# Summer Program for Undergraduate Research (SPUR) — Summer 2016

For a sample of students’ work on previous research projects, including the ongoing *Analysis on Fractals* project, see Undergraduate Research Programs.

### PROJECT 1: Analysis on Fractals, directed by Robert Strichartz

Students in this project will study properties of functions defined on fractals. For certain fractals, including the Sierpinski gasket, the Sierpinski carpet, and some of the classical Julia sets, there is now a theory of “differential equations.” (See my book, *Differential Equations on Fractals, a tutorial*, Princeton University Press, 2006.) One of the goals of this project is to obtain more information about solutions of these fractal differential equations, following up on work that has been done by past REU students. Most of the work on this project will involve both computer experimentation and theoretical study, but individual students may put more emphasis on one or the other. We expect that students will be involved in all stages of the process: planning what examples to study, doing the programming for the computations, and interpreting the results (and attempting to prove the conjectures that come out of the process).

### PROJECT 2: Topological methods in discrete geometry, directed by Florian Frick

This project will focus on the interplay among combinatorics, discrete geometry, and algebraic topology. Given a convex body in $\mathbb{R}^4$ can it be partitioned into sixteen equal-volume pieces by four affine hyperplanes? Given three red, three green, three blue, and one yellow point in the plane, can these points always be partitioned into four sets with no repeated colors whose convex hulls intersect? These are two of many "simple to state" but open questions in discrete geometry. We will explore a multitude of possible approaches to such intersection and partition results including elementary combinatorial arguments as well as more sophisticated topological methods. The topological methods usually yield continuous generalizations of these convex-geometric results. Recent developments have shown that there are important differences between the affine world of discrete geometry and the continuous world of the corresponding topological generalizations. We will further explore this affine-continuous dichotomy and the role of primes and prime powers in delimiting these two worlds from one another.

Some of these topics could benefit from computer experiments. Prior knowledge of topology is not strictly required. Students who enjoy combinatorial reasoning or want to learn (more) about techniques from algebraic topology are particularly encouraged to apply.

### PROJECT 3: Nonlinear heat equations, directed by Xiaodong Cao

Students in this project will study differential equations arising from differential geometry, also known as geometric flows. We will be focusing on some nonlinear heat equations, the Ricci flow on surface, curve shortening flow and the cross curvature flow. In most cases, these are partial differential equation systems, but if we are on a homogeneous manifold, they become ordinary differential equation systems. We will study both the analytic and geometric properties of such equations. Most of the work in this project will involve both computer experimentation and theoretical study. Individual students may choose to put emphasis on one or the other. Students with strong programming (such as Mathematica and Matlab) are encouraged to apply, but such background is not required.

**WHEN**: June 20 – August 12, 2016 (8 weeks)

**WHERE**: Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853-4201.

**SUPPORT**: Students not from Cornell will need to come with outside funding or be self-funded. Participants will arrange for their own housing; we will assist with local contact information.

**HOW TO APPLY**: Please submit an application (through the SPUR web site) that includes a statement about your background, educational goals, and your scientific interests. Include whatever further information you consider relevant. (Be sure to include information about your computer experience.) You will be able to upload letters of recommendation and transcripts (unofficial transcripts are accepted) via the SPUR website. Two letters of recommendation are required. Uploading your letters and transcripts is preferred and is the most efficient means of processing. The reference writer is notified to upload their letter once you enter their information on the application.

If necessary you can send letters and transcript (unofficial transcripts accepted) via email to mathreu@cornell.edu or mail them to the SPUR Program, Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853-4201. Please avoid sending duplicate copies of recommendation letters and transcripts.

**DEADLINE**: **February 25, 2016**. All materials must be received by this date. Late applications will not be accepted. You will receive notification some time in March. We will be adhering to the uniform first reply date for REU programs, so no student will be asked to reply to an acceptance offer before March 8.

If you have comments, questions, or concerns, please send e-mail to the SPUR coordinators at mathreu@cornell.edu.