Query: The ricci flow techniques and applications

title: The Ricci Flow: Techniques and Applications

title: The Ricci Flow: Techniques and Applications: Part III: Geometric Analytic Aspects

title: The Ricci flow: techniques and applications Part I: Geometric aspects (Mathematical Surveys and Monographs 135)

title: The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects
abstract: Contents Preface ix What Part II is about ix Highlights and interdependences of Part II xi Acknowledgments xiii Contents of Part II of Volume Two xvii Notation and Symbols xxiii Chapter 10. Weak Maximum Principles for Scalars, Tensors, and Systems 1 1. Weak maximum principles for scalars and symmetric 2 tensors 2 2. Vector bundle formulation of the weak maximum principle for systems 9 3. Spatial maximum function and its Dini derivatives 24 4. Convex sets, support functions, ODEs preserving convex sets 32 5. Proof of the WMP for systems: time dependent sets and avoidance sets 43 6. Maximum principles for weak solutions of heat equations 47 7. Variants of maximum principles 56 8. Notes and commentary 65 Chapter 11. Closed Manifolds with Positive Curvature 67 1. Multilinear algebra related to the curvature operator 69 2. Algebraic curvature operators and Rm 77 3. A family of linear transformations and their effect on R 2 R# 89 4. Proof of the main formula for D a (R) 94 5. The convex cone of 2 nonnegative algebraic curvature operators 105 6. A pinching family of convex cones in the space of algebraic curvature operators 116 7. Obtaining a generalized pinching set from a pinching family and the proof of Theorem 11.2 126 8. Summary of the proof of the convergence of Ricci flow 134 9. Notes and commentary 136 Chapter 12. Weak and Strong Maximum Principles on Noncompact Manifolds 139 vi CONTENTS

title: The Ricci Flow: Techniques and Applications: Part IV: Long Time Solutions and Related Topics

title: An introduction to Ricci flow and volumetric approximation with applications to shape modeling
abstract: Extending a shape driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level. This survey discusses the main volumetric approximation schemes for both 3D shapes and d dimensional data, and provides a unified discussion on the integration of surface based and volume based shape information. Then, it describes the application of shape based and volumetric techniques to shape modeling through volumetric parameterization and polycube splines; feature driven approximation through kernels and radial basis functions. We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal shape structure and to design Riemannian metrics of manifolds by prescribed curvatures. We conclude the presentation by discussing applications to shape analysis and medicine.

title: Higher Order Computational Electromagnetics, Uncertainty Quantification, and Meshing Techniques with Applications in Wireless Communication, Medicine, and Meteorology
abstract: We present some advances in several major components of computational electromagnetics (CEM) including (1) higher order method of moments, finite element method, ray tracing, and hybrid techniques, (2) uncertainty quantification techniques for CEM modeling of RF and microwave devices and systems featuring a posteriori error estimation, sensitivity analysis, and intelligent model refinement based on adjoint methods, and (3) automatic surface meshing in CEM by the discrete surface Ricci flow method enabling generation of high quality meshes and adaptive iterative mesh refinement. We also show how these methodologies and techniques can be effectively applied to solving general real world problems with impacts on wireless communication, medicine, and meteorology. The applications include (A) smart underground mining with an integrated wireless cyber physical framework using CEM modeling and measurements of wireless propagation in underground mines, (B) design of RF coils for next generation high and ultra high field magnetic resonance imaging (MRI) scanners based on CEM and MRI experiments, and (C) accurate characterization of winter precipitation using multi angle snowflake camera, visual hull image processing, advanced scattering methods, and polarimetric radar.

title: OPTIMAL TRANSPORTATION THEORY AND APPLICATIONS
abstract: Part I. Short Courses: 1. Introduction to optimal transport theory Filippo Santambroggio 2. Models and applications of optimal transport in economics, traffic and urban planning Filippo Santambroggio 3. Logarithmic Sobolev inequality for diffusions and curvature dimension condition Ivan Gentil 4. Lecture notes on variational methods for incompressible Euler equations Luigi Ambrosio and Alessio Figalli 5. Ricci flow: the foundations via optimal transportation Peter Topping 6. Lecture notes on gradient flows and optimal transport Sara Danieri and Guiseppe Savare 7. Ricci curvature, entropy, and optimal transport Shin Ichi Ohta Part II. Survey and Research Papers: 8. Computing the time continuous optimal mass transport without Lagrangian techniques Olivier Besson, Martine Picq and Jerome Poussin 9. On the duality theory for the Monge Kantorovich transport problem Mathias Beiglbock, Chrsitian Leonard and Walter Schachermayer 10. Optimal coupling for mean field limits Francois Bolley 11. Functional inequalities via Lyapunov conditions Patrick Cattiaux and Arnaud Guillin 12. Size of the medial axis and stability of Federer's curvature measures Quentin Merigot.

title: Uncertainty Quantification for River Flow Simulation Applied to a Real Test Case: The Garonne Valley
abstract: Sensitivity analysis techniques have been widely used in multitude of applications to quantify the impact of inputs variables imprecision on the accuracy of the model output variables. Depending on the problem at hand, an appropriate method of sensitivity analysis should be selected. Direct and adjoint sensitivity analysis are two complementary approaches known to be efficient. While the direct approach provides an assessment of the propagation of the error of a given input parameter in the studied system, the adjoint approach enables to identify the source of the uncertainty of a given output variable with respect to several input parameters. Direct methods have been extensively investigated in different geophysical applications, particularly in the context of the hydraulic modeling. In this work, several methods will be described and applied to the same benchmark during over flooding events. The effect of uncertainties in the boundary conditions, the spatially distributed functions (bed level, river width, friction, etc. and the numerical parameters on the model state variables (discharge, water surface elevation, etc. is examined. This study has been carried out on the Garonne River test case, along a 50 km downstream reach, using 1D full Saint Venant hydraulic models SIC2 (Irstea) or Mascaret (EDF) and 2D Telemac model (EDF) Results illustrate the influence of individual and combined contributions of input variables uncertainties.

title: Perelman's Monotonicity Formula and Applications
abstract: This article relies on  that the author wrote with Gang Tian and Xiaodong Wang. In view of Hamilton's important work on the Ricci flow and Perelman's paper on the Ricci flow where he developes the techniques that he will later use in completing Hamilton's program for the geometrization conjecture, there may be more interest in the area. We will also discuss the author's theorem which says that the curvature tensor stays uniformly bounded under the unnormalized Ricci flow in a finite time, if the curvatures are uniformly bounded. We will prove that in the case of a Kahler Ricci flow with uniformly bounded Ricci curvatures, for each sequence of flows g(ti t) for ti \to \infty there exists a subsequence of metrics converging to a solution to the flow outside a set of codimension 4.