"Ginzburg–Landau Functionals in the Large-Graph Limit"

Another of my papers has appeared in final form. Here are some details.

Title: Ginzburg–Landau Functionals in the Large-Graph Limit

Authors: Edith J. Zhang, James Scott, Qiang Du, and Mason A. Porter

Abstract: Ginzburg–Landau (GL) functionals on graphs, which are relaxations of graph-cut functionals on graphs, have yielded a variety of insights in image segmentation and graph clustering. In this paper, we study large-graph limits of GL functionals by taking a functional-analytic view of graphs as nonlocal kernels. For a graph W_n with n nodes, the corresponding graph GL functional GL_ϵ^{W_n} is an energy for functions on W_n. We minimize GL functionals on sequences of growing graphs that converge to functions called graphons. For such sequences of graphs, we show that the graph GL functional Γ-converges to a continuous and nonlocal functional that we call the graphon GL functional. We investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total-variation (TV) functional. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the minimization problem in the large-graph limit. Finally, to develop intuition about graphon GL functionals, we determine the GL minimizer for several example families of graphons.

A Non-Maleficence Aphorism for Mentorship: "First, Don't be an Asshole"

A Non-Maleficence Aphorism for Mentorship: "First, don't be an asshole." pic.twitter.com/4ne87JmG4K

— Mason Porter (@masonporter) October 22, 2017

Congratulations to Hall Daily on his Retirement!

Hall Daily, Caltech's Director for Government Relations, is retiring after a long tenure on the job.

Back in my undergrad era, Hall Daily was also the advisor for The California Tech (our student newspaper), of which I was a writer and co-editor. I learned a ton about journalism from Hall. He too someone who was someone who Autumn Looijen and I purposely earmarked as getting one of our free copies of Legends of Caltech III. He was an awesome advisor, and I owe a lot to him for my communication and journalistic skills. My experience with The Tech was probably more important for my career than any math or science class I ever took.

Bringing Mathematics to the Traumatized and the Perplexed

As many of you know, Steve Strogatz is one of my mathematical heroes. One of the reasons is that he's a master teacher and expositor. (Whenever I get his seal of approval for one of my articles or other efforts, I always feel like I did a good job on it.) Take a look at this new article.

Quoting Steve from this article: "So many of the things that we do in math education—and maybe more generally in education—are giving students answers to questions that they would never think of asking. By definition, that's what it is to be boring."

Conversely, this also speaks to why so many students find networks exciting from the start. They're already asking many of these questions! They just need the words and concepts to make the questions more precise to be able to answer them in a better way.

Unearthed: A Report I Wrote on the KdV Equation During College

Oh, wow. From some 'random' googling, I just found the rough draft of a report on the KdV equation that I wrote my senior year in college. This apparently got uploaded when various past things from courses that Jerry Marsden taught were collected, scanned, and uploaded.

My exposition (including of the KdV equation!), my citation practices and being very thorough about giving proper credit for results and the exposition of others (right now, I think that my old report should have included specific statements of whose discussions I was following and not just citations to those papers), and my work on nonlinear waves has advanced a lot since that report -- though I think that one can see many hints of my current style in it. Certainly one can see some of my current research interests. I had completely forgotten about this old report. (My handwriting in parts of the report is almost exactly the same as now, though! :P)

If you want to compare my old exposition to my current exposition, see the Scholarpedia entry on solitons that I wrote with Norm Zabusky.

I should perhaps use this as before/after to show my students for certain assignments. I think this could be helpful for some interesting teaching opportunities.

I'm not sure why the rough draft was uploaded, though. Maybe Jerry's copy of the final draft got lost?

RIP Jerry Marsden (1942-2010)

I just received an e-mail (via Caltech's Control & Dynamical Systems mailing list, which is probably the one list where I still get relevant e-mails via my UGCS address) that Jerry Marsden died of cancer last night. I knew he was sick, but apparently things got extremely bad lately.

Here is what I just wrote as my Facebook status: RIP Jerry Marsden. Your supervision of my first research project, excellent lectures, and tons of recommendation letters, and much more meant so much to my career. (Not the e-mail I wanted to get...)

This bothers me a lot. Jerry was one of my biggest mentors, especially in the early stages. (I ultimately decided that my research style and preferences were different from his, but of course we all ultimately decide our path based on our various experiences. This is a sad, sad day.)

9/23/10: One of my collaborators has posted an appropriate quote from MacBeth, and I'll just assume (though perhaps incorrectly... I didn't check and I don't plan to ask) that it's for the same piece of news as reported in this blog entry. I knew that Jerry's death would bother me, but it's definitely hit me harder than I was expecting. Our relationship was entirely professional, but that is meaningful all the same.

9/24/10: Here is (a version that had some 1997 modifications of) my SURF Report from my project with Jerry.

9/26/10: There is now a remembrance page set up at Caltech.

10/01/10: Caltech has now posted an obituary.