- Latham Boyle - Perimeter Institute for Theoretical Physics
Title: The standard model of particle physics from non-commutative geometry: a new perspective
Abstract: Non-commutative geometry (NCG) is of physical interest since it offers a novel geometric explanation for certain otherwise-unexplained features of the standard model of particle physics. In two recent papers (arXiv:1401.5083 and arXiv:1408.5367) we suggested a reformulation of NCG in which the fundamental object was a certain algebra B. Although this reformulation had certain mathematical and physical advantages, it also had two unresolved problems. Recently, Brouder, Bizi and Besnard (arXiv:1504.03890) have pointed out a natural improvement on our formalism in which these two problems are automatically resolved. I will review this story, and then present results from our forthcoming paper (Boyle and Farnsworth, to appear) in which we build upon Brouder et al's insight to suggest that B should actually be replaced by a related algebra B_D -- a certain differential graded star-algebra that I will introduce.
- Eric van Erp - Dartmouth College
Title: What does it mean to solve an index problem?
- Shane Farnsworth - Perimeter Institute for Theoretical Physics
Title: The symmetries of non-commutative geometry: a new perspective
Abstract: The key idea of non-commutative geometry (NCG) is to shift attention from geometric spaces, to instead focus on the algebra of functions defined on them. In this way NCG allows one to explore geometries where one has only the algebra and there is no known analogue of space whatsoever. In two recent papers (arXiv:1401.5083 and arXiv:1408.5367) we suggested a reformulation of NCG in which the fundamental object was a certain algebra B. Not only does this reformulation see immediate application in associative NCG, it readily extends to non-associative NCG. I will review this story, and then present results from our forthcoming paper (Boyle and Farnsworth, to appear) in which we give a number of example non-associative geometries.
- Asghar Ghorbanpour- Western University
Title: The Curvature of the Determinant Line Bundle for the Noncommutative Two Torus
- Mohammad Hassanzadeh - University of Windsor
Title: Nonassociative geometry, Hom-associative algebras, and cyclic homology
Abstract: We recall some motivations behind nonassociative geometry. Among all important examples of nonassociative algebras we review Hom-associative algebras. The Hom-associative algebras first appeared in contexts related to physics. The study of q-deformations, based on deformed derivatives, of Heisenberg algebras, Witt and Virasoro algebras, and the quantum conformal algebras reveals a generalized Lie algebra structure in which the Jacobi identity is deformed by a linear map. In this talk, we develop homological tools to study this interesting class of nonassociative algebras. Specifically we introduce Hochschild and cyclic homology and cohomology for Hom-associative algebras. This is Joint work with Ilya Shapiro and Serkan Sutlu.
- Bruno Iochum - Aix-Marseille University
Title: Spectral geometry, complex analysis and Berezin-Toeplitz quantization Abstract: Motivated by the noncommutative geometry of manifolds with boundaries and its applications to physics, I will select few facts on the Boutet de Monvel pseudodifferential approach, and then consider the CR-geometry for the case of bounded strictly pseudoconvex domains. Spectral triples will be exhibited in this context with finally an application to the Berezin-Toeplitz quantization.
- Masoud Khalkhali- Western University
Title: On The spectral action principle
- Bahram Rangipour - The University of New Brunswick
Title: Topological Hopf algebras and their Hopf cyclic cohomology
Abstract: Based on several occasions we show how nicely topological Hopf algebras is a better kingdom for Hopf cyclic cohomology. Part of the project is an ongoing joint work with H. Moscovici and the other part is joint with S. Sutlu.
- Ilya Shapiro - University of Windsor
Title: Induction and the Clifford-Mackey Theory
Abstract: Fusion categories form a relatively manageable subclass of tensor categories, furthermore they all occur as categories of representations of weak Hopf algebras. Considering them as a categorification of rings, it is natural to look at their module categories. We will discuss a notion of induction for these modules and see what light it sheds on the Clifford-Mackey Theory by considering some fusion categories and their modules naturally associated with a finite group.
- Baris Ugurcan - Western University
Title: Commutative and the Non-commutative: Dilation theory, Noncommutative Stochastic Processes and Analysis on Non-smooth spaces Abstract: We start with the well-known correspondence between Dirichlet forms, Laplacian operators and operator semigroups and explain how this is useful to do analysis on non-smooth settings (such as fractals) through spectral convergence. We briefly touch Kigami's analysis on fractals, an up-to-date account of spectral triple approach by various authors and connections between two approach. After that, we proceed to an explanation, including our own results, of dilation theorems in various frameworks and highlight how they appear in noncommutative Stochastic Processes: as a connection of NC-semigroups and NC-processes.
- Mitsuru Wilson - Western University
Title: Isospectral deformation: scalar curvature of noncommutative 3-spheres
Abstract: There had been a fairly recent discovery of an immense collection of one parameter family of non-commutative manifolds [Connes-Landi 01] by de-forming the algebra of smooth functions along the action of the maximal torus in the isometry group, cf [Rieffel 89]. However, the explicit computations of geometric invariants, such as the scalar curvature, are only a recent achievement in NCG. Indeed, the local expression for the scalar curvature of the non-commutative tori was computed for the conformally perturbation of the flat metric. This amounts to evaluating the spectral zeta function $\zeta_a(s)$ at s = 0 as a linear functional on $A_\theta$ and was done by Khalkhali and Fathizadeh and independently Connes and Moscovici in 2011. A purely non-commutative novelty of their formula is the appearance of the modular automorphism group from the type III factors in the final expression of the scalar curvature. Although non-commutativity prohibits many naive generalizations of classical geometry, it is possible in some cases to define curvature type invariants. In my joint work with Joakim Arnlind, we developed a sufficiently general geometric setting for the non-commutative 3-spheres and computed its scalar curvature. In my talk, I will survey the recent development in this field and present our setup for the computation of the scalar curvature of the non-commutative 3-spheres.