Undergraduate and Graduate Research Opportunities

I have projects for undergraduate and graduate students in the areas of partial differential equations, numerical analysis, and optimization.

Specific projects include:
How can we quantify implicit biases in letters of recommendation?
How can we numerically simulate swarms of robots and colonies of slime mold?
What happens in the limit of very, very slow diffusion? Can we use this to design new numerical schemes for pedestrian crowds?

Teaching

During spring quarter 2017, I am teaching Math 117, Methods of Analysis.
Congratulations to the winners of the extra credit math movie competition!
You can find links to their excellent videos here: 1st place, 2nd place, 3rd place, honorable mention.

On my curriculum vitae, I link to material from recent courses I have taught.


Research Program

I am interested in nonlinear PDE, optimal transport, calculus of variations, and numerical analysis.

I received my Ph.D. at Rutgers University, where my advisor was Eric Carlen. I spent the first year of my NSF postdoc at UCLA, where my postdoctoral mentor was Andrea Bertozzi. I then spent a year at UCSB as a UC President’s Postdoctoral Fellow, under the mentorship of Bjorn Birnir. I am now an assistant professor at UCSB.

On my curriculum vitae, I link to slides from recent talks I have given.


Publications and Preprints

1) K. Craig, I. Kim, and Y. Yao, Congested aggregation via Newtonian interaction, arXiv: 1603.03790, accepted to Archive for Rational Mechanics and Analysis.

2) K. Craig,
Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions, arXiv:1512.07255, Proc. London Math. Soc., (2017), no. 114, 60–102.

3) K. Craig and I. Topaloglu,
Convergence of regularized nonlocal interaction energies, arXiv: 1503.04826, SIAM J. Math. Anal. 48 (2016), no. 1, 34-60.

4) K. Craig and A. Bertozzi,
A blob method for the aggregation equation, arXiv:1405.6424, Math. Comp. 85 (2016), no. 300, 1681-1717.

5) K. Craig,
The exponential formula for the Wasserstein metric, arXiv:1310.2912, ESAIM COCV 48 (2016), no. 1, 169-187.

6) E. Carlen and K. Craig,
Contraction of the proximal map and generalized convexity of the Moreau-Yosida regularization in the 2-Wasserstein metric, arXiv: 1205.6565, Math. and Mech. of Complex Systems 1 (2013), no. 1, 33-65.


In Preparation

  1. 1)J.A. Carrillo, K. Craig, and F. Patacchini, A blob method for degenerate diffusion

  2. 2)K. Craig and I. Topaloglu, A numerical method for height constrained aggregation, with applications to nonlocal shape optimization.


Thesis

  1. K.Craig, The exponential formula for the Wasserstein metric




 
Simplify!http://stanfordchaparral.com/magazine/simplifyhttp://stanfordchaparral.com/magazine/simplifyshapeimage_2_link_0