SUMMER@ICERM: 2013 Undergraduate Summer Research Program

Geometry and Dynamics (June 17 - August 9, 2013)

 

2013 Organizing Committee:

Summer@ICERM 2013
Funding Includes*
  • $3,000 stipend
  • Support towards travel within the U.S.
  • Dormitory housing
  • Meal plan
  • Fun activities
*Brown University students must also apply for an UTRA in order to be eligible to receive the $3,000 stipend (in lieu of the Summer@ICERM funding package)

Images
[Images: C. Goodman-Straus]

'Geometry and Dynamics' was the topic of conversation for these
2012 Summer@ICERM undergraduate researchers and
organizers Sergei Tabachnikov and Patrick Hooper.


Imagine spending eight-weeks on the beautiful Brown University campus in historic Providence, RI, working in a small team setting to solve mathematical research problems developed by faculty experts in their fields.

Imagine creating career-building connections between peers, near peers (graduate students and postdocs), and academic professionals.

Imagine spending your summer in a fun, memorable, and intellectually stimulating environment.

Now, imagine having this experience with support for travel within the U.S., room and board paid, plus a $3,000 stipend*.



The 2013 Summer@ICERM program is designed for a select group of 10-12 undergraduate scholars. Students will work in small groups of two or three, supervised by faculty advisors and aided by teaching assistants. The faculty advisors will provide a list of interesting open problems in geometry and in dynamical systems. These problems include Euclidean, hyperbolic and projective geometry, iteration of geometric constructions, and mathematical billiards. Students will form research groups to work on the problems, will give talks about their findings, and will write up their research into a paper at the end of the program.


Applications for the 2013 Summer@ICERM are no longer being accepted. Please check our website and MathPrograms.org in the fall of 2013 for next year's program details.

Applicants must not yet have graduated with an undergraduate degree at the time of the program start. Funding is available for a limited number of students who are not US citizens or permanent residents.


 

Topics for Research Projects for SUMMER@ ICERM 2013

In this Research Program, the faculty organizers will stress open questions which are accessible via mathematical methods available to undergraduates. In addition, participants are expected to be able to gain intuition into some of these problems via computer experimentation and visualization. The complete problem list for the 2013 program is available as a PDF document. PDF

Problems were selected in each of three categories. The full problem list is given at the bottom of the page. The following is a brief introduction to the three categories of problems:

Dynamical systems associated to billiards are of interest in physics, geometry and dynamics. These billiard systems are idealized versions of the physical system. For instance, we always play on a frictionless table, and we assume that both momentum and energy are conserved in the system.

Mathematicians have become interested in variations of billiard systems. For instance, outer billiards is a dynamical system in which points orbit around a convex object according to a simple geometric rule, and magnetic billiards describe the motion of a charge in a magnetic field, subject to elastic collision with the boundary of the domain.

The participants will help make progress on open questions about billiards and related dynamical systems. For instance, in many cases the structure of periodic orbits of these systems is not well understood. It is expected that elementary approaches to these problems will help to make progress in these topics, and to expose new mathematical phenomena.

The problem list is available as a PDF document (see above).

Physical objects and systems represent a natural source for interesting mathematics. For instance, the participants could investigate the possible paths the wheels of a bicycle can take. Another possibility is to investigate the behavior of folded paper. How can we understand the geometry of the folded piece of paper given the folding pattern? Another intriguing question is how to model a surface of negative curvature in 3-dimensional space. These and other problems are offered under the geometry rubric.

The problem list is available as a PDF document (see above).

We offer a variety of research projects on quasi-periodic tilings and quasi-periodic structures. Aperiodic tilings were discovered by mathematicians in the early 1960s. These structures have symmetry different from the familiar crystallographic ones, that is, symmetry of orders other than two, three, four, or six. For example, the famous Penrose tiling, discovered in 1976, has fivefold symmetry. The discovery of these aperiodic forms in nature in the 1980s was awarded Nobel Prize in Chemistry (Dan Shechtman, 2011).

There are numerous ways of generating quasi-periodic structures, and their study is related to many areas in mathematics: Euclidean and non-Euclidean geometry, Fourier transform, dynamical systems, complexity theory, mathematical logic, and others. One common feature is that the objects of study are visually stunning.

The problem list is available as a PDF document (see above).