The theory of error-correcting codes, also known as Coding Theory, was invented by R. Hamming and C. Shannon around 1948. Since then, we can communicate information or data reliably. It has been applied to satellite communication, mobile phone, compact disc, high definition TV, and Artificial Intelligence. 

In this talk, we will give two other applications of error-correcting codes. One is a game based on the binary Hamming [7,4,3] code. The other is code-based cryptography based on product codes. We only assume graduate level Algebra. 

This is a gentle introduction to the Langlands program based on selected historical developments. In particular attention will be drawn to the birth of the Langlands program in a letter of Langlands to Weil in 1967. Time permitting a snapshot of some current developments will be given.

Given real valued polynomials $P$ on $mathbb{R}^n$ and various unbounded domains $D subset mathbb{R}^n$, we consider the oscillatory integrals $$ I(P, D, lambda) = int_D e^{ilambda P(t)} dt. $$ We establish a criterion on $(P, D)$ to determine the convergence of these integrals, and find the oscillation indices when they converge. These indices are described in terms of a generalized notion of Newton polyhedra associated with $(P, D)$. When $(P, D)$ for $D=mathbb{R}^n$ satisfies the criterion of the vector polynomial version $(t_1, cdots, t_n, P(t))$, we obtain the Strichartz estimates for the following general linear propagators: $ e^{it P(D)}(f)(x) text{ where } D=left(frac{partial_{x_1}}{i}, cdots, frac{partial_{x_n}}{i} right). $