Thursday, January 20John Roe, The Pennsylvania State University
2:30pmMorley’s theorem
ABSTRACTMORLEY’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 21George Andrews, The Pennsylvania State University
2:30pmRamanujan, Fibonacci numbers, and continued fractions
ABSTRACTThis 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 3Sergei Tabachnikov, The Pennsylvania State University
2:30pmEquiareal dissections
ABSTRACTIf 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 17Anatole Katok, The Pennsylvania State University
2:30pmBilliard table as a mathematician's playground
ABSTRACTThe 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 3Vladimir Dragovic, MI SANU Belgrade/ GFM, University of Lisbon
2:30pmTheorems of Poncelet and Marden -- two of the most beautiful theorems
ABSTRACTWe 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 17Vaughn Climenhaga, University of Maryland visiting the Pennsylvania State University
2:30pmThe bigness of things
ABSTRACTIt 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 31Omri Sarig, The Pennsylvania State University
2:30pmSymbolic 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.