In processes involving multi-phase flow in highly heterogeneous media, such as oil recovery by gas injection, mobile phases will seek high-permeability flow paths. Therefore, it is essential that models for such processes effectively account for these paths. For this purpose, we have developed a computational framework for flow solvers based on adapted Cartesian grids that are equipped with multi-point flux approximations obtained with specialized transmissibility upscaling methods.

For gridding, we propose using Cartesian Cell-based Anisotropically Refined (CCAR) grids, which inherit the ease of Cartesian grids while providing rapid transition between coarse and fine scales to resolve fine-scale features accurately and efficiently. We present an iterative algorithm for automatically generating such grids based on geological data and information from global coarse-scale flow simulations.

For upscaling, we discuss a local transmissibility upscaling method, called Variable Compact Multi-Point (VCMP), that uses spatially varying and compact multi-point flux stencils. The stencil weights are chosen so as to reproduce generic local flow problems accurately, while remaining as close as possible to a two-point flux for the sake of robustness. The inherent flexibility of VCMP can also be exploited to ensure that the solution of the resulting system satisfies a discrete maximum principle.

We conclude with application of these gridding, upscaling and discretization methods, originally designed for single-phase flow, to two-phase flow, which requires enhancing our adaptive mesh refinement scheme in order to accurately resolve rapidly advanacing saturation fronts. We show that adaptivity allows such accurate resolution by upscaling only single-phase parameters, thus avoiding the significant computational expense of multi-phase upscaling.

In this talk we will discuss refined class number formulas conjectured by Gross and by Darmon. We will prove (a slight variant of) Darmon's conjecture, using the theory of Kolyvagin systems. This is joint work with Barry Mazur.

Inspired by the canonical neighborhood theorem of G. Perelman in 3 dimensional, we study the weak compactness of sequence of ricci flow with scalar curvature bound, Kappa non-collapsing and integral curvature bound.

All of these constraints are natural in the Kahler ricci flow in Fano surface and as an application, we give a ricci flow based proof to the Calabi conjecture in Fano surface.

For an ergodic system, the theorem of Shannon-McMillan-Breiman states that for every finite generating partition the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. In 1962 Ibragimov showed that the distribution of the measure of cylinder sets is lognormally distributed provided the measure is strong mixing and its conditional entropy function is sufficiently well approximable.
Carleson (1958) and Chung (1960) generalised the theorem of SMB to infinite partitions (provided the entropy is finite). We show that the measures of cylinder sets are lognormally distributed for uniformly strong mixing systems and infinite partitions and show that the rate of convergence is polynomial. Apart from the mixing property we require that a higher than fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.