SCIENTIFIC PROGRAMS AND ACTIVITIES

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UNDERGRADUATE SUMMER RESEARCH PROGRAM

July 4 to August 26, 2011

The Fields Institute will be hosting a summer research program for undergraduates to be held in July and August of 2011. The program will support up to thirty students to take part in research projects supervised by leading scientists from Fields partner universities or thematic programs.

PROGRAM

9:30-10:15 a.m. Orientation Meeting: Students meet with Fields program staff
Re: computer accounts, offices, expense reimbursements, and overview of Fields facilities.

10:15 a.m. Coffee break

10:30 a.m. Introductory Session: Introduction and presentation of the program (Fields Director, Edward Bierstone)

Introduction to supervisors, and overview of theme areas and projects:

12:30 p.m. Lunch provided at Fields for students and supervisors
Afternoon open for students to meet informally with supervisors.
*By 4 p.m. Hand in ranking sheet to Members Liaison, Sharon McCalla, Room 330*

July 5-11

Tuesday July 5 (or Wednesday July 6) afternoon: Introduction to the Fields SMART board and video conferencing facilities which are useful for remote collaboration.

Friday July 8

Daniel Pellicer will meet with students involved with the "Symmetries of Euclidean Tessellations and Their Covers" Project in Room 210, 10 a.m.-12 noon.

Lecture by Moshe Vardi, Rice University Stewart Library at 1:30 pm
Title: P vs NP
The question of P vs. NP is one of the central questions in computer science and mathematics. (It is one of the Clay Institute Millenial Problems whose solution would yield an award of $1,000,000.) In the first half of August 2010, an HP researcher claimed to have solved the problem, using tools from mathematical logic and statistical physics, including a theorem proved by the speaker in 1982.
The claim generated a huge buzz in computer science, with coverage also in the New York Times. This talk will explain what the P-vs-NP problem is, what tools were employed in the claimed proof, and what the status of the claim is.

July 8-11

The Mitacs Globalink students are invited to the Ontario Conference for the Globalink Industry Conference. Opening gathering on Thursday July 7

Saturday July 23

Students are invited to participate in a Fields Undergraduate Network (FUN) event in Ottawa:
http://www.fields.utoronto.ca/programs/outreach/11-12/undergradnet/

August 22-27

During the final week, students are requested to prepare a report on their projects and their experience in the Program to be emailed to programs(PUT_AT_SIGN_HERE)fields.utoronto.ca before August 27. These reports will be used in the Fields Newsletter and Annual Report.

August 25

Mini-Conference: Undergraduate research students will present their work. This Conference will form a special Fields Undergraduate Network (FUN) event.

August 25

An excursion - sponsored and organized by Fields - is planned for all students.

Project Overview

Symmetries of Euclidean tessellations and their covers
Supervisors: Isabel Hubard, Mark Mixer, Daniel Pellicer, Asia Ivic Weiss.

The study of Euclidean symmetries has origins in antiquity, for instance with the Platonic solids. In the 19th and 20th century, new ideas emerged, which revitalized the eld and created numerous interesting topics of active research.
A symmetry of an Euclidean tessellation is an isometry of the space that preserves it. Our study of these symmetries will be focused on the number of orbits of some elements of the tessellation, such as vertices, edges, ags, etc., under a given isometry subgroup.
Initially we start with a geometric approach, that will lead into algebraic, combinatorial, and topological ones. Various open problems related to Archimedean tessellations in 2 and 3 space will be explored.
Students should consult the following literature:
1. First three chapters of "Regular Complex Polytopes" by H.S.M. Coxeter.
2. "Uniform Tilling of 3-Space" by B. Grunbaum, Geombinatorics 4(1994), 49 - 56.
3. The attached notes by I. Hubard.

Constraint Satisfaction (slides)
Supervised by : Ross Willard, Matt Valeriote, and Libor Barto

Many interesting and important questions from computer science, combinatorics, logic, and database theory can be expressed in the form of a constraint satisfaction problem. Recently an algebraic approach to settling some central conjectures in this area have been developed and have led to the investigation of some novel properties of finite algebraic and combinatorial systems. The proposed project will involve experimenting with small algebraic and combinatorial systems to test several conjectures and open problems that are concerned with solving associated instances of the constraint satisfaction problem.

Ideally students will have an interest and background in abstract algebra and also in combinatorics, logic, and computational complexity, but this is not essential.

Mathematical Finance - Understanding Financial Crisis
Matheus Grasselli, McMaster University presenting and Oleksandr Romanko
Abstract to follow

Model Theory of Operators (slides)
Supervised by Bradd Hart & Ilijas Farah

Model theory is a branch of mathematical logic which studies the logical theories of classes of structures or models. Traditionally this logic has been classical first order logic and the techniques of first order model theory have been used successfully in many areas of algebra, number theory and geometry. Recently a new logic called continuous logic has been developed and it is more suited for applications in analysis. One area of application is operator algebras (algebras of operators acting on a Hilbert space). A concrete problem in this area is studying the asymptotic behaviour of sentences in continuous logic in matrix algebras.

Some familiarity with basic logic would be helpful and a solid grounding in linear algebra and analysis would be an asset.

Study of the development of glaucoma
Supervisors: Irwin Pressman, Carleton (presenting) and Siv Sivologathan, University of Waterloo

Study of the development of glaucoma, this is a condition in which the optic nerve is damaged and is often associated with increased fluid pressure in the eye (occular hypertension). The proposal is to model the flow of aqueous humor in the eye using the partial differential equations governing buoyancy-driven flows (Navier-Stokes equations coupled to the heat equation). Of course the full set of equations are too difficult to solve (except numerically), and we will make some approximations from lubrication theory that will lead to a set of equations that are analytically tractible.

Combinatorial Rigidity And Graph Constructions (course introduction)
Supervisors: Tony Nixon, Fields Institute (presenting )

Rigidity theory is motivated by diverse applications in computer aided design, materials science and structural engineering. We consider realisations of graphs as physical objects (called frameworks) where the vertices represent joints and the edges bars between pairs of joints. A framework is rigid if the only edge-length-preserving continuous motions of the vertices are induced by isometries. For almost all frameworks it is the properties of the graph that determine rigidity rather than the specific realisation.

The natural classes of graphs that arise in rigidity theory are graphs G=(V,E) for which |E|=k|V|-l (for natural numbers k,l) together with a corresponding subgraph inequality. Despite their innocent appearance these graphs have a rich combinatorial flavour that the students can explore. Particularly the proposed project would be based around one or more of the following: inductive constructions, spanning subgraph decompositions, algorithms, matroids and simple versus multigraphs.

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Out of town students accepted into the program will receive financial support for travel to Toronto, student residence housing on the campus of the University of Toronto from July 4 to August 26, 2011, and a per diem for meals. Non-Canadian students will receive medical coverage during their stay.

Students will work on research projects in groups of three or four. Some projects will be related to the Fields Thematic Programs on The Mathematics of Constraint Satisfaction and on Discrete Geometry and Applications. In addition, supervisors will suggest other topics outside of these fields. In some cases students may also have the opportunity to spend a week off site at the home campus of the project supervisor(s).

Undergraduate students in mathematics and related disciplines are encouraged to apply.
Note: Students requiring visas for travel to Canada will need to make their own arrangements to obtain the necessary documents.

Confirmed Program Students

Ferenc Bencs --Eötvös Loránd University
Lucas Bentivenha -- UNESP
Zoltan Blazsik --Eötvös Loránd University
Luke Paul Broemeling --University of Calgary
Kostiantyn Drach -- V.N. Karazin Kharkiv National University
Qian (Linda) Liu --University of Toronto
Hao Liu -- Nanjing University
Hyung-Bin Ihm --University of Toronto
Euijun Kim -- University of Toronto
Maximilian Klambauer -- University of Toronto
Avinash Kulkarni -- University of Waterloo
Fernando Lenarduzzi -- Universidade Estadual Paulista “Julio de Mesquita Filho”
Daniel Perkins --Bowie State University
Nikita Reymer --University of Toronto
Rafael Rocha -- Universidade Estadual Paulista “Julio de Mesquita Filho”
Nigel Sequeira -- McMaster University
Vishal Siewnarine -- University of Waterloo
Maksym Skoryk --V.N. Karazin Kharkiv National University
Garence Staraci --Stanford University/McGill
Rebecca Tessier -- Queen's University
Louis-Philippe Thibault -- University of Montreal
Anna Tossenberger -- Eötvös Loránd University
Yiyang (Young) Wu -- University of Waterloo

To apply

We need the following by April 30, 2011
(Note late applications are accepted dependent on funding)

(1) Brief covering letter outlining your background and experience, sent by email to programs(PUT_AT_SIGN_HERE)fields.utoronto.ca

(2) Copy of your academic transcript sent in .pdf format as an attachment to (1)

(3) An official copy of your transcript send by issuing institution either by email to programs(PUT_AT_SIGN_HERE)fields.utoronto.ca or by hard copy mailed to "Fields Manager of Scientific Programs, 222 College Street, Toronto M5T 3J1"

(4)Two letters of reference. Please ask referees to send letters directly to programs(PUT_AT_SIGN_HERE)fields.utoronto.ca in .pdf format as an attachment.

Late applications will be accepted funds allowing.