MASS Colloquia 2011

Thursday, September 8 Pat Hooper, CUNY
2:30pm Patterns of curves in the plane and piecewise translations
ABSTRACT A curve in the plane is called simple if it has no self intersections. In this talk we will consider various methods of constructing random simple curves in the plane. These curves can either close up or be unbounded. We consider the question "What is the probability that a random curve closes up?" This question shows up in Percolation theory and also in the study of piecewise translations, which we empasize.
Thursday, September 22 Tristan Needham, University of San Francisco
2:30pm Visual Differential Geometry and the Hyperbolic Plane
ABSTRACT Differential Geometry" contains the word "Geometry", yet the standard texts are filled with proofs (versus explanations) that are nothing more than ungeometric calculations. We will instead give a few elementary examples of how the concepts and results of this important subject can be made accessible to visual intuition. In particular we will show how this Visual Differential Geometry sheds light on the hyperbolic plane, yielding a simplified proof of the key fact that it has constant negative Gaussian curvature. [No prior knowledge of Differential Geometry will be assumed. Brief prior exposure to Hyperbolic Geometry would be helpful, though even that is not essential.]
Thursday, September 29 Michael Gekhtman, Notre Dame
2:30pm Combinatorics of Total Positivity
ABSTRACT A totally positive matrix is a matrix with all positive minors. We will review origins and examples of total positivity and show how the key properties of totally positive matrices can be encoded by a purely combinatorial tool - directed planar graphs.
Thursday, October 6 Aaron Abrams, MSRI/Emory University
2:30pm Cutting a square into triangles
ABSTRACT Suppose you want to cut the unit square into triangles. What are the possibilities for the set of areas of the triangles? It turns out there are some restrictions: for instance, a theorem of P. Monsky says that if the number of triangles is odd, it is impossible for them all to have the same area. 

In general, what happens is that once you decide on the combinatorics of the triangulation, there will always be a polynomial relation that the areas are guaranteed to satisfy. This polynomial is different for different combinatorial triangulations, and it tends to be quite complicated; in particular the polynomial will have one variable for each triangle in the triangulation. The degree of the polynomial, however, is an integer invariant of the underlying triangulation. In this talk we will discuss this polynomial and an algorithm for computing its degree.

Thursday, October 20 Frank Sottile, Texas A&M
2:30pm The Real, Complete Story of Conics
ABSTRACT In 1848 Steiner asserted that, given 5 general conics in the plane (ellipses, hyperbolae, or parabolae), there are exactly 7776 (=6^5) other conics tangent to the given 5. He was unfortunately wrong and the correct answer of 3264 (=2^5 times 102) was given by Chasles in 1864. More recently, Fulton, and Ronga, Tognoli, and Vust have separately shown it is possible to choose 5 real conics so that each of the 3264 tangent conics are also real. 
While this talk will explain the obvious numerological questions (where do 7776 and 3264 come from?), my goal is to convince you that 3264 is the correct number. This explanation will also show why this works over the real numbers. 
For more information and some pictures:
Thursday, October 27 Alexandre Kirillov, University of Pennsylvania
2:30pm Geometry and analysis on fractals
Thursday, November 3 Chaim Goodman-Strauss, University of Arkansas
2:30pm “The more things change...”
ABSTRACT In two dimensions, there are three model geometries: the sphere, the Euclidean plane, and the hyperbolic plane. In each we can ask various kinds of combinatorial questions (“What are the possible discrete symmetry groups?” is a pretty good example). Although the spaces have pretty different properties, they’re not so different as they might seem. We’ll walk through several hands-on constructions, exploring the underlying topological, unifying perspective of orbifolds.
Thursday, November 10 George Andrews, Penn State
2:30pm The Lost Notebook of Srinivasa Ramanujan
ABSTRACT In 1976 quite by accident, I stumbled across a collection of about 100 sheets of mathematics in Ramanujan's handwriting; they were stored in a box in the Trinity College Library in Cambridge. I titled this collection "Ramanujan's Lost Notebook" to distinguish it from the famous notebooks that he had prepared earlier in his life. On and off for the past 35 years, I have studied these wild and confusing pages. Some of the weirder results have yielded entirely new lines of research. I will try to provide some highlights of where these efforts have led. If time permits, I will conclude with a couple of accounts of associated TV and film projects that arose because of this discovery.
Thursday, November 17 Charles Conley, University of North Texas
2:30pm Trigonometric identities and the Weyl character formula
ABSTRACT You may have seen trigonometric identities relating ratios of sine functions to sums of cosine functions, for example, 
sin 6θ /sin θ = 2 cos 5θ + 2 cos 3θ + 2 cos θ. 
Such identities are special cases of the Weyl character formula, which describes representations of compact Lie groups and relates ratios of certain generalized sine functions to sums of generalized cosine functions. These generalized trigonometric functions depend on more than one angle, and have symmetry properties under permutation of their arguments analogous to the symmetries of sine and 
cosine under θ → −θ. 
In this talk we will give a gentle introduction to groups and their representations. We will exhibit the Weyl character formula for the group U(3) (the 3 by 3 unitary matrices), and for the unit sphere S 3 in 4-dimensional space, which is a group under Hamiltonian (quaternionic) multiplication. In the case of S^3 , we will use integration in 4-dimensional spherical coordinates to describe the approach Weyl used to arrive at his formula.