SUMMER@ICERM: Undergraduate Summer Research Program

Geometry and Dynamics
June 18 - August 10, 2012
Organizing Committee

Funding Includes*
  • $3,000 stipend
  • Support towards US travel
  • Dormitory housing
  • Meal plan
  • Access to fun local events

Images

[Image: R. Schwartz]

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 the fields of low dimensional geometry and dynamical systems.

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 Now, imagine having this experience with all expenses for travel, room and board paid, plus a $3,000 stipend.

The 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 a faculty advisor and aided by a teaching assistant. The faculty advisors will describe a variety of enticing open questions in geometry and in dynamical systems of geometric origin. Topics discussed will include Euclidean, hyperbolic and projective geometry, iteration of geometric constructions, and mathematical billiards. A variety of activities around these research themes will allow participants to engage in collaborative research, communicate and examine their findings in formal and informal settings, and report-out their findings with a finished product.

Applications for the 2012 Summer@ICERM program have closed. Applicant notifications will be emailed by or around March 15, 2012.

*Accepted Brown University UTRA students receive only a $3,000 stipend.


Topics for Research Projects for SUMMER@ ICERM

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

Problems have been 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 preserving the local metric structure of the plane.

Participants in this program 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, participants could investigate the possible paths the two 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?

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

The history of algebraic geometry is full of beautiful elementary theorems. Participants may investigate still unknown aspects of theorems such as Poncelet's Porism, Desargues Theorem, and Pascal's Theorem. These theorems expose surprising structure coming from easy to describe constructions. We hope participants will prove new theorems about these topics and further our understanding of these important results.

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