Physical Sciences and Mathematics Commons^™

Index Theory For Toeplitz Operators On Algebraic Spaces, Mohammad Jabbari

Arts & Sciences Electronic Theses and Dissertations

This dissertation is about the abstract Toeplitz operators obtained by compressing the multishifts of the usual Hilbert spaces of analytic functions onto co-invariant subspaces generated by polynomial functions. These operators were introduced by Arveson in regard to his multivariate dilation theory for spherical contractions. The main technical issue here is essential normality, addressed in Arveson's conjecture. If this conjecture holds true then the fundamental tuple of Toeplitz operators associated to a polynomial ideal $I$ can be thought as noncommutative coordinate functions on the variety defined by $I$ intersected with the boundary of the unit ball. This interpretation suggests operator-theoretic techniques …

Limits And Singularities Of Normal Functions., Tokio Sasaki

Arts & Sciences Electronic Theses and Dissertations

On a projective complex variety $X$, constructing indecomposable higher Chow cycles is an interesting question toward the Hodge conjecture, motives, and other arithmetic applications. A standard method to determine whether a given higher cycle is indecomposable or not is to consider it as a general fiber of a degenerate family of higher cycles, and observe the asymptotic behaviors of the associated higher normal functions.

In this thesis, we introduce some known examples of indecomposable cycles and a new method to detect the linearly independence of $\mathbb{R}$-regulator indecomposable $K_1$-cycles which is based on the singularities and limits of admissible normal functions …

A Q-Analogue And A Symmetric Function Analogue Of A Result By Carlitz, Scoville And Vaughan, Yifei Li

Arts & Sciences Electronic Theses and Dissertations

We derive an equation that is analogous to a well-known symmetric function identity: $\sum_{i=0}^n(-1)^ie_ih_{n-i}=0$. Here the elementary symmetric function $e_i$ is the Frobenius characteristic of the representation of $\mathcal{S}_i$ on the top homology of the subset lattice $B_i$, whereas our identity involves the representation of $\cS_n\times \cS_n$ on the top homology of Segre product of $B_n$ with itself. We then obtain a q-analogue of a polynomial identity given by Carlitz, Scoville and Vaughan through examining the Segre product of the subspace lattice $B_n(q)$ with itself. We recognize the connection between the Euler characteristic of the Segre product of $B_n(q)$ with …