Undergraduate Math Club

Abstract: This talk will give a glimpse of some of the ideas and motivations underlying the new, so called, ``noncommutative geometry.'' We will begin with familiar objects from calculus (such as ellipses and parabolas), and proceed from there to ``quantizations of the plane'' and (perhaps) beyond.

The famous Euler's formula relates the number of vertices $V$, edges $E$ and faces $F$ in any polyhedron by a beautiful formula \[ V-E+F=2. \] For example in a cube $V=8$, $E=12$ and $F=6$, so that $8-12+6=2$. It turns out that there is a deep and mysterious connection between the Euler's theorem and another famous theorem due to Gauss. When Gauss discovered it, he was so amazed by its beauty that he called it "Theorema Egregium" ('remarkable theorem') in a very uncharacteristic for Gauss self-praise. I will explain how this connection gave rise to an extensive and important topic of current research.

Abstract:  In this talk, we'll describe what futures and options are, their uses and misuses, what kind of math is involved in their analysis, and the celebrated Black-Scholes formula that was worth a Nobel prize.

The first set of interesting functions we meet in mathematics (after polynomials) are circular functions $\sin x$ and $\cos x$. They are related to the circle group and to the ordinary differential operator $\frac{d^2}{dx^2}+1$, which is a polynomial in $\frac{d}{dx}$ with constant coefficients. The next in this hierarchy is the class of hypergeometric functions. They are solutions of the equation of the form \[P_1(\frac{d}{dx})=xP_2(\frac{d}{dx}),\] where $P_1$ and $P_2$ are polynomials in $\frac{d}{dx}$ with constant coefficients. The simple factor $x$ makes a world of a difference. In particular it changes the abelian group (circle group) into a non-abelian group.

Abstract: Using a finite model we will explore devices such as CAT Scanners, and MRI from a mathematical perspective. Using counting arguments and linear algebra we'll consider questions such as the recovery of an image from X-rays, uniqueness, minimality, and more.

 Abstract: It is well-known (to most people) that the so-called Bible Codes are not at all amazing. But Parrondo's "paradox" is even less amazing. The only amazing thing is that many people find it amazing.

Abstract: Using a finite model we will explore devices such as CAT Scanners, and MRI from a mathematical perspective. Using counting arguments and linear algebra we'll consider questions such as the recovery of an image from X-rays, uniqueness, minimality, and more.

This theorem, originally due to Hilbert (1890) and Noether (1916), is widely regarded as a starting point of the branch of modern ``abstract'' algebra now called ring theory. I will discuss this theorem, with some background, and present a very simple proof that was found in 2000. 
 

Abstract: The theorems of Perron (1907) and Frobenius (1909) deal with eigenvalues and eigenvectors of nonnegative matrices. In this short talk we will give the ideas behind different proofs of the theorems, and present some applications of the Perron-Frobenius Theory to economics, population dynamics and the study of iterative solution of linear systems.