| Thursday, April 18 | Joel C. Langer, Case Western Reserve University |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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. |