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SCU Mathematics Colloquium Series Schedule
Winter 1998
SCU Mathematics Colloquium Series Schedule
Winter 1998
Talks will be at 4:00 Tuesdays in O'Connor 207. There will be refreshments before all talks in O'Connor 31 starting around 3:40pm.
Title: Curvature without (much) calculus
Curvature of smooth surfaces in space can be understood from many different perspectives, most of which require enough smooth structure to use the tools of calculus. But some characterizations of curvature are general enough to apply to more general structures, including Cayley graphs of infinite discrete groups. I will discuss several characterizations of curvature, focusing on some qualitative ones that generalize well. In particular, I will talk about what it means for a metric space to be negatively curved and discuss hyperbolic groups.
Title: Why does "real" physics need "imaginary" numbers? A history of physical applications of geometric algebras.
Certain mathematical systems: complex algebra, Gibbs vectors (1881) and Sylvester's matrices (1850), have wide applications in physics, engineering and chemistry. However there were many siblings (Hamilton's Quaternions 1843, Grassmann's Algebra of Extension 1844, Cayley's Octonions 1845) that fell into obscure disuse but have recently found new applications which we will review. In particular, William Kingdon Clifford proposed (1876) a universal geometric algebra which combined features of all the above. Due to his untimely death this extraordinary system, in which one can add a vector to a scalar to a plane, was ignored for nearly 100 years. Recent revival (last 20 years) has shown that undergraduates can more quickly learn and apply Clifford vectors than Gibbs vectors; becoming quite excited with the interpretation of "i" as the volume of 3 space, seeing "planes" as things that cause rotations (or Lorentz transformations), and being able to do divergence, curl and gradient in a single equation. New extensions of a generalized geometric calculus hold promise for completely new approaches to unified physical theories.
Title: From basic diophantine approximation to a conjecture of Littlewood
Here we will begin with a survey of the fundamental themes involved in diophantine approximation and some classical results from the field. We will then turn towards what is referred to as the Littlewood Conjecture regarding simulaneous diophantine approximation of two real numbers and consider some new related results. No specialized number theory background is required.
DUE TO INCLEMENT WEATHER AND ROAD CLOSURES, THIS PRESENTATION IS CANCELLED AND WILL BE RE-SCHEDULED FOR THE SPRING.
The Cake-Cutting Problem, the Ham Sandwich Theorem and other topics in culinary mathematics.
The Intermediate Value Theorem of freshman calculus has many interesting corollaries and we explore some of them in this talk. In particular we show that it is always possible to divide a cake fairly and that, if different people value different parts differently, everyone can get more than their fair share.
q-Catalan Numbers
A brief introduction to some q-objects (q-binomials,q-Catalan Numbers) will be followed by some generalizations of Polya's description of q-binomials in terms of lattice paths. The talk should be accessible to math majors and will contain some new material.
1001 Hikes on the Gaussian Primes
"Everybody" knows that there are infinitely many prime numbers. What should we do with this plethora of primes? To start, let's play a game. Stand on the number 0 and start walking. The only rule to this game is, as you take each step, you are ONLY allowed to step on the primes. A curious question: Can you walk to infinity on primes in steps of bounded length? The answer is NO, and the solution is an elegant tidbit (which I will divulge) from Elementary Number Theory.
Now that I have your attention, let's ask the same question in another setting. The Gaussian integers Z[i] (the ring of integers in C) are of the form a+bi where a and b are (rational) integers. The Gaussian primes are those elements of Z[i] that are prime. Starting at the origin, can you walk to infinity on Gaussian primes in steps of bounded length? This seemingly innocent question, posed in 1962 by Basil Gordon, remains unsolved to this day. I will give a brief tour of some helpful related results from Analytic Number Theory, and will talk about my work with Griff Elder, Carl Pomerance, Harold Stark, Stan Wagon, and Brian Wick on various successful approaches to some of the "foothills" of this problem.
Title: Isoperimetric conditions and modern probability
This talk concerns problems that lie in the intersection of three branches of modern mathematics: geometry, probability, and partial differential equations. At the heart of the work is the relationship between the notion of ``random motion'' and the classical isoperimetric condition: amongst all simply connected planar domains of fixed perimeter, a circle has the largest volume. In the talk we will study how this geometric condition determines a number of properties of ``random motion.'' Time permitting, we will also discuss a partial converse: Given the correct definition of random motion, what geometric properties of the underlying space are determined?
The talk is intended for undergraduates and will be reasonably self-contained. No knowledge of probability or differential geometry will be assumed. In the course of the lecture, we will construct a simple random walk and indicate how such a construction leads to Brownian motion, the ``correct'' definition of random motion in a variety of contexts. We will indicate how immediate combinatoric properties of random walks imply interesting analytic properties of Brownian motion and how these analytic properties in turn restrict the geometry of the underlying space.
Last Updated: 4 February 1998
Maintainer: dsmolarski@scu.edu