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 long-term neurodevelopmental sequelae. There is evidence for a combined role of fetal infection and inflammation and hypoxic-acidemia. Concomitant hypoxia and acidemia (umbilical cord blood pH < 7.00) during labour increase the risk for neonatal adverse outcomes and longer-term sequelae including cerebral palsy. The main manifestation of pathologic inflammation in the feto-placental 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 one-week 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, Se-jin 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 self-absorbing 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.univ-lyon1.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/singapore-final-r.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 non-SAI 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