Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G_2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on $\Lambda(x,t)=\left(|\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\right)^{\frac 12}$ will imply bounds on all covariant derivatives of Rm and T. (2). We show that $\Lambda(x,t)$ will blow up at a finite-time singularity, so the flow will exist as long as $\Lambda(x,t)$ remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2). (5). Finally, we study compact soliton solutions of the Laplacian flow.Comment: 59 pages, v2: minor corrections and additions, accepted version for GAF

Extracting Functional Modules from Biological Pathways

It has been proposed that functional modules are the fundamental units of cellular function. Methods to identify these modules have thus far relied on gene expression data or protein-protein interaction (PPI) data, but have a few limitations. We propose a new method, using biological pathway data to identify functional modules, that can potentially overcome these limitations. We also construct a network of these modules using functionally relevant PPI data. This network displays the flow and integration of information between modules and can be used to map cellular function

Lattice QCD with 12 Quark Flavors: A Careful Scrutiny

With a substantial amount of simulations, we have explored the system across a wide range of lattice scales. We have located a lattice artifact, first order bulk transition, have studied its properties, and found that the flavor-singlet scalar meson mass vanishes at the critical endpoint. We will discuss the lattice phase diagrams and the continuum limits for both a spontaneous chiral symmetry breaking phase and an infrared conformal phase, and compare results with other groups.Comment: 7 pages, 7 figures; Contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya Universit