MASS Colloquia 2013

Thursday, September 5 Mark Levi, Penn State University
1:25pm Mathematics through physics
ABSTRACT 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.

Thursday, September 12 Joel Hass, UC Davis
1:25pm Some applications of geometry to problems in biology
ABSTRACT Many problems arising in biology are geometric, since many biological properties 
are reflected in the shapes of biological objects. <br> We will explain recent developments in 
the use of conformal mappings and hyperbolic geometry to study several biological problems.<br> 
1. The study of cortical surfaces. In particular, the question of how similar are two brains. <br> 
2. The classification of proteins. <br> 
3. Deducing the evolutionary tree of old-world monkeys from fossilized skulls.
Thursday, September 19 Dmitri Burago, Penn State University
1:25pm Counting collisions in hard ball gas models and singular geometry of non-positive curvature
ABSTRACT 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.
Wednesday, September 25 Robert Lang, Alamo, CA
4:00pm From flapping birds to space telescopes: the mathematics of origami
ABSTRACT 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.
Thursday, October 3 Diane Henderson, Penn State University
1:25pm The William Pritchard Fluid Mechanics Laboratory
ABSTRACT 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.
Thursday, October 10 Roger Howe, Yale University
1:25pm About the numbers 12 and 24
ABSTRACT 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.
Thursday, October 17 Vadim Kaloshin, University of Maryland
12:00pm Kirkwood gaps and instability for three body problems
ABSTRACT 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.
Thursday, October 31 Krishnaswami Alladi, University of Florida
1:25pm Paul Erdos - one of the most influential mathematicians of our times
ABSTRACT Paul Erdos (1913-1996) was one of the most influential 
mathematicians of the twentieth century. This is his 100-th birthday 
year. A Hungarian by birth, Erdos 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, Erdos 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.
Thursday, November 7 Ken Ono, Emory University
1:25pm Adding and counting
ABSTRACT 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.
Thursday, November 7 Ken Ono, N/A
4:30pm Ramanujan
Thursday, November 14 Sarah Koch, University of Michigan
1:25pm Mating habits of polynomials
ABSTRACT 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.
Thursday, November 21 Don Saari, UC Irvine
1:25pm We vote, but do we get what we want?
ABSTRACT 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.