FRG: Collaborative Research: Noncommutative Dimension Theories
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abstract

Approximation defines our world. For example, the letters on this screen have smooth curves and bends. But zoom in and all you see are squares: pixels. The smaller the pixels (i.e., the finer the approximation), the smoother a curve looks. Add three colors and suddenly we can approximate a rainbow. In a sense, science is all about refining and improving approximations to reality. Take Newtonian physics, for example. It works great at medium scales, but breaks down when things are too big or too small. Einstein''s relativity and quantum mechanics work much better at those scales. And these theories require sophisticated mathematics. This focused research group project addresses several outstanding questions in operator algebras and their analogies in other areas of mathematics.Operator algebras arose as a framework for quantum mechanics. Over the years many classical theories were extended to this noncommutative context: geometry, topology, probability and more. The PIs will spearhead an international effort to capitalize on recent connections between operator algebras and other areas such as dynamics, measure theory, coarse geometry and K-theory. Specifically, the PIs shall push analogies between nuclear dimension and asymptotic dimension, two notions defined via approximation and encompassing a huge swath of examples, to address K-theoretic questions such as the Universal Coefficient Theorem and the Baum-Connes and Farrell-Jones conjectures.