## Homework

Homework will be due roughly every other Friday, according to the schedule below. Each homework will consist of several problems, but students are only required to submit solutions to one problem.

Number |
Due Date |
Questions |
Solutions |

1 | Sep 21 | .pdf | .tex | |

2 | Oct 5 | .pdf | .tex | |

3 | Oct 19 | .pdf | .tex | |

4 | Nov 2 | .pdf | .tex | |

5 | Nov 16 | .pdf | .tex |

## Suggested papers for projects

Below is a list of suggested papers for term-end presentations.

- Kohn, Robert, and Sylvia Serfaty. "A deterministic‐control‐based approach motion by curvature." Communications on Pure and Applied Mathematics. 59.3 (2006): 344-407. [.pdf]
- Manfredi, Juan, Mikko Parviainen, and Julio Rossi. "An asymptotic mean value characterization for p-harmonic functions." Proceedings of the American Mathematical Society 138.3 (2010): 881-889. [.pdf]
- Levine, Lionel, Wesley Pegden, and Charles K. Smart. "Apollonian structure in the Abelian sandpile." Geometric and functional analysis 26.1 (2016): 306-336. [.pdf]
- Also, see the prior work: Pegden, Wesley, and Charles K. Smart. "Convergence of the Abelian sandpile." Duke Mathematical Journal 162.4 (2013): 627-642. [.pdf]
- Bardi, Martino, and Lawrence C. Evans. "On Hopf's formulas for solutions of Hamilton-Jacobi equations." Nonlinear Analysis: Theory, Methods & Applications 8.11 (1984): 1373-1381. [.pdf]
- Ishii, Hitoshi, and Paola Loreti. "Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems." SIAM Journal on Mathematical Analysis 37.2 (2005): 411-437.[.pdf]
- Evans, Lawrence C., H. Mete Soner, and Panagiotis E. Souganidis. "Phase transitions and generalized motion by mean curvature." Communications on Pure and Applied Mathematics 45.9 (1992): 1097-1123.[.pdf]
- Cao, Frédéric. "Partial differential equations and mathematical morphology." Journal de mathématiques pures et appliquées 77.9 (1998): 909-941.[.pdf]
- Barles, Guy, and Christine Georgelin. "A simple proof of convergence for an approximation scheme for computing motions by mean curvature." SIAM Journal on Numerical Analysis 32.2 (1995): 484-500.[.pdf]
- Oberman, Adam M. "A convergent monotone difference scheme for motion of level sets by mean curvature." Numerische Mathematik 99.2 (2004): 365-379.[.pdf]
- Deckelnick, Klaus, and Charles M. Elliott. "Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities." Interfaces and free boundaries 6.3 (2004): 329-349.[.pdf]
- Zhao, Hongkai. "A fast sweeping method for eikonal equations." Mathematics of computation 74.250 (2005): 603-627.[.pdf]
- Sethian, James A. "A fast marching level set method for monotonically advancing fronts." Proceedings of the National Academy of Sciences 93.4 (1996): 1591-1595.[.pdf]

## Writing Mathematics

In mathematical and scientific writings, equations should occur naturally within full sentences and flow with the surrounding text. The goal is to communicate your mathematical reasoning to another person, and this usually requires more than just a string of equations. It is strongly recommended that students practice this style of writing in their homework. The homework solutions posted on this website can be used as examples, and a some good resources are

- Kevin P. Lee's A Guide to Writing Mathematics
- Michael Hutchings' Introduction to Mathematical Arguments (for writing proofs).

## LaTeX

A great way to write mathematics electronically is to use LaTeX. LaTeX is widely used for the publication of scientific documents in many fields, including mathematics, physics, computer science and engineering. Some good resources for getting started with LaTeX are

- Information on LaTeX, MiKTeX, etc.
- Getting MiKTeX.
- The not so short introduction to LaTeX by Tobias Oetiker.
- Wikibook on LaTeX.

For preparing LaTeX documents, I use the command line editor vim, along with vim-LaTeX, and a PDF-viewer that automatically updates when the pdf file changes (e.g., Skim on Mac, or Evince in Linux). I have used this setup on both Mac OSX and Linux operating systems. Beware that while vim is a powerful text editor, it has a very steep learning curve.