The department has a diverse mix of research active faculty members in the most active phases of their careers. Faculty research falls into a variety of areas, including partial differential equations, dynamical systems, mathematical neuroscience, numerical analysis, approximation theory, combinatorics, functional analysis, representation theory, geometry and topology, matrix and operator theory, inverse problems, and stochastic analysis.
The department hosts several vibrant research seminars, as well as a colloquium series, in which experts from both inside and outside of Drexel communicate novel mathematical ideas. In addition, we have a Distinguished Visitor Lecture Series, where annually we invite a top expert in an active research area. Our research has been supported by government agencies such as the National Science Foundation and the National Security Agency.
The abstract mathematical training of our research faculty members allow them to apply mathematical techniques to diverse application areas ranging from control theory, mirror design, computer graphics, data compression, fluid mechanics and modeling of biological phenomena. To learn more about the research in our department, please visit the Faculty Research page.
The mathematical structures that we study typically have their origins in very applied problems, such as:
- Why is it possible to filter out the 'noise' in a digital signal or image while preserving the important features in the signal? Why is it possible to compress a 5 megabyte high resolution digital photo all the way down to 300 K, without losing much of the visual perceptions?
- With the advance of nanotechnology, it is increasingly important to be able to simulate quantum effects on computers. Can it be done as accurately and efficiently as the way we simulate fluid flows around a submarine or an airfoil?
- Why is it so hard to forecast weather accurately for just a few days, despite all the advances in satellite technologies?
- However, why are satellites and GPSs doing so well in guiding almost every single driver on the charted map?
- Why is it possible to efficiently compute certain eigenvectors of the gigantic – and time-varying -- matrix modeling the whole Internet, thus allowing us to search almost anything we need (and not really need) in a matter of seconds?
- However, why can't any computer program read distorted text, while a human can do so almost effortlessly?
- Can we quantify the complicated dynamical behavior of human neurons?
- Can we decide whether there is oil beneath the water and rocks somewhere on this planet, without first spending millions of dollars to drill a hole only to find out that we drill at the wrong place?
This list of questions, while far from exhaustive, motivate the fundamental mathematical questions studied in our department. For instance, the study of partial differential equations provides understanding to fluid, materials and wave phenomena.
The study of number theory, algebra, and combinatorics have applications in cryptography and in network engineering.
The study of geometry and topology is crucial, among many other things, for the cross-fertilization of physics and mathematics.
The quantitative modeling of 'noise' and 'regularity', and the associated mathematical analysis, are the key to efficient signal compression, denoising and data analysis. The detailed mathematical study of dynamical systems and ordinary differential equations play a fundamental role in complex systems arising from neuroscience, climate modeling and financial mathematics. Studies in functional and harmonic analysis shed important lights on solving inverse problems arising from remote sensing, medical imaging and oil exploration.