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SUMMER@ICERM: 2012 Undergraduate Summer Research Program

Geometry and Dynamics
June 18 - August 10, 2012

 

SUMMER@ICERM: 2013

Organizing Committee

Funding in 2012 Included
  • $3,000 stipend
  • Support towards US travel
  • Dormitory housing
  • Meal plan
  • Access to fun local events

What to Expect
Frequently Asked Questions

[Image: R. Schwartz]

The Summer@ICERM program
is designed for a select group of 10-12 undergraduate scholars. Students in the 2012 program worked in small groups of two or three, supervised by a faculty advisor and aided by a teaching assistant. The faculty advisors described a variety of enticing open questions in geometry and in dynamical systems of geometric origin. Topics discussed included Euclidean, hyperbolic and projective geometry, iteration of geometric constructions, and mathematical billiards. A variety of activities around these research themes allowed 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.


Summer@ICERM 2012 Participants

Summer@ICERM 2012 Participants: Front row, l-r: Dmitri Gekhtman, Desislava Nikolov, Julienne LaChance, Clara Redwood, Amy Nesky. Middle row, l-r: Kevin Casto, Diana Davis, Pat Hooper, In-Jee Jeong, Ian Alevy. Back row, l-r: Tarik Aougab, Sergei Tabachnikov, Andrew Kimball, Emmanuel Tsukerman, Francisc Bozgan, Yilong Yang, Hai Bin Chang, Katherine Engelman



  • Ian Alevy
    (University of Chicago)
  • Tarik Aougab
    Teaching Assistant
    (Yale University)
  • Francisc Bozgan
    (UCLA, Rumania)
  • Kevin Casto
    (Brown University)
  • Hai Bin Chang
    (National University of Singapore)
  • Diana Davis
    Teaching Assistant
    (Brown University)
  • Katherine Engelman
    (Bryn Mawr College )
  • Dmitri Gekhtman
    (Harvard University)
  • In-Jee Jeong
    (Brown University)
  • Andrew Kimball
    (Western Carolina University)
  • Julienne LaChance
    (Rensselaer)
  • Amy Nesky
    (Boston College)
  • Desislava Nikolov
    (University of the Free State, South Africa)
  • Clara Redwood
    (Reed College)
  • Emmanuel Tsukerman
    (Stanford University)
  • Yilong Yang
    (Brown University)

First Week Outline
Monday, June 18th: ICERM Orientation and Welcome Lunch from 9AM-1PM
Tuesday, June 19 - Friday, June 22: Research Topics Introductions, Times TBA
Wednesday, June 20: BearCore Training from 8:30AM-12:30PM

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

Wednesdays will occur at 10:30 AM every Wednesday

Coffee & Tea Break at 3:00-3:30, Monday - Friday

Wednesday Special Talk Speakers
(every Wednesday at 2:00pm unless otherwise noted)

Rich Schwartz, Brown University

Govind Menon, Brown University

Jill Pipher, ICERM

Bjorn Sandstede, Brown University

Jeremy Kahn, Brown University

Mini-Course Schedule
Java Training Sessions led by Pat Hooper
Mon 6/25, Tu 6/26, Th 6/28, Fr 6/29 (9:30 - 11:00)

"Introduction to Complex Dynamics" led by Sarah Koch, Harvard University
Mon 7/2, Tu 7/3, Th 7/5, Fr 7/6 (10AM - 11AM)

"Introduction to Hyperbolic Geometry" led by Jeff Brock, ICERM
7/16-7/20 (10AM - 11AM)

Special Events
See schedule

Closing Reception: August 10th



MondayJune 18, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 9:30Summer@ICERM Non-Brown Student Meet & Greet
9:30 - 10:00Non-Brown Student Orientation
10:00 - 11:30All Summer@ICERM students orientation
11:30 - 1:00ICERM Welcome Lunch11th Floor Collaborative Space
1:00 - 4:00Introductions11th Floor Lecture Hall

TuesdayJune 19, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 12:00Research Topic Introductions11th Floor Lecture Hall
12:00 - 1:30Break for Free Lunch
1:30 - 3:00Research Topic Introductions11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

WednesdayJune 20, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
8:30 - 12:30BEARCORE TrainingMedical Education Building - 222 Richmond Street, Providence RI
2:30 - 3:00IT TutorialIsani Cayetano, ICERM11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

ThursdayJune 21, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 12:00Research Topic Introductions11th Floor Lecture Hall
1:30 - 3:00Problem Discussions11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJune 22, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:00 - 12:00Research Topic Introductions11th Floor Lecture Hall
1:30 - 3:00Form Research Groups11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJune 25, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00The JAVA programming languagePat Hooper, City College of New York11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJune 26, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Mathematical graphicsPat Hooper, City College of New York11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

WednesdayJune 27, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Working Seminar11th Floor Lecture Hall
11:00 - 12:30Square grids, double lattice PETS, and renormalizationRich Schwartz, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

ThursdayJune 28, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Creating interactive programsPat Hooper, City College of New York11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJune 29, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Sharing programs with othersPat Hooper, City College of New York11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJuly 2, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJuly 3, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space
ICERM Closed

ThursdayJuly 5, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJuly 6, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Introduction to Complex DynamicsSarah Koch, Harvard University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJuly 9, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJuly 10, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

WednesdayJuly 11, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 12:00Working Seminar11th Floor Lecture Hall
2:00 - 3:00Building polyhedra by self-assemblyGovind Menon, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

ThursdayJuly 12, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJuly 13, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJuly 16, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Introduction to Hyperbolic GeometryJeff Brock, ICERM, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJuly 17, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Introduction to Hyperbolic GeometryJeff Brock, ICERM, Brown University11th Floor Lecture Hall
3:00 - 3:30Yellow Pig Day Celebration11th Floor Collaborative Space

WednesdayJuly 18, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
2:00 - 3:00Geometric Discrepancy TheoryJill Pipher, ICERM11th Floor Lecture Hall
3:00 - 3:05Summer@ICERM Group Photo11th Floor Lecture Hall
3:05 - 3:35Coffee/Tea Break11th Floor Collaborative Space

ThursdayJuly 19, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
9:30 - 11:00Introduction to Hyperbolic GeometryJeff Brock, ICERM, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJuly 20, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:00Introduction to Hyperbolic GeometryJeff Brock, ICERM, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJuly 23, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:30Presentations11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJuly 24, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:30Presentations11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

WednesdayJuly 25, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
2:00 - 3:00Snakes and laddersBjorn Sandstede, Brown University, ICERM11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

ThursdayJuly 26, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 11:30Presentations11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayJuly 27, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
1:00 - 2:00Presentations11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayJuly 30, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

TuesdayJuly 31, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
2:00 - 2:20Converse Desargues TheoremKevin Castro, Brown Univeristy and Desislava Nikolov, University of the Free State, South Africa11th Floor Lecture Hall
2:20 - 2:40Snaking under radical perturbationsNadejda Drenska, Brown University11th Floor Lecture Hall
2:40 - 3:00Bug dynamics on manifoldsDmitri Gekhtman, Harvard University11th Floor Lecture Hall
3:00 - 3:20A model of chip firingAlexander St. Laurent, Brown University11th Floor Lecture Hall
3:20 - 3:40Outer Billiards with ContractionFrancisc Bozgan, UCLA, Rumania, In-Jee Jeong, Brown University and Julienne LaChance, Rensselaer Polytechnic Institute11th Floor Lecture Hall
3:40 - 4:00From spots to target patterns in Swift-Hohenberg equationChukiat Phonsom, Brown University11th Floor Lecture Hall
4:00 - 4:20A warm little pondNash Rochman, Brown University11th Floor Lecture Hall
4:20 - 4:40Bicycle Polygons, Solitons and the Darboux TransformationIan Alevy, University of Chicago and Emmanuel Tsukerman, Stanford University11th Floor Lecture Hall
4:40 - 5:00Instabilities of Euler flowZuhal Yildirim, Brown University11th Floor Lecture Hall
5:00 - 5:30Coffee & Dessert Reception11th Floor Collaborative Space

WednesdayAugust 1, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
11:00 - 12:00Random InterfacesRick Kenyon, Brown University11th Floor Lecture Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

ThursdayAugust 2, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
11:30 - 2:002012 UTRA SymposiumSayles Hall
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

FridayAugust 3, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
3:00 - 3:30Coffee/Tea Break11th Floor Collaborative Space

MondayAugust 6, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space
10:30 - 11:30Presentations: Unicycle problemAmy Nesky, Boston College and Clara Redwood, Reed College10th Floor Collaborative Space
1:00 - 2:00Presentations: Outer billiards with contraction (and, perhaps, without :)Francisc Bozgan, UCLA, Rumania, In-Jee Jeong, Brown University and Julienne LaChance, Rensselaer Polytechnic Institute10th Floor Collaborative Space
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

TuesdayAugust 7, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space
10:30 - 11:30Presentations: Billiards in near squaresYilong Yang, Brown University and Hai Bin Chang, National University of Singapore10th Floor Collaborative Space
1:00 - 2:00Presentations: Bicycle curves and bicycle polygonsIan Alevy, University of Chicago and Emmanuel Tsukerman, Stanford University10th Floor Collaborative Space
2:30 - 2:45Coffee/Tea Break11th Floor Collaborative Space

WednesdayAugust 8, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space
10:30 - 11:30Presentations: Negative Snell propagationKatherine Engelman, Bryn Mawr College and Andrew Kimball, Western Carolina University10th Floor Collaborative Space
1:00 - 2:00Presentations: Converse Desargues theoremDesislava Nikolov, University of the Free State and Kevin Casto, Brown University10th Floor Collaborative Space
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

ThursdayAugust 9, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space
1:00 - 2:00Presentations: Beetle problemDmitri Gekhtman, Harvard University10th Floor Collaborative Space
2:30 - 3:00Coffee/Tea Break11th Floor Collaborative Space

FridayAugust 10, 2012
TimeDescriptionSpeakerLocationAbstractsSlides
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space
1:00 - 1:30Closing Remarks10th Floor Collaborative Space
1:30 - 2:00Coffee/Tea Break10th Floor Collaborative Space


Outer Billiards with Contraction

This program was developed by Julienne LaChance, for the use of my REU research project with Francisc Bozgan and In-Jee Jeong. The research is funded by the NSF through ICERM, in collaboration with Brown University. This program gives the user a basic intuition for the dynamics of our new piecewise contractive system, called "outer billiards with contraction".

"We encourage the interested user to play around with the program. It has been extremely useful to our research." -Julienne LaChance

Example and and code can be found here: http://sourceforge.net/projects/outercontract/

Geometry of bicycle curves and bicycle polygons:
Three papers resulted from the study of bicycle polygons and the related Darboux transformation:



100% of the Summer@ICERM 2012 student survey respondents indicated they would recommend the Summer@ICERM program to their friends.

Participant quotes:

"Perfectly organized research program, projects were very accessible and we had the chance to have real progress on our research. Professors were very helpful and the meetings were very useful. ICERM looks wonderful and it is a great place to do research."

"I enjoyed the program thoroughly; it was well organized and would highly recommend it!"

"I loved the workspace. It was particularly stimulating and exciting to work there. I also enjoyed the varied exposure to different topics by the different advisors and TA's, and the fact that they were easily approachable and willing to help at all times."

Topics for Research Projects for SUMMER@ ICERM 2012

In this Research Program, the faculty organizers stressed open questions which were accessible via mathematical methods available to undergraduates. In addition, participants were expected to be able to gain intuition into some of these problems via computer experimentation and visualization. The complete problem list for the 2012 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 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).

Geometry

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).

Algebraic Geometry

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).