I am inspired and motivated by the interaction of analysis and geometry. I also find applications to data modelling and analysis to be a very rich source of new ideas.
I was first introduced to geometric measure theory by David Caraballo, who I met at the 1999 joint meetings in San Antonio, in a short course on Nonlinear Control Theory. David Caraballo convinced me (he was the pied piper — I was completely swept away with his beautiful obsession) that GMT was the thing for me to study and I went home to Los Alamos and worked rather quickly through a majority of Evans and Gariepy’s Measure Theory and Fine Properties of Functions.
I was hooked.
At this same time I was beginning to work on image analysis problems related to very sparse tomography problems. The image analysis, that ended up drawing me more deeply into the theory of sets of finite perimeter, also drew Bill Allard into collaborations with me on data analysis problems. During a visit to Duke University hosted by Andrea Bertozzi, Andrea introduced me to Bill Allard and I invited him to Los Alamos to visit me. He began working with my team on the rather fascinating data analysis problems we had at the lab, while at the same time he began writing the definitive papers on the properties of minimizers of TV regularized image functionals. Over a period of several years, some of his deep knowledge in geometric measure theory began rubbing off on me. The 2008 move to academia and the resulting need to mentor students in the dual arts of (1) unravelling intricate mathematical problems and (2) developing a research vision, has deepened my focus on geometric analysis and geometric measure theory.
Geometric measure theory has a reputation for being difficult, but if you learn to translate difficult looking prose into pictures, using the lowest dimensional example that will work, the task is made easier, sometimes by a very large amount. Here are a couple of blog articles (you can find them here and here), the first introduces you to GMT reference books and the second opens up the world of geometric measure theory using the vehicle of the derivative.
If you are a potential graduate student and you are passionate about analysis and geometry, with an interest in interaction with data analysis applications, please contact me! If you have any questions about anything you read here feel free to email me.
You can find more information about classes and research and papers I have written, conferences organized, etc. by going to my geometricanalysis.org website. My contact information, including website and personal blog is listed here:
Photography for this page by Levi K. Vixie
Explaining 4.2.25 to Levi (and Obi) …
Here is the first of three videos containing a tutorial on metrics and regularization in image analysis. Lots of analysis, geometric analysis, image analysis. The tutorial was one of about 10 short courses at the 2005 Graduate summer School at the Institute for Pure and Applied Mathematics (IPAM) at UCLA.