Derek L. Smith

• F. Linares, G. Ponce, D. L. Smith, On the regularity of solutions to a class of nonlinear dispersive equations, submitted for publication October 2015. arXiv:1510.02512
• J. Segata, D. L. Smith, Propagation of regularity and persistence of decay for fifth order dispersive models. Journal of Dynamics and Differential Equations, arXiv:1502.01796

My dissertation research focuses on smoothing properties of solutions to dispersive nonlinear partial differential equations. For the Korteweg-de Vries (KdV) equation, it was recently shown that appropriate regularity present in the initial data on the positive, or right, half-line travels to the left with infinite speed. In collaboration with J. Segata, we extended this propagation of regularity phenomenon to equations in the KdV hierarchy. However, propagation of regularity does not depend on the equation being completely integrable or even having constant coefficients. In a joint work with F. Linares and G. Ponce, we demonstrated the result for a family of quasilinear KdV-type equations. More generally, smoothing effects play a crucial role in the well-posedness and control theory of nonlinear dispersive equations, topics which I intend to pursue in future research. I have also identified potential research projects related to my dissertation work which are suitable for undergraduates.

I will speak about the quasilinear results during the upcoming Joint Mathematics Meetings in Seattle.

My teaching philosophy is encapsulated by the following definition:

Learning is the iterative process of asking and answering questions.

In my teaching and service I aim for three goals, each of which addresses an aspect of this definition. First, I share problems--both mathematical and applied--which build intuition and inspire students to integrate the course material into the broader context of their education, to convey the ``why". Second, I structure my courses to encourage students with a diverse array of learning styles to spend the time necessary to master computational aspects of the material, to absorb the ``how". Finally, I strive to build accessible interdisciplinary mentoring networks, in particular through my position in the UCSB Student Chapter of the SIAM. These goals are shaped by having instructed three classes at UCSB:

• Spring 2013 - Math 6A Vector Calculus (129 students, survey)
• Summer 2014 - Math 6A Vector Calculus (74 students, survey)
• Summer 2015 - Math 6B Vector Calculus and Intro to PDEs (37 students, survey)

As detailed by the survey results, 69-87% of students in each course responded with the top rating when asked to ``rate the overall quality of the instructor's teaching." This compares with the department average over time of 42% for graduate student instructors.

Taking advantage of my software development background, I have collected problems with a familiar quality around which I could build a compelling mathematical narrative. For instance, describing touchscreen interaction as an application of the geometry of curves in the plane, discussing a self-driving car to introduce constrained optimization and highlighting the use of linear algebra in data science. I package these problems as demonstration modules to build intuition and encourage small group experimentation. An example can be found in this handout discussing the Laplacian matrix of a social network.

The following is a collection of notes and slides from talks.

• The Fast Fourier Transform - Talk given in our SIAM Student Chapter's seminar on algorithms in computational science. Contains basic explanation of MP3 compression algorithm.
• Smoothing Effects for Linear PDEs - A talk given to introduce fellow graduate students to my dissertation research. Portions of this talk could easily be adapted to an advanced undergraduate level.
• Parabolic Smoothing for the Linear Heat Equation - The linear heat equation provides a canonical example of a smoothing effect for a solution to an evolution equation. These notes describe how, for positive times, solutions to the heat equation possess infinitely many derivatives. In fact, three proofs are given.
• Exact Control of the Linear Korteweg-de Vries Equation - Talk given in the GSCubed engineering seminar. This talks also describes an interesting, though not directly related problem, of data assimilation for weather models. One of the main reasons US weather models have not kept pace is the superior data assimilation schemes of the European models.