### MASS Colloquium    Math 497E    Fall 2013

#### Thursdays 1:25-2:25  in 114 McAllister

Date Speaker Title Abstract Notes
9/5 Mark Levi
Penn State
web page
Mathematics through physics Physics often provides mathematics not only with a problem, but sometimes also with the idea of a solution. Some calculus problems can be solved by a physical argument more quickly and easily than by the "standard" approach used in college courses. This simplification can be quite striking in some cases. Quite a few theorems which may seem somewhat mysterious become completely obvious when interpreted physically (the trick is to find a suitable interpretation). This is the case for some "elementary" theorems (the Pythagorean Theorem, Pappus' theorems, some trig identities (e.g., cos(x+y)=..., Euler's famous formula V-E+F=2, and more) and for some less elementary ones (no familiarity with any of these is assumed): Green's theorem, the Riemann Mapping Theorem, the Gauss-Bonnet theorem, Noether's theorem on conserved quantities, Poincare integral invariance, and more. I will describe a miscellaneous sampling of problems according to the audience's preferences. Notes by
Max Goering
and
Wilson Jarrel
pdf
9/12 Joel Hass
UC Davis
web page
Some applications of geometry to problems in biology Many problems arising in biology are geometric, since many biological properties are reflected in the shapes of biological objects. We will explain recent developments in the use of conformal mappings and hyperbolic geometry to study several biological problems.
1. The study of cortical surfaces. In particular, the question of how similar are two brains.
2. The classification of proteins.
3. Deducing the evolutionary tree of old-world monkeys from fossilized skulls.
Notes by
Aaron Calderon
and
Leah Frederick
pdf
9/19 Dmitri Burago
Penn State
Counting collisions in hard ball gas models and singular geometry of non-positive curvature We will discuss a solution to a long-standing problem. The problem itself can be explained to school kids. Basically, one asks if there exists an N that not more than N elastic collisions can occur between 10 identical balls before they fly away from each other (no gravity, no external forces). Of course, there is nothing special about 10, it could be any M and then N=N(M). The solution is also elementary modulo a few well-known facts. The problem was that the tools and facts lie in an area of math which was thought to be very far from the original problem. We will not get into any technicalities, so the talk should be easily accessible. Notes by
Andrew Hanlon
and
pdf
9/25
4-5 p.m.
Robert Lang
Alamo, California
web page
From flapping birds to space telescopes: the mathematics of origami The last decade of this past century has been witness to a revolution in the development and application of mathematical techniques to origami, the centuries-old Japanese art of paper-folding. The techniques used in mathematical origami design range from the abstruse to the highly approachable. In this talk, I will describe how geometric concepts led to the solution of a broad class of origami folding problems -- specifically, the problem of efficiently folding a shape with an arbitrary number and arrangement of flaps, and along the way, enabled origami designs of mind-blowing complexity and realism, some of which you'll see, too. As often happens in mathematics, theory originally developed for its own sake has led to some surprising practical applications. The algorithms and theorems of origami design have shed light on long-standing mathematical questions and have solved practical engineering problems. I will discuss examples of how origami has enabled safer airbags, Brobdingnagian space telescopes, and more. Notes by
Samantha Fairchild
and
pdf
10/3 Diane Henderson
Penn State
web page
The William Pritchard
Fluid Mechanics Laboratory
Penn State is one of the few Departments of Mathematics that houses a physical laboratory. In this talk I will describe some of the activities going on in the lab. Then we will walk down to the basement for a tour.
10/10 Roger Howe
Yale University
web page
About the numbers 12 and 24 The numbers 12 and 24 come up often in contexts that involve symmetry. For example, a cube has 12 edges, and a dodecahedron has 12 faces. This talk will discuss the extent to which various appearances of 12 and 24 should be considered "the same". We will argue that some appearances should definitely be considered the same, and speculate about others. Notes by
Liuquan Wang
and
Zheyi Xu
pdf
10/17
noon-
1 pm
room 113
University of Maryland
web page
Kirkwood gaps and instability for three body problems It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. We study the dynamics of the Newtonian Sun-Jupiter-Asteroid problem near such resonances. We construct a variety of diffusing orbits which show a drastic change of the osculating eccentricity of the asteroid, while the osculating semi-major axis is kept almost constant. We shall also discuss stochastic aspects of dynamics in near mean motion dynamics. This might be an explanation of presence of Kirkwood gaps. This is a joint work with J. Fejoz, M. Guardia, and P. Roldan. Notes by
Zhiying Xu
and
Hongyuan Zhan
pdf
10/24 No MASS colloquium this week
Marker Lecture Series

University of Florida
web page
Paul Erdos at 100 -- reflections on his life and work Paul Erdös (1913-1996) was one of the most influential mathematicians of the twentieth century. This is his 100-th birthday year. A Hungarian by birth, Erdös had no permanent home. He traveled around the world constantly, lecturing at hundreds of universities, and seldom staying at a place for more than a week. On these trips he collaborated with both mathematicians and students. Of his research papers that exceed 1500 in number, more than half are in collaboration. While traveling, he was constantly on the look out for very young and talented mathematicians with whom he would collaborate and mold their careers. In a remarkable career that spanned the entire twentieth century, Erdös made pioneering contributions to number theory, combinatorics, graph theory, set theory and geometry. After describing his unusual life and some of his charming idiosyncrasies, we will discuss some of his most fundamental contributions and ideas in prime number theory and probabilistic number theory. Both the story of the elementary proof of the prime number theorem and the creation of probabilistic number theory are fascinating, and will be described. Finally, I will also briefly describe how I met him, and how we collaborated. Notes by
Juan Vargas
and
Justin Miller
pdf
11/7 Ken Ono
Emory University
web page
Adding and counting This lecture will only be about adding and counting. There are many difficult problems related to these seemingly simple tasks. Here we address the problem of finding an exact formula for p(n), the number of partitions of n, and we explain a comprehensive theory of congruence properties. Notes by
Joseph Kraisler
and
Charles Walker
pdf
11/14 Sarah Koch
University of Michigan
web page
Mating habits of polynomials Given two suitable complex polynomial maps, one can construct a new dynamical system by mating the polynomials; that is, by "gluing" the polynomials together in a dynamically meaningful way. In this talk, we focus on quadratic polynomials -- we begin with a brief discussion of parameter space for quadratic polynomials (the Mandelbrot set), we then define the mating of two quadratic polynomials, and finally we explore examples where the mating does exist, and examples where it does not. Steven Metallo
and
Minh Nguyen
11/21 Don Saari
UC Irvine
web page
We vote, but do we get what we want? We vote to select the choice of a pizza, of a member to a social group, of congress, of rankings of football teams, of almost everything. But, does the outcome reflect what the voters really wanted? Mathematics is providing that there is much to worry about. Indeed, by the end of this presentation, some in the audience will worry about some personal election in which she or he was involved. Notes by
Michael Miller
pdf