Goursat's Theorem and its proof by Marshall Whittlesey
A function f(z) of a complex variable z is said to be differentiable at z if the limit of [f(z+h)-f(z)]/h as h approaches 0 exists. If this limit exists and equals L, we write that f '(z)=L. This is much the same as the definition for a function of a real variable - the only serious difference being that h is a complex number converging to zero.
In calculus, students learn that a function of a real variable can be differentiable, but there may not exist a second derivative. In this talk we will discus Goursat's remarkable theorem (proven in 1900) that if f '(z) exists in some open domain, then f '(z) is continuous in z. Earlier work of Cauchy then shows that f must be differentiable infinitely many times.

Finite Simple Groups and tests for non-simplicity by Mary Stewart
A simple group G is a nontrivial group that has only the group G and the group {e} as normal subgroups. In this talk, we will mention some examples of finite simple groups and discuss the Classification Theorem, which classifies all such groups. We will also provide some interesting examples for proving certain groups are not simple, based on results and tools of group theory.

The Combinatorics of higher order chain rules by Tejinder Neelon
Suppose y = f(x) and w=g(y) are two functions with sufficient number of derivatives. The chain rule states that (g o f)'(x) = g'(f(x)).f'(x). What is the formula for the nth derivative of the composition g o f in terms of the derivatives of g anf f? The well known answer is the Faa Di Bruno formula. The Faa di Bruno formula can be understood from many different points of view. We will focus on three different versions: the set-partition version, Riordan's Bell polynomial version, and the determinant version. We will also show that the proof becomes quite straight-forward if we consider the analogous formula for power series.

An introduction to Number Theory from a p-adic point of view. by Douglas Haessig, Mathematics Department, UC Irvine
Many number theorists are interested in the solution sets of multivariable polynomial equations. For example, Fermat's last theorem asks about integer solutions of the polynomial equation xⁿ + yⁿ = zⁿ. In this talk, we will see how the solution set of a quadratic equation depends on the domain of the variables. This will lead us into defining the "p-adic numbers," and as an application of these p-adic numbers, we finish with a discussion on counting the number of solutions to a famous cubic equation using d ifferential equations! The talk will be accessible to all math students.

Caratheodory's definition of the derivative by Heydar Zahedani,
Mathematics Department, CSUSM

The function f, defined on the interval U, is said to be differentiable at the point a in U if there exits a function g that is continuous at x=a and satisfies the relation: f(x)-f (a) = g(x) (x-a) for all x in U.
Constantin Caratheodory (1873-1950) gave this formulation of the derivative for complex-valued functions in his last textbook. We show that how Caratheodory's characterization of differentiability simplifies the proofs of the basic differentiability theorems for real-valued functions of one variable, in particular the proof of the Chain Rule. We also consider the extension to functions of several variables and discuss some of the advantages it has over Frechet's characterization of differentiability.