Mathematical models are widely used throughout science and engineering in fields as diverse as physics, bioinformatics, robotics, image processing, and economics. Despite the broad range of applications, there are a few essential techniques used in addressing most problems. The Applied Mathematics major provides a foundation in these mathematical techniques and trains the student to use them in...

The complexity, unpredictability and evolving nature of financial markets continues to provide an enormous challenge to mathematicians, engineers and economists to identify, analyze and quantify the issues and risks they pose. This has led to problems in stochastic analysis, simulation, differential equations, statistics and big data, and stochastic control and optimization (including dynamic g...

It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We m...

These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the “tropical approach” in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon ...

A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. The class of sets B which contain the same information as A under Turing computability ( T) is the (Turing) degree of A, and a degree is c.e. if it contains an c.e. set. The extension of embedding problem for the c.e. degrees R = (R; <...

The rise of the field of ‘‘experimental mathematics’’ poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, nonempirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathemati...

We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics.

I t is indeed an honor to be invited to this platform. One has only to read the names of my distinguished predecessors to realize that, as the saying goes, I have thirty-seven acts that are hard to follow. As a mostly pure mathematician my concern is increased when I note that thirty-two of the lectures have dealt with applications of mathematics to other sciences. The true desperation of my pr...