I graduated with BS degrees in Mathematics and Physics in 1978. At that time, a lot of the job openings for Math majors were for programmers, which didn't interest me. Professor Roger Lautzenheiser suggested I tell interviewers that I was interested in control systems, an area of engineering that uses a lot of math. Using that advice I was able to get a job with Rockwell-Collins, working on the design of aircraft navigation algorithms. At the same time I took a few night classes in aerospace control, nonlinear systems, and Kalman filtering.
Roger was correct -- the general area of control and estimation uses a lot of mathematics and one can find jobs! I soon applied to graduate school at the University of Illinois, where I graduated in 1984 with an MS and PhD in electrical engineering, and a specialty in feedback control theory. Since that time I have been a Professor in the Depatment of Electrical Engineering and Computer Science at the University of Michigan.
While at Illinois I took many graduate mathematics courses, and even the engineering classes I took had a significant math content. Although my degree is in electrical engineering, the field of control is very multidisciplinary -- at the University of Michigan, where I now teach, there are control-related projects, students, and faculty in at least 5 departments in the college of engineering.
You might be interested to know that the technical achievement I am most known for is an application of complex variable theory to the theory of linear feedback systems. Originally developed in the 1940s, this field had lain dormant until the 1980s, in part because of a textbook on the subject that had a mistake due to an incorrect closure of a contour integral around a logarithmic singularity! The engineering implications of this mistake were quite profound, and the correct version of the result now appears in most textbooks, and is the object of further research.
Advice: If you have an undergraduate degree in mathematics, and are willing to learn some engineering, then the closely related areas of control, communications, and/or signal processing may be of interest. All use a fair bit of mathematics, and graduate students typically take several math courses during their studies. I know the most about the control area, in which an MS degree is desired to work in R&D at most companies. I work with people at the major automotive companies and their suppliers (this being near Detroit), Whirlpool, whose R&D center is in Michigan, and General Dynamics Land Systems, who make military vehicles. There are engineers at all these places with MS and PhD degrees in control systems.
With an undergraduate Math background, you are well placed to study controls, but my advice is to learn some engineering as well. Eventually the mathematical models we use deteriorate due to physical constraints, and it is important to have some intuition for when this happens. Moreover, it is often advisable to consider the design and implementation of a control algorithm simultaneously, and these days algorithms are going to be implemented on embedded microprocessors.
I hope that your math background serves you as well as has mine. Currently I am studying probability theory, in particular Shannon's theory of reliable communication, because I want to understand performance limitations imposed on a feedback system by the presence of a communication channel in the feedback path. Sound interesting? Apply to grad school, or do like I did and work a year or two first to learn all the possibilities!