday, October 11 Sheldon Newhouse, Michigan State University
2:30pm On the use of computers in Mathematics and Science: the good, the bad, and what we should do about them
ABSTRACT Computers have become a fundamental part of our lives. They are used in appliances ranging from cell phones to rocket launchers, in controlling our air travel, in describing models for financial management, and in large parts of Mathematics and Science. Can we trust what computers tell us? How can we improve the verification of various types of computation? These are questions of fundamental importance to all of us. In this lecture, we will explore some ideas and examples related to computation as it applies to Dynamical Systems--the branch of mathematics which deals with how various systems evolve in time. We will see that modern methods for verified computation can provide tests for finding certain computational errors as well as providing proofs of theorems which are, at the present time, unattainable with traditional methods.
Thursday, October 18 Anton Petrunin, Penn State
2:30pm Number of collisions of n balls on the infinite billiard table after Burago-Ferleger-Kononenko.
ABSTRACT Consider a system of n hard balls moving freely and colliding elastically in Euclidean space. We will give an upper bound for the number of collisions in such a system. The proof is a simple application of Alexandrov geometry. The talk is based on work of D. Burago, S Ferleger and A. Kononenko
Thursday, October 25 Peter Winkler, Dartmouth College
2:30pm Mathematical Puzzles that S-t-r-e-t-c-h Your Intuition
ABSTRACT Humans are not born with perfect mathematical intuition, to say the least, yet most decisions we make are based on "feel", not calculation. Today you will hear some mind-boggling puzzles (some with solutions, some without) that are designed to help you adjust your intuition when it's about to run off the rails.
Thursday, November 1 Alice Medvedev, UC Berkeley
2:30pm Some polynomial dynamics systems
ABSTRACT Consider a (discrete) dynamical system F(x, y, z) := ( f(x), g(y), h(z) ) for polynomials f, g, and h, acting on the three-dimensional space over complex numbers. What subsets S are invariant under F, in the sense that F(S) is a subset of S? In particular, what algebraic sets, that is solution sets of systems of polynomial equations, are invariant under F? I will describe the tools from modern model theory, a branch of mathematical logic, that reduce this question to understanding composition of one-variable polynomials, and an old theorem of Ritt that supplies this understanding.
Thursday, November 8 Peter Kuchment, Texas A&M University
2:30pm Seeing Invisible: Mathematics of Medical Imaging
ABSTRACT In this talk I will attempt to give the attendees the idea of what kind of mathematics is involved into contemporary medical imaging. Most of you have probably heard of CAT scan, MRI, and maybe some other CT (computed tomography) types. However, it is not that often that people know that tomographic images are not actual images, but rather results of intricate mathematical procedures. Advanced mathematical tools play a huge role. The mathematics of this subject is beautiful, hard, and diverse. Just to give you an idea, it deals in particular with differential equations, differential and integral geometry, group representations, harmonic analysis, numerical analysis, and surely, computer programming. This is what has kept the speaker and many others hooked up on tomography for years. 

Several brand new cheap, effective, and safe tomographic methods are being developed by engineers right now, which requires new mathematical techniques. Similar techniques are used for industrial non-destructive testing and geophysical imaging.

Thursday, November 15 Joel Langer, Case Western Reserve University
2:30pm A short look at the long history of the lemniscate of Bernoulli
ABSTRACT Could a beautiful plane curve launch a thousand propositions? 

To make the case for the lemniscate, this talk will romp through hundreds (ok thousands!) of years of history: From the spiric sections of Perseus, to the planetary orbits of Cassini, to mechanical linkages of Watt and others; not to mention the ``Enigma of Viviani's Temple" and the early investigations on the theory of elasticity by Bernoulli himself. But above all, the lemniscate is tied to the birth of elliptic functions and their connections to the theory of equations and numbers through the discoveries of Count Fagnano, Euler, Gauss and Abel. 

All of this predates the fuller view of the lemniscate as a Riemann surface of genus zero--i.e., a sphere--sitting in the complex projective plane. In this setting, the ever-elegant lemniscate turns out to have octahedral symmetry. In fact, with the help of pictures, this talk will attempt to provide some impression of the lemniscate as a disdyakis dodecahedron.