PMASS Colloquia 2013

Thursday, April 18 Joel C. Langer, Case Western Reserve University
2:30pm A short history of length
ABSTRACT A handy old device called a waywiser - basically a wheel and axle mounted 
on a handle - may be used to measure the length of a path, straight or curved. If the wheel 
is one meter in circumference, the waywiser measures the length of the path in meters by 
counting revolutions of the wheel as it is walked from beginning to end of the path. It works 
well enough in practice - but does it also work in theory? 
In fact, the waywiser and the concept of arc length may be used to illustrate both successes 
of ancient geometers and some of the struggles faced by subsequent mathematicians and 
philosophers in coming to terms with innity, innite processes and associated computations. 
The story of arc length alternates between geometry and the theory of numbers, between 
the continuous and the discrete, over two thousand years.
Thursday, March 14 Andrey Gogolev, Binghamton University
2:30pm Prime number theorem
ABSTRACT The primes have been boggling the minds of humans since the ancient times. We will discuss the statistical distribution of primes within the set of positive integers. This distribution is described by the prime number theorem. Interestingly, finer versions of the prime number theorem are conjectures that are related to the core of modern mathematics such as the Riemann hypothesis.
Thursday, March 21 Karl Schwede, Penn State
2:30pm Bezout's theorem
ABSTRACT How many times can a line intersect a parabola? You probably said the answer is zero, one, or two. How many times can two ellipses intersect? What if you count imaginary or complex intersection points? Do parallel lines ever intersect? What if we change the rules of the game and consider parallel lines in a twisted pac-man-like world? Bezout's theorem tells us exactly how many times two implicitly defined polynomial curves intersect, provided you count complex intersections, intersections at infinity (where parallel lines meet up), and double/triple/multiple roots correctly.
Thursday, February 28 Alexandre Kirillov, University of Pennsylvania
2:30pm Geometry and arithmetic of Apollonian gasket
ABSTRACT One of oldest and beautiful examples of a fractal set is the Apollonian gasket, arising as a maximal tiling of 2-dimensional sphere by non-intersecting discs. There are (at least) two remarkable facts: 
1. The Descartes theorem, which gives the algebraic relation between the curvatures of four pair-wise tangent circles on the plane. I shall give the proof based on the geometry of Minkowski space.
2. The existence of integral realisations of the Apollonian gasket, where all curvatures are integers. I show some arithmetic properties of these curvatures and formulate several open questions. 

I attached a picture of one integral realisation.

Thursday, February 14 Victoria Sadovskaya, Penn State
2:30pm Fractals and dimension
ABSTRACT Fractals can be described as self-similar sets that have a fine 
structure at arbitrarily small scales. We will consider a variety 
of fractals and discuss their properties, which will lead us to 
their non-integer dimension. We will define and explore the 
notion of box dimension.
Thursday, January 17 Sergei Tabachnikov, Penn State
2:30pm Proofs from the Book
ABSTRACT The famous mathematician of the last century, Paul Erdos, often referred to "The Book" in which God keeps the most elegant proofs of mathematical theorems. So, he would say: "This is a proof from the Book", or "This is a correct proof, but not from the Book". In fact, a book called "Proofs from the Book" was written by M. Aigner and G. Ziegler, and this is one of the most popular mathematical books ever. In this talk, I shall present some proofs of great theorems from this book, and some other ones that, in my opinion, belong there.
Thursday, January 31 Eii Byrne, Penn State
2:30pm Game Theory 101: numerical modeling of life events
ABSTRACT This talk will illustrate the process of assigning numerical payoffs to 
model outcomes of multi-agent decision profiles that are not necessarily 
numerical in nature. The emphasis will be on modeling; i.e. the logic of 
how inequalities between the numerical payoffs within a game model agent 
preferences over outcomes. I will present example problems requiring no 
technical background.