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[PAST EVENT] Mathematics Colloquium/CSUMS Lecture
April 20, 2012
2pm - 3pm
Title: Toeplitz Problems on Singular Spaces
Abstract:
A Toeplitz operator is the compression of a multiplication operator over a domain D in C^n to the Bergman space. Toeplitz operators are natural examples of non-normal operators that generate interesting non-commutative C^*-algebras. Moreover, the simplest examples of Toeplitz operators reveal a relationship between the topological invariant known as the winding number and the analytical index or the Fredholm index. The Fredholm index is a very important numerical invariant, for instance, it plays an important role in the classification of essentially normal operators. Guentner and Higson's work on Toeplitz operators over strongly pseudoconvex domains shows that a Fredholm Toeplitz operator can be "identified" with an elliptic operator for which the Atiyah-Singer index theorem applies. This work extends our understanding that Toeplitz operators can be formulated in terms of K-theory pairings.
My research (with Peter Haskell) takes the first steps in using embedding and K-theory to achieve a definition of and calculation for Toeplitz operators on singular spaces such as the intersection of algebraic varieties with the unit ball in C^n, for example the zero set of z_1 z_3 in the unit ball in C^3.The approach consists of using Kasparov's bivariant K-theory to express and analyze a Toeplitz problem on the singular variety by using analysis on the ambient unit ball. The relationship between Toeplitz index theory and K-theory pairings motivates the use of Kasparov's bivariant K-theory also known as KK-theory. The intersection product of KK-theory (KK-product) is a flexible tool for calculating K-theoretic products involving differential operators.
Abstract:
A Toeplitz operator is the compression of a multiplication operator over a domain D in C^n to the Bergman space. Toeplitz operators are natural examples of non-normal operators that generate interesting non-commutative C^*-algebras. Moreover, the simplest examples of Toeplitz operators reveal a relationship between the topological invariant known as the winding number and the analytical index or the Fredholm index. The Fredholm index is a very important numerical invariant, for instance, it plays an important role in the classification of essentially normal operators. Guentner and Higson's work on Toeplitz operators over strongly pseudoconvex domains shows that a Fredholm Toeplitz operator can be "identified" with an elliptic operator for which the Atiyah-Singer index theorem applies. This work extends our understanding that Toeplitz operators can be formulated in terms of K-theory pairings.
My research (with Peter Haskell) takes the first steps in using embedding and K-theory to achieve a definition of and calculation for Toeplitz operators on singular spaces such as the intersection of algebraic varieties with the unit ball in C^n, for example the zero set of z_1 z_3 in the unit ball in C^3.The approach consists of using Kasparov's bivariant K-theory to express and analyze a Toeplitz problem on the singular variety by using analysis on the ambient unit ball. The relationship between Toeplitz index theory and K-theory pairings motivates the use of Kasparov's bivariant K-theory also known as KK-theory. The intersection product of KK-theory (KK-product) is a flexible tool for calculating K-theoretic products involving differential operators.
Contact
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