CSULA Math Club abstracts

05/21/08    Mike Krebs    The "No WAY!" moment in mathematics

Much has been written, and many tales told, of the so-called "Aha!" moment in mathematics.  In this talk, we discuss a subtly different but no less pleasurable instant, namely the "No WAY!" moment.  By way of illustration, we consider an unexpected turn of events the speaker once encountered while exploring a small variation on Cantor's original proof of the uncountability of the real numbers.  This talk will take about fifteen (15) minutes.

04/30/08    Jamie Pommersheim    Euler-Maclaurin summation for polytopes

04/16/08    Lenny Fukshansky    Sphere packing, lattices, and Epstein zeta function

The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current knowledge. If, however, one only considers lattice packings, i.e. arrangements of spheres with centers at points of a lattice, more is known.

In this talk, I will introduce the sphere packing problem, briefly surveying its history and known results. I will then restrict to lattice packings, describing a connection between the problem of finding an optimal lattice packing in a given dimension and minimization problem for Epstein zeta function on the space of unimodular lattices in this dimension. I will also introduce some important classes of lattices which are expected to solve these related problems, and will demostrate these concepts on the well understood 2-dimensional case. If time allows, I will conclude with a certain approximation lemma which shows how good of a packing density one can expect from lattices with rational bases coming from one of these important classes of lattices in dimension 2.

04/09/08    Angel Pineda    To Bin or Not to Bin? Using Mathematics to Improve CT scans

Mathematics plays an important role in x-ray medical computer tomography
(CT) scans. In fact, one of the recipients of the Nobel Prize for Medicine
in 1979 for his work in this area was the mathematician Allan Cormack.
This will be an introductory talk (assuming no previous background) giving
a description of the mathematics of CAT scans. A CAT scan uses x-ray
projections acquired at many angles (a chest x-ray would be one projection)
to reconstruct the images we typically see. Reducing the size of the
projections stored would significantly accelerate the image reconstruction
process. We will use mathematics to make a practical design decision on
whether to reduce the data stored from large digital detectors by adding
pixels together (binning) before storing them. This is analogous to reducing
the number of pixels of a digital photo. The answer as to whether we should
bin or not bin will come from understanding the noise amplification of the
reconstruction process, the human visual system and how binning affects
tumor detection.

04/02/08    Professor Hrushikesh N. Mhaskar    Analysis of local features of a function using spectral data

We discuss the question of identifying local features of a function, such as the discontinuities in its derivatives, membership in local smoothness classes, etc., given global information about the function in the form of its Fourier coefficients with respect to an orthonormal system. We present a unifying theme for some of the recent work on trigonometric and algebraic polynomial frames on the circle, the unit interval, the Euclidean sphere, and a smooth manifold in general.  Applications include direction finding in phased array antennas, estimation of the velocity of the gulf stream, and semi–supervised learning of hand written digits.

02/20/08    Tony Shaheen    Zeta Functions of Graphs

In this talk we will discuss the Ihara zeta function of a graph. I will begin by defining the Ihara zeta function of a graph. This definition is usually impossible to compute, but there is a nice formula due to Ihara that allows one to compute the zeta function in terms of a determinant.  I will also tell you what the "Riemann Hypothesis" is for regular graphs. It turns out that it is true if and only if the graph is "Ramanujan". We will also discuss how the zeta function tells you how many spanning trees a graph has.

02/13/08    Gary Brookfield    What you should know about LaTeX

02/06/08    David Beydler    Cryptography: Some History and the Math Behind the RSA Cipher

Cryptography is the study of message secrecy. Historically, one of the driving motivations
for studying cryptography has been war. A more recent motivation has come
from the rise of computers and, in particular, the Internet. Everytime you make a credit
card purchase, or enter a password to access private records, you want to make sure that
the information is kept secret. Thankfully, many algorithms (called ciphers) have been
invented to address these new needs. One of the most prominent of these ciphers is called
RSA. This talk will cover some history of cryptography, review relevant number theory, give
an overview of the RSA cipher, and then prove that it works.

01/30/08    Art Benjamin    Combinatorial Trigonometry (and a method to DIE for)

Many trigonometric identities, including the Pythagorean theorem, have
combinatorial proofs. Furthermore, some combinatorial problems have
trigonometric solutions. Most of these problems reduce to
alternating sums, and are attacked by a technique we call
D.I.E. (Description, Involution, Exception). This technique offers new
insights to identities involving binomial coefficients, Fibonacci
numbers, derangements, zig-zag permutations, and Chebyshev
polynomials.