Mathematics and Statistics Seminar: A learner-centered approach to teaching computational modeling, data analysis, and programming
Mathematics and Statistics Seminar
From Symmetry to Linear Operators to Noncommutative Geometry: A Journey through Representation Theory
Dr. Angel Roman Martinez
Lecturer, Science and Mathematics Department
National Technical Institute for the Deaf
Abstract: Symmetry is ubiquitous and groups are the mathematical objects that allow us to describe it. However, they are notoriously difficult to study. I will introduce representation theory: a “linearization” of group action, through linear operators on vector spaces. A major motivation in representation theory is to classify all irreducible representations. Topologically, the space of irreducible representations is badly behaved. The active and exciting research field of noncommutative geometry studies algebras of linear operators associated with such pathological spaces. This leads us to introduce C*-algebras. Of particular interest in representation theory is the Mackey analoy. Consider two related groups: rigid motions on the hyperbolic space (which has negative curvature) and rigid motions on the euclidean (flat) space, which can be viewed as a “flattening” of the former. Structurally, the two groups are different, but in the 1970s George Mackey observed a striking similarity between their representations. We obtain a new result in C*-algebras, which we call the Mackey embedding, arising from the Mackey analogy.
Intended Audience: Beginners, undergraduates, graduates. Those with interest in the topic.
Interpreters have been requested.
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This is an RIT Only Event
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