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Joshua Sussan

Assistant Professor

Mathematics, Medgar Evers College
Topological and Geometric Invariants from Representation Theory

The critical question in topology, the mathematical study of shapes, is identifying whether two shapes are the same. This seemingly simple question is difficult for mathematicians to answer. Sussan is investigating this problem in the fourth dimension with his three-year, $145K NSF award, Topological and Geometric Invariants from Representation Theory. “The approach is to find an invariant of a shape, to mathematically name it. If two different shapes have different invariants we prove they are distinct shapes.” Invariants in three dimensions have been well studied using representation theory, but finding four dimensional invariants requires a more sophisticated theory, categorification. Broadly, the philosophy of categorification is taking a mathematical structure and finding a similar but richer analogue. Sussan is using representation theory and categorical techniques to construct four dimensional invariants. Knowledge of invariants in four dimensions may have deep physical applications.

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