Research Scientist, Schlumberger
I work as a research scientist at Schlumberger, an oilfield services company that provides measurements of the Earth acquired while prospecting for oil and developing oilfields. A good example of these measurements are seismic surveys where seismic waves are propagated into the Earth's surface, their reflections are recorded, and then these recorded reflections are turned into structural images showing various rock layers and fluids under the Earth's surface. A lot of physics is needed to understand how waves propagate through rocks of various types (sandstone, limestone, clay, salt,...) that may or may not be filled with fluids (water, oil, gas), but a lot of mathematics is needed to take the seismic measurements and the physics equations and turn them into usable images which can be used by people drilling oil wells and trying to produce oil in an economically and environmentally sensible way.
Computing these images accurately and efficiently requires learning how to implement the linear algebra, geometry, and calculus one learns in school on the computer. As a simple example, one can represent the Earth's subsurface as a layered model where each layer is given a number describing how fast sound can propagate through the layer: call this list of say 250 numbers m. The recorded seismic reflections can be also be represented by a list of say 125 numbers that we'll label as d. The physics equation that describe how these two list of numbers are related is often described by a simple integral equation that gives rise to a matrix of numbers G (here with 250 columns and 125 rows). Producing an image of our layered Earth, m, from our seismic measurements, d, then boils down to solving the equation d = G m for m on the computer. There is a lot of wonderful and practical mathematics involved in accomplishing this task.
My experience of working in industry has shown me that collaborating effectively with other people who are well versed in other disciplines is an important skill to improve whenever possible, but especially while one is in school. I would recommend seeking out internships and summer jobs that provide opportunities to see how math is used everyday. I spent two summers while in college working in the Director's Summer Program at the National Security Agency at Ft. Meade, Maryland where a lot of interesting math is needed to effectively and efficiently process signals to find useful information. I also spent two summers working as an intern at Schlumberger while in graduate school and this also provided an excellent introduction to what a mathematical career in industry might be like.
If you are an undergraduate student who might be interested in graduate school, I would recommend looking into the many Research Experience for Undergraduate programs held at several universities. I participated in one held during the summer at the University of Tennessee and learned how interesting working on mathematics that is not already printed in books can be.
If you are an undergraduate student who might be interested in studying abroad, I would recommend the Budapest Semesters in Mathematics program. Their signature problem solving course "Conjecture and Proof" and their courses on combinatorics showed me some of the rich diversity of cultures within the mathematics community. This awareness has served me well.