PMASS Colloquia 2011

Thursday, January 20 John Roe, The Pennsylvania State University
2:30pm Morley’s theorem
ABSTRACT MORLEY’S THEOREM is a result in “classical” plane geometry - but its first proof was given in modern times! It states that for any triangle at all, the trisectors of successive angles meet at the vertices of an equilateral triangle. I’ll explain how the great French mathematician Alain Connes was motivated to give a new proof by a lunch-time conversation - and how Napoleon Bonaparte comes into the story as well.
Thursday, April 21 George Andrews, The Pennsylvania State University
2:30pm Ramanujan, Fibonacci numbers, and continued fractions
ABSTRACT This talk focuses on the famous Indian genius, Ramanujan. The first part of the talk will give some account of his meteoric rise and early death. Then we shall lead gently from some simple problems involving Fibonacci numbers to a discussion of some of Ramanujan's achievements.
Thursday, February 3 Sergei Tabachnikov, The Pennsylvania State University
2:30pm Equiareal dissections
ABSTRACT If a square is dissected into triangles of equal areas then the number of triangles is necessarily even. This "innocently" looking result is surprisingly recent (about 40 years old), and its only known proof is surprisingly non-trivial: it involves ideas from combinatorial topology and number theory. I shall outline a proof and discuss various variations on this theme.
Thursday, February 17 Anatole Katok, The Pennsylvania State University
2:30pm Billiard table as a mathematician's playground
ABSTRACT The title of this lecture may be understood in two ways. Literally, in 
a somewhat lighthearted way: mathematicians play by launching 
billiard balls on tables of various forms and observe (and also 
try to predict) what happens. In a more serious sense, the 
expression ``playground'' should be understood as ``testing grounds'': 
various questions, conjectures, methods of solution, etc. in the theory 
of dynamical systems are ``tested'' on various types of billiard 
problems. I will try to demonstrate using some accessible examples that at least the second 
interpretation deserves a serious attention.
Thursday, March 3 Vladimir Dragovic, MI SANU Belgrade/ GFM, University of Lisbon
2:30pm Theorems of Poncelet and Marden -- two of the most beautiful theorems
ABSTRACT We are going to present cases of the Siebeck-Marden theorem, 
from the geometric theory of polynomials and of the Poncelet theorem, one 
of the most important results about pencils of conics. We are going to 
discuss also recently observed relationship between these statements.
Thursday, March 17 Vaughn Climenhaga, University of Maryland visiting the Pennsylvania State University
2:30pm The bigness of things
ABSTRACT It is very natural to ask how "big" something is, but answering this question properly in various settings often requires some new ideas. We will explore this question for the Cantor set, for which I'll explain why some more familiar notions of "bigness" are unsatisfactory and how a concept of "fractional dimension" arises.
Thursday, March 31 Omri Sarig, The Pennsylvania State University
2:30pm Symbolic dynamics
ABSTRACT "Symbolic dynamics" is a technique for studying chaotic dynamical systems. The idea is to associate to every orbit a sequence of symbols and then study the combinatorial properties of the resulting sequences. I will describe a particular example: movement on a straight line on a negatively curved surface.