Browsing by Author "Yu, Guoliang"

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FRG: Collaborative Research: Noncommutative Dimension Theories
Mathematics; TAMU; National Science Foundation
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.
K-Theory of Operator Algebras and Its Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/255; National Science Foundation
In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made towards the solutions of long-standing problems in classical geometry and topology. K-theory, higher index theory, and secondary index theory serve as bridges between noncommutative geometry and classical geometry and topology. The principal investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology. The K-groups of certain operator algebras are receptacles of higher indices and secondary index invariants of elliptic differential operators and have important applications to problems in differential geometry and in the topology of manifolds. Examples of such applications include estimation of the size of the moduli space of all Riemannian metrics with positive scalar curvature and questions concerning the rigidity or nonrigidity of a manifold. The principal investigator and his students plan to apply the techniques of quantitative operator K-theory and dynamic complexity to study K-theory of operator algebras, higher index theory, and secondary index theory. The principal investigator and his students also intend to apply quantitative techniques to study the isomorphism conjectures in algebraic K-theory.
K-Theory of Operator Algebras and Its Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/255; National Science Foundation
In classical geometry, one studies geometric objects whose coordinates commute. Noncommutative geometry is a mathematical theory specifically designed to study "geometric objects" whose coordinates do not commute but which do occur naturally in mathematics and physics. In the last decade or so, with the help of new ideas from noncommutative geometry, great advances have been made towards the solutions of long-standing problems in classical geometry and topology. K-theory, higher index theory, and secondary index theory serve as bridges between noncommutative geometry and classical geometry and topology. The principal investigator and his students plan to apply noncommutative geometry methods to study problems in differential geometry and topology. The K-groups of certain operator algebras are receptacles of higher indices and secondary index invariants of elliptic differential operators and have important applications to problems in differential geometry and in the topology of manifolds. Examples of such applications include estimation of the size of the moduli space of all Riemannian metrics with positive scalar curvature and questions concerning the rigidity or nonrigidity of a manifold. The principal investigator and his students plan to apply the techniques of quantitative operator K-theory and dynamic complexity to study K-theory of operator algebras, higher index theory, and secondary index theory. The principal investigator and his students also intend to apply quantitative techniques to study the isomorphism conjectures in algebraic K-theory.