MASS Colloquia 1998

Fri, Sep 4 Karl Rubin (Stanford) 4:00 pm Rational numbers, right triangles, and elliptic curves Abstract: Many mathematical problems, including Fermat's Last Theorem, turn out to be related to polynomial equations called elliptic curves. Motivated by one such question, this talk will introduce elliptic curves and discuss the current state of knowledge about them, including some open questions.

Thur, Sep 17 Mark Levi (Penn State) 3:00 pm Bicycle wheels, Gauss-Bonnet formula and Berry's phase Abstract: I will show how some simple physics of the bike wheel leads to easy insights into topics such as the Gauss-Bonnet theorem, Berry's phase and the writhing number.

Thur, Oct 1 Alexandre Kirillov (University of Pennsylvania) 3:00 pm Why representation theory is interesting and useful? Abstract: 1. Universality: in all domains of mathematics and applications you encounter various types of symmetry and RT is an adequate mathematical tool to study and use it. 2. Many asymmetric problems have symmetric particular cases where the solution can be obtained using RT. After studying this particular cases sometimes the general results were found. 3. There is a lot of beautiful symmetric objects (such as sporadic simple groups or root systems) several useful formulae (e.g. Weyl-Kac algebraic formula and my integral formula for characters of unirreps). Some examples of all this will be discussed.

Thur, Oct 8 Gregory Galperin (Eastern Illinois University) 3:00 pm Playing pool with π Abstract: Is it possible to find the number pi with any accuracy you wish without any calculations? It turns out, yes! A special dynamical system of billiard balls is going to be constructed in the talk. This system will "count" pi to any number of decimal digits, n. The concepts of a configuration space and phase space will be introduced for the investigation of the problem, and we may consider some related questions to which these notions can be applied.

Thur, Oct 22 Bruce Berndt (University of Illinois) 3:00 pm The Life and Work of India's Greatest Mathematician Srinivasa Ramanujan Abstract: Ramanujan was born in southern India in 1887 and died there in 1920 at the age of 32. He had only one year of college, but his mathematical discoveries, made mostly in isolation, have made him one of this century's most influential mathematicians. An account of Ramanujan's life will be presented. Most of Ramanujan's mathematical discoveries were recorded without proofs in notebooks, and a description and history of these notebooks will be given. Lastly, we describe some of the topics that are found in the notebooks, and a sample of some of the more fascinating entries will be provided.

Thur, Oct 22 Yakov Pesin (Penn State University) 3:00 pm Mathematics of Fractal Images Abstract: Fractals have immeasurably enlarged our ability to describe nature. The abstract constructions going back to Bolzano, Cantor, and Peano have furnished us with models of reality much more realistic than the Euclidean empire of integer exponents and smooth shapes. In my talk I will discuss the notion of a fractal object (image) versus smooth Euclidean shapes, its appearance, and role in describing various phenomena. I will also outline some mathematical ideas and notions from Fractal Geometry which lead to rigorous mathematical study of fractals.

Thur, Nov 5 Paul Baum (Penn State University) 3:00 pm An Introduction to K-theory

Thur, Dec 3 Fedor Bogomolov (Courant Institute) 3:00 pm Arithmetic Curves and Triangulations Abstract: Compact orientable surfaces appear in several areas of mathematics under different disguise and with different natural additional structures. For instance, in topology the most natural additional structure is a triangulation of the surface. In algebraic geometry one should treat those surfaces as complex projective curves (e.g., the projective line or an elliptic curve) which may be viewed as compact orientable surfaces with an additional structure of a (one-dimensional) complex manifold (Riemannian surface). Quite often, an additional feature which is natural in one area of mathematics has an interesting interpretation in the different area. In my lecture I shall explain how a special triangulation of a Riemannian surface (viewed as a topological object) defines the structure of a complex projective curve on this surface. In addition, we will see that the triangualtion gives rise to a natural family of elliptic curves parametrized by the points of the corresponding Riemannian surface. I will also discuss interrelations between the so called Szpiro inequality for this family and the Euler characteristic of the surface expressed in terms of the triangulation.

Friday, Dec 4 Simon Gindikin (Rutgers University) 3:00 pm Social Life of Curves Abstract: All individual (smooth) curves are locally identical, but when a curve is a member of a family of curves it can be provided a nontrivial intrinsic structure. Curves of a 2-parametric family on a 2-dimensional manifold carry canonical structures of local lines. An interesting old problem: when a 2-parametric family of curves on a 2-dimensional manifold is diffeomeorphic to the family of lines. Such kind considerations were popular in the beginning of our century between mathematicians around Lie and Cartan. In last 20 years there were appeared new problems about families of curves in integral geometry and nonlinear differential equations, including Einstein's equation. There are several interesting results and surprising connections with the classical geometry.