Projects
Project 1  Spectral Geometry in Fuzzy Domains
Supervisor: Masoud Khalkhali (Western University)
Research Group:
Xiaozhu Li, Leanne Mezuman, Kateryna Tatarko, Minliu Wu, Ruoqi Yu
Spectral geometry is a branch of mathematics which studies those properties of a space that can be encoded in terms of eigenvalues of operators like Laplacian. In a nutshell one wants to know what one can hear about the shape of a space. The simplest such invariant is the volume. This was discovered by Herman Weyl about 100 years ago. Recently there has been some progress in extending techniques of spectral geometry beyond its tradiational doamin and to discrete objects like graphs, to fractals, or even to much more singular objects. A concrete problem is to develop these techniques for singular spaces that are defined as limits of matrix algebras.
Resources for Research Group.
1. Our primary background text will be this.
It has a good selection of material on aspects of spectral geometry for noncommutative spaces. I shall cover all background and preparatory stuff in my lectures and provide many examples. A good textbook on classical spectral geometry is this. During our meetings we shall also discuss recent papers directly related to our project.
2. More advanced texts on noncommutative geometry for your own self study include: this, and this.
Project 2  Modelling of Fetal Neurovascular Coupling
Supervisor: Huaxiong Huang, Qiming Wang (York University)
Research Group:
Mark Freeman, Adam Mauskopft, Shuyan Mei, Kimberly Stanke, Zihao Yan
Brain injury acquired antenatally remains a major cause of postnatal longterm neurodevelopmental sequelae. There is evidence for a combined role of fetal infection and inflammation and hypoxicacidemia. Concomitant hypoxia and acidemia (umbilical cord blood pH < 7.00) during labour increase the risk for neonatal adverse outcomes and longerterm sequelae including cerebral palsy. The main manifestation of pathologic inflammation in the fetoplacental unit, chorioamnionitis, affects 20% of term pregnancies and up to 60% of preterm pregnancies and is often asymptomatic.
During the first two weeks of the summer program, students will be introduced to a recently developed mathematical model that couples blood circulation with neural responses to investigate the effect of umbilical cord occlusion on heart rate variation as well as the development of acidemia in the fetus. Stuednts will be asked to use the model and run computer simulations under a variety of occulsion conditions.
In third week of the program, students will be asked to participate in a problem solving workshop on neurovascular coupling and developing brain, and work with other participants of the workshop to develop mathematical models, based on their work during the first two weeks of the program and experimental observations. They will present a preliminary report at the end of the oneweek workshop and continue to refine their model during the reminder of the program, by comparing them with experiment data when possible, and produce a final report and present their findings during the final week of the program.
Project 3  The model theory of C*algebras
Supervisor: Bradd Hart (McMaster) and Ilijas Farah(York University)
Research Group:Jonathan Berger, Diego Caudillo Amador, Jamal Kawach, Sejin Kim, Yushen Zhang
Model theory is a branch of mathematical logic which studies classes of structures or models of theories in the sense of logic. 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 that of C*algebras (algebras of operators acting on a Hilbert space) and a concrete example of a problem in this area is understanding the model theory of strongly selfabsorbing algebras, a class of C*algebras that have a central place in classification program.
Some familiarity with basic logic would be helpful and a solid grounding in linear algebra and analysis would be an asset.
Reading list:
Introduction to continuous model theory
1. Ben Ya'acov, Berenstein, Henson and Usvyasatov, http://math.univlyon1.fr/~begnac/articles/mtfms.pdf
2. Bradd Hart lecture notes: http://ms.mcmaster.ca/~bradd/courses/math712/index.html
Introduction to operator algebras with a logic perspective
1. Ilijas' notes: http://www.math.yorku.ca/~ifarah/Ftp/singaporefinalr.pdf
More advanced reading:
1. Farah, Hart and Sherman: http://arxiv.org/pdf/1004.0741v5.pdf
Project 4  Metric Arens irregularity
Supervisor: Matthias Neufang, (Carleton University) and Juris Steprans (York University)
Research Group: Eric Hu, Georg Maierhofer, Roberto Hernández Palomares, Pranav Rao
Banach algebras are fundamental objects in functional and harmonic analysis. One can think of the product in a Banach algebra as a generalization of matrix multiplication. In recent years the study of bidual Banach algebras has been a very active field. Given a Banach algebra A, its second dual carries two natural products extending the multiplication on A, called the left and right Arens products. If these coincide, A is called Arens regular. If, on the contrary, A equals the set of elements in the bidual for which multiplication with respect to both products is the same, A is called strongly Arens irregular (SAI). All operator algebras are Arens regular, while the group algebra of any locally compact group is SAI. In this context, Z. Hu, M. Neufang and Z.J. Ruan have introduced and started to develop the concept of 'metric Arens irregularity' which measures the degree of Arens irregularity through a numercial value g(A), forming an isometric invariant of the algebra A: indeed, g(A) is the supremum of the norms of all differences of left and right Arens products formed by elements in the unit ball of the bidual. Obviously, g(A) is a
number between 0 and 2, and g(A)=0 precisely when A is Arens regular. We have shown that g(A)=2 for many SAI algebras A, but also that there exist nonSAI algebras A with g(A)=2. Our study gives rise to fascinating questions to be explored, e.g., can g(A) lie strictly between 0 and 2, and which values are produced by Beurling algebras or algebras of operators, particularly those that are neither Arens regular nor SAI?
Program
Week of July 25  
Jul 2

Introductory Session: Introduction and presentation of the program Introduction to supervisors, and overview of theme areas and projects Open time for students to meet informally with supervisors. Orientation Meeting: Students meet with Fields program staff Lunch provided at Fields for students and supervisors 
Jul 35 
Students will meet informally with supervisors and in their groups to work on research project. 
Week of July 711  
Students will meet informally with supervisors and in their groups to work on research project.  
Week of July 1418  
Students will meet informally with supervisors and in their groups to work on research project.
Introduction to the Fields SMART board and video conferencing facilities which are useful for remote collaboration. 

Week of July 2125  
Students will meet informally with supervisors and in their groups to work on research project. 

Week of July 28August 1  
Students will meet informally with supervisors and in their groups to work on research project.  
Week of August 58 (Note Aug.4 is a Civic Holiday)  
Students will meet informally with supervisors and in their groups to work on research project.  
Week of August 1115  
Students will meet informally with supervisors and in their groups to work on research project.  
August 13: Joint Meeting with Waterloo Women in math Summer School  
Week of August 18 22  
Students will meet informally with supervisors and in their groups to work on research project.  
Week of August 2529  
During the final week, students are requested to prepare a report on their projects and their experience in the Program to be emailed to
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// > " before August 30. These reports will be used in the Fields Newsletter and Annual Report. 

Aug 27

MiniConference: Undergraduate research students will present their work. 
Aug 28

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