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SCU Mathematics Colloquium Series Schedule
Winter 2001
SCU Mathematics Colloquium Series Schedule
Winter 2001
Talks will be at 4:00 Tuesdays, Room O'Connor 106. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.
Title:Theory of Computation and Computational Complexity: An Overview
Theory of computation and complexity is a formalization of our understanding of computation (i.e., the notion of an "algorithm"). It is a field of study which investigates the solvability/unsolvability (existence/non-existence of algorithms) of computational problems. It also aims to develop techniques for designing efficient algorithms as well as to explain why some problems are inherently difficult to solve in terms of their time and/or space requirements.
We present an overview of the field.
Title: Classifying some hyperbolic 3-manifolds
A nice class of 3-dimensional manifolds can be obtained by gluing together copies of the dodecahedron. Some nice features of these manifolds are (1) they are highly symmetric, (2) their standard topological invariants are easy to compute, and (3) they have natural hyperbolic structures. We shall describe how these properties can be used to give a topological classification of these manifolds.
Title: A smooth space of tetrahedra
A classical question in enumerative geometry asks: given two curves in the plane, how many triangles can be simultaneously inscribed in one curve and circumscribed about the other? Cayley gave the correct answer to this question around 1870, but it was Schubert's argument ten years later that put Cayley's calculation in the context of modern intersection theory. Schubert's argument relies on his construction of a particularly nice (meaning smooth) space that parametrizes the set of all triangles in the plane. Unfortunately, Schubert's construction does not generalize to higher dimensions (for example, it does not yield a smooth parameter space for tetrahedra in 3-space). In this talk, I will first discuss the ideas behind Cayley's and Schubert's solutions, and then describe a parameter space for plane triangles that does generalize to higher dimensions.
Title: The Sun, the Moon, and Convexity
I will present a simple model of a moon's orbit about a sun, and discuss the conditions under which such an orbit is convex. I will attempt to convince the audience that the path our moon makes about our sun is convex.
The talk will chiefly consist of Quicktime animations of lunar paths that were created with Mathematica.
Title: L-functions and Random Matrices
One of the fundamental questions in all of mathematics is the Riemann Hypothesis which postulates that all of the zeros of the Riemann zeta-function have real part 1/2. Riemann made this hypothesis because of a connection between these zeros and the prime numbers. We still do not know how to prove the Riemann Hypothesis but we have some interesting clues and connections. For more than 25 years it has been suspected that the zeros of the Riemann zeta-function have spacings like the spacings of eigenvalues of large random Hermitian matrices, appropriately scaled. Recent developments have made this connection much more precise. Moreover, it is now believed that the statistics of zeros of families of L-functions - (functions like the Riemann zeta-function which are associated with various arithmetic objects of interest to Number Theorists) - also behave like eigenvalues of orthogonal, unitary, or symplectic matrices, depending on the family. I'll talk about some of these developments.
Title: Lies, Damn Lies, and Statistics (with apologies to Mark Twain)
This talk describes some things which I have personally found fascinating under the umbrella heading of probability and statistics. The presentation is intentionally pitched at a level that does not require any heavy duty analytic machinery. The first topic considers the elementary question of determining the probability that a "random" chord of the unit circle exceeds the square root of 3 in length. Three plausible definitions of random chord length are given which are shown to lead to three different answers. This raises the question of a "natural" definition of randomness, and some comments are made regarding probabilistic theorems on convex figures in the plane which are due to Barbier, Crofton and Sylvester. The second topic considered is the importance of query dependent information in computing conditional probabilities. Illustrations are given using the three and the one thousand prisoner problem and the Monty Hall problem, made famous by Marilyn Vos Savant. The final topic is "Simpson's Paradox" which has played a part in recent equal opportunity suits and which remains a topic of controversy after more than a century.
Title: Concurrent Cevians in Tetrahedra and some related aspects of the Geometry of the Tetrahedron
The Moorish king Yussuf al Mut'aman ibn Hud (murdered 1095) of Zaragoza (now Spain) presented in his "Book of Perfection" a theorem which is usually attributed to Giovanni Ceva (1647-1734). It gives a necessary and sufficient condition for the concurrency of three straight lines each passing through the three vertices of a triangle. Unfortunately such a condition does not exist one dimension higher, i.e., for four straight lines passing through the four vertices of a tetrahedron. We (Roland Eddy from Memorial University of Newfoundland and myself) are studying such arrangements of lines arising from familiar geometric settings and give conditions for their concurrency. One interesting result describes the class of tetrahedra such that the lines connecting the vertices with the circumcenters of the opposite faces concur.
The list of talks from previous quarters are available via this archive link.
Last Updated: 25 February 2001
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