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SUMMER@ICERM: 2013 Undergraduate Summer Research Program

Faculty: Lead a S@I program!

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

2013 Organizing Committee:

Summer@ICERM 2013
Funding Includes*
  • $3,000 stipend
  • Support for 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)

What to Expect

Brown Students: UTRA Instructions
Frequently Asked Questions
View Summer@ICERM 2012


[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.



  • Tarik Aougab *
    (Yale University)
  • Eric Chen
    (Princeton University)
  • Matthew Cole
    (University of Notre Dame)
  • Diana Davis *
    (Brown University)
  • Benjamin DeMeo
    (Williams College)
  • Kelsey DiPietro
    (University of Illinois)
  • Emily Fischer
    (Harvey Mudd College)
  • Stephanie Ger
    (Boston College)
  • Chaim Goodman-Strauss
    (University of Arkansas)
  • Ryan Greene *
    (Ohio State University)
  • Nicholas Lourie
    (Brown University)
  • Nakul Luthra
    (Brown University)
  • Cristian Munteanu
    (Jacobs University)
  • Jenny Rustad
    (Luther College)
  • Alex St. Laurent
    (Brown University)
  • Xidian Sun
    (Wabash College)
  • Sergei Tabachnikov
    (Pennsylvania State University)
  • Ananya Uppal
    (University of Illinois at Urbana-Champaign)
  • Kamron Vachiraprasith
    (Brown University)
  • Yuwen Wang
    (Swarthmore College)
  • Zijian Yao
    (Brown University)

First Week Outline
Monday, June 17th: ICERM Orientation, Scavenger Hunt & Welcome Lunch from 10AM-2PM
Monday, June 17th: "Piecewise Circular Space Curves" from Professor Tom Banchoff, Brown University (2PM)
Tuesday, June 18 - Friday, June 21: Research Topics Introductions, Times TBA
Tuesday, June 18: BearCore Training from 8:30AM-12:30PM (Smith Buonanno)
Students will be assigned to their Research Groups by end of the first week


Weekly Events Outline
Every Monday, Tuesday, Thursday & Friday
Meetings between research groups and supervisors
(See Research Group Meeting Schedule, TBA)

Working Seminars will occur at 10:30 AM every Wednesday

Wednesdays will also be reserved for any special talks or unscheduled meetings
Coffee & Tea Break at 3:00-3:30, Monday - Friday


Mini-Course Schedule
"Introduction to Complex Dynamics" led by Sarah Koch, Harvard University
Mon 7/1, Tu 7/2, Wed 7/3, (10:00 - 2:00)

SVG Training Sessions led by Pat Hooper, City College of New York
Mon 7/15, Tu 7/16, Th 7/18, Fr 7/19, (9:30 - 11:00)

Mini-series to be scheduled, one or two day lectures


Special Events (with more to be scheduled)
6/17: "Piecewise Circular Space Curves" from Tom Banchoff, Brown University (2:00 PM)

6/26: "Letting Loops Loose" from Yuliy Baryshnikov, University of Illinois Urbana-Champaign (1:00 PM)

7/24: Title TBA, from Jill Pipher, ICERM (1:00 PM)

7/31: "Snakes and Ladders" from Bjorn Sandstede, Brown University (1:00 PM)

2013 UTRA symposium: Thursday, August 1 from 11:30 am to 2:00 pm at Sayles Hall

Group Talks on Research: Monday, August 5 - Thursday, August 8

Closing Reception: August 9th, Time TBA


MondayJune 17, 2013
TimeDescriptionSpeaker LocationAbstractsSlides
10:00 - 10:30Welcome/Meet Greet121 S. Main St., 11th Floor
10:30 - 11:45Scavenger Hunt
12:00 - 1:30Welcome Lunch11th Floor Conference Room
2:00 - 3:00Piecewise Circular Space CurvesTom Banchoff, Brown University 11th Floor Lecture Hall
3:00 - 5:00Research Topic Introductions 11th Floor Lecture Hall

Tuesday June 18, 2013
TimeDescriptionSpeakerLocation AbstractsSlides
8:30 - 12:30BEARCORE Training Smith Buonanno Hall, 95 Cushing St.
12:30 - 1:30Break for Lunch
1:30 - 3:00Research Topic Introductions 11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break 11th Floor Collaborative Space

WednesdayJune 19, 2013
TimeDescriptionSpeaker LocationAbstractsSlides
9:00 - 12:00Research Topic Introductions 11th Floor Collaborative Space
2:30 - 3:00IT TutorialIsani Cayetano, ICERM11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break 11th Floor Collaborative Space

ThursdayJune 20, 2013
TimeDescriptionSpeaker LocationAbstractsSlides
9:00 - 12:00Research Topic Introductions 11th Floor Lecture Hall
3:00 - 3:30 Coffee/Tea Break11th Floor Collaborative Space

FridayJune 21, 2013
TimeDescription SpeakerLocationAbstractsSlides
9:00 - 12:00Research Topic Introductions11th Floor Lecture Hall
1:30 - 3:00Form Research Groups11th Floor Lecture Hall
3:00 - 3:30 Coffee/Tea Break11th Floor Collaborative Space

MondayJune 24, 2013
TimeDescription SpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
2:30 - 3:00Coffee/Tea Break 11th Floor Collaborative Space
3:00 - 4:00 Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan
3:00 - 4:00Research Group Meeting: Negative Snell Supervisors: Chaim, Diana

TuesdayJune 25, 2013
TimeDescription SpeakerLocation AbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise Circular Supervisors: Sergei, Diana, Ryan
1:30 - 2:30Research Group Meetings: Hyperbolic Billiards 1 Supervisors: Chaim, Tarik
2:30 - 3:00Coffee/Tea Break 11th Floor Collaborative Space
3:00 - 4:00 Research Group Meeting: Hyperbolic Billiards 2 Supervisors: Ryan, Tarik

WednesdayJune 26, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:00 - 2:00Letting Loops LooseYuliy Baryshnikov, University of Illinois Urbana-Champaign 11th Floor Lecture Hall
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

ThursdayJune 27, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:30Meet at ICERM for AMS Field Trip
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana

FridayJune 28, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Hyperbolic Billiards 1Supervisors: Chaim, Tarik
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic Billiards 2Supervisors: Ryan, Tarik

MondayJuly 1, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 12:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
PDF
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

TuesdayJuly 2, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 12:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
PDF
1:30 - 2:30Research Group Meetings: Hyperbolic Billiards 1Supervisors: Chaim, Tarik
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic Billiards 2Supervisors: Ryan, Tarik

WednesdayJuly 3, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 12:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
PDF
2:30 - 3:30Meet and Greet with the French Students11th Floor Collaborative Space
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

ThursdayJuly 4, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
8:00 - 5:00ICERM Closed

FridayJuly 5, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Hyperbolic Billiards 1Supervisors: Chaim, Tarik
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic Billiards 2Supervisors: Ryan, Tarik

MondayJuly 8, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

TuesdayJuly 9, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic Billiards Supervisors: Ryan, Tarik, Chaim

WednesdayJuly 10, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
7:00 - 9:30Summer@ICERM Math Movie Night11th Floor Lecture Hall

ThursdayJuly 11, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana

FridayJuly 12, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

MondayJuly 15, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00The JAVA programming language.Pat Hooper, City College of New York10th Floor Classroom
11:00 - 12:00Progress Seminar11th Floor Lecture Hall
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

TuesdayJuly 16, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Mathematical graphicsPat Hooper, City College of New York10th Floor Classroom
11:00 - 12:00Progress Seminar11th Floor Lecture Hall
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic Billiards Supervisors: Ryan, Tarik, Chaim

WednesdayJuly 17, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space

ThursdayJuly 18, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Creating interactive programsPat Hooper, City College of New York10th Floor Classroom
11:00 - 11:25Progress Seminar: Constant Curvature10th Floor Classroom
11:25 - 11:50Progress Seminar: Negative Snell10th Floor Classroom
11:50 - 12:15Progress Seminar: Piecewise Circular10th Floor Classroom
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana

FridayJuly 19, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Sharing programs with othersPat Hooper, City College of New York10th Floor Classroom
11:00 - 11:25Progress Seminar: Hyperbolic Billiards10th Floor Classroom
11:25 - 11:50Progress Seminar: Nonconvex Tripod10th Floor Classroom
11:50 - 12:15Progress Seminar: Equiangular Chords10th Floor Classroom
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

MondayJuly 22, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:30Presentations10th Floor Classroom
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

TuesdayJuly 23, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

WednesdayJuly 24, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
11:45 - 12:45Lunch with the REUF Workshop11th Floor Collaborative Space
1:00 - 2:00Lattices and Public Key CryptographyJeff Hoffstein, Brown University10th Floor Classroom
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
6:00 - 8:00Game Night

ThursdayJuly 25, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

FridayJuly 26, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break10th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

MondayJuly 29, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan

TuesdayJuly 30, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

WednesdayJuly 31, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:30 - 11:30Graduate Student PanelSergei Tabachnikov, Penn State and ICERM; Amalia Culiuc, Brown University; and Ian Alevy, Brown University11th Floor Lecture Hall
1:00 - 2:00Spiral waves: from experiments to theoryBjorn Sandstede, Brown University11th Floor Lecture Hall
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
6:00 - 8:00Summer@ICERM Game Night11th Floor Lecture Hall

ThursdayAugust 1, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meeting: Equiangular ChordsSupervisors: Sergei, Tarik
1:30 - 2:30Research Group Meeting: Constant CurvatureSupervisors: Chaim, Diana
2:30 - 2:45Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Nonconvex TripodSupervisors: Sergei, Ryan
3:00 - 4:00Research Group Meeting: Negative SnellSupervisors: Chaim, Diana

FridayAugust 2, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
1:30 - 2:30Research Group Meetings: Piecewise CircularSupervisors: Sergei, Diana, Ryan
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space
3:00 - 4:00Research Group Meeting: Hyperbolic BilliardsSupervisors: Ryan, Tarik, Chaim

MondayAugust 5, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

TuesdayAugust 6, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

WednesdayAugust 7, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
11:30 - 12:30Summer@ICERM Lunch with Directors11th Floor Conference Room
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

ThursdayAugust 8, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:45Student Presentations: Tripod 11th Floor Lecture Hall
11:00 - 11:45Student Presentations: Equiangular Chords11th Floor Lecture Hall
12:00 - 1:30Break for Lunch
1:30 - 2:50Student Presentations: Outer Billiards11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space
3:30 - 4:15Student Presentations: Constant Curvature11th Floor Lecture Hall

FridayAugust 9, 2013
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:45Student Presentations: Piecewise Circular11th Floor Lecture Hall
11:00 - 11:45Student Presentations: Negative Snell11th Floor Lecture Hall
12:00 - 2:00Break for Lunch
2:00 - 2:30Closing Remarks & Refreshments11th Floor Lecture Hall and Collaborative Space



A paper resulted from the study of curves and polygons:
A paper resulted from the study of the periodicity of outer billiards in the hyperbolic plane:
A paper resulted from the study outer billiards:

Participant quote:

The program exceeded my expectations by any standard I can think to apply -- the students were by and large strong and motivated, the ones I interacted with regularly were clearly learning quite a bit, the people involved in organizing the program were a pleasure to work with, and the outcomes in terms of research have been impressive. It was both fun and rewarding to be involved in this program.

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:

Billiards and related systems

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).

Geometry

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).

Quasi-periodic structures

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).