Given a blurry image, the goal is to find the most clear image. There are many methods to solve this inverse problem in the case of cartoon images containing rather piecewise smooth objects. However, in the presence of oscillations the blurring process removes those oscillations in the images and that makes this inverse problem harder to solve. We approach this problem by minimizing a convex functional whose domain is the product of the space of functions of bounded variation and the homogeneous Sobolev space. As we will see, the homogeneous Sobolev space turns out to be a good space to capture oscillations. We will talk about the existence of a minimizer and characterization of the minimizers and PDE based numerical scheme and then briefly discuss a noisy case. If time permits, we will also talk about a medical image denoising application.

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.