Speaker: Ji Li, Macquarie University
Title: Weak factorization of Hardy spaces and characterization of BMO spaces in the Bessel setting
Abstract: It is well-known that the classical Hardy space $H^p$, $0
This factorization plays an important role in studying function theory and operator theory connected to the spaces $H^1(\mathbb{D})$, $H^2(\mathbb{D})$ and the space $BMOA(\mathbb{D})$ (analytic BMO). The analogue of the Riesz factorization theorem, sometimes referred to as strong factorization, is not true for real-variable Hardy space $H^1(\mathbb{R}^n)$. Nevertheless, Coifman, Rochberg and Weiss provided a suitable replacement that works in studying function theory and operator theory of $H^1(\mathbb{R}^n)$, the weak factorization via a bilinear form related to the Riesz transform (Hilbert transform in dimension 1).
We study the analogue of the result of Coifman, Rochberg and Weiss for the Hardy spaces associated with differential operators $L$ developed in recently ten years. Then we further provide a characterization of BMO spaces associated with $L$ in terms of the commutators related to the Riesz transform $\nabla L^{-1/2}$. Examples of such operators $L$ include the Neumann Laplacian and the Bessel operators.
This is joint work with Xuan Duong, Brett D. Wick and Dongyong Yang.
Speaker: Romain Tessera (Université Paris-Sud)
Title: Local-to-Global rigidity of affine buildings
Abstract: It is well-known that every simply connected homogeneous Riemannian manifold M is “characterized up to isometry” by its ball of radius 1. Precisely: if N is another (not necessarily homogeneous) simply connected Riemannian manifold whose balls of radius 1 are all isometric to a ball of radius 1 in M, then M and N must be isometric.
In this talk, we investigate an analogous property for singular metric spaces, such as Cayley graphs of finitely generated groups, affine buildings…
This is joint work with Mikael de la Salle.
Title: Mathematical Methods for Analyzing Genomic Data
Speaker: J. B. Nation, University of Hawaii
We describe how some simple modifications of basic methods
give a useful analysis of genetic data, and apply this to
ovarian cancer. A refined singular value decomposition
allows one to look for common biological signals in
gene expression, microRNA and methylation site data.
A clustering technique based on lattice theory identifies
microRNA signals that predict survival. A neural network
using a wavelet transform can be used to choose candidates
for different treatments. Finally, we can identify some
of the biology behind these results, indicating treatment
possibilities.
This is joint work with Gordon Okimoto, Ashkan Zeinalzadeh,
Jenna Maligro, Tammy Yoshioka, and and Tom Wenska of the
University of Hawaii Cancer Center.
Speaker: Anthony Walter, University of Hawaii
Title: Introduction to the Ising model.
Abstract: We will define the Ising model, one of the simplest statistical mechanics
models which exhibits a phase transition. We are looking at a configuration
space on a discrete finite lattice, which is extended to the infinite lattice limit. Depending on some boundary conditions and a parameter $\beta$, in two or more dimensions it can be shown there is a phase transition from a disordered to an ordered phase. In particular we will look at the ferromagnetic case with positive boundary conditions.
We will prove a lower bound of the critical phase transition point.
Title: Bounds on the Number of Covers for Lattices and Related Posets
Abstract: How many covers can there be in a lattice of order n? This question has gone unanswered for decades. In the pursuit of a couple of conjectures, one of which is nearly 40 years old, we have obtained novel results primarily related to asymptotic bounds for lattices and related posets. This talk will involve lattice theory, order theory, combinatorics, graph theory, and analysis.