Thursday, April 18Joel C. Langer, Case Western Reserve University
2:30pmA short history of length
ABSTRACTA 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 14Andrey Gogolev, Binghamton University
2:30pmPrime number theorem
ABSTRACTThe 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 21Karl Schwede, Penn State
2:30pmBezout's theorem
ABSTRACTHow 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 28Alexandre Kirillov, University of Pennsylvania
2:30pmGeometry and arithmetic of Apollonian gasket
ABSTRACTOne 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 14Victoria Sadovskaya, Penn State
2:30pmFractals and dimension
ABSTRACTFractals 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 17Sergei Tabachnikov, Penn State
2:30pmProofs from the Book
ABSTRACTThe 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 31Eii Byrne, Penn State
2:30pmGame Theory 101: numerical modeling of life events
ABSTRACTThis 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.