We survey some of the recent progress on complete gradient shrinking Ricci solitons, including the classifications in dimension three and asymptotic behavior of potential functions as well as volume g...
We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.
We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^...
We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Reid–Wang–Zhou [...
We propose the study of Markov chains on groups as a “quasi-isometry invariant” theory that encompasses random walks. In particular, we focus on certain classes of groups acting on hyperbolic spaces i...
We propose a surface meshing approach for computational electromagnetics (CEM) based on discrete surface Ricci flow (DSRF) with iterative adaptive refinement (AR) in the parametric domain for the auto...
We present the estimates of the order O(α s) QCD corrections to R(s) = σtot(e e → hadrons)/ σ(ee → μμ), Rτ = Γ(τ → ντ +hadrons)/Γ(τ → ντνee) and to the deep inelastic scattering sum rules, namely to t...
We present methods for the automated generation of high-order generalized quadrilateral surface meshes for computational electromagnetics (CEM) using discrete surface Ricci flow and parametric domain ...
We perceive the world through images formed by scattering. The ability to interpret scattering data mathematically has opened to our scrutiny the constituents of matter, the building blocks of life, a...
We introduce LCL covers of closed n-dimensional manifolds by n-dimensional disks and study their properties. We show that any LCL cover of an n-dimensional sphere can be converted to the minimal LCL c...
We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also cons...
We discuss some of the key ideas of Perelman's proof of Poincare's conjecture via the Hamilton program of using the Ricci flow, from the perspec- tive of the modern theory of nonlinear partial differe...
Two recent articles [1, 2] suggested an interesting dynamical mechanism within the framework of the vacuum Einstein flow (or Einstein-Λ flow if a positive cosmological constant Λ is included) which su...
This study aims at generating textile patterns and innovative artistic and constructional designs for the artists and architects. It is the development of programming procedure for the analysis and de...
This is a survey about Thurston’s geometrization conjecture of three manifolds and Perelman’s proof with the Ricci flow. In particular we review the essential contribution of Hamilton as well as some ...
This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanic...
These notes are intended as an introduction to the theory of Coxeter groups. They closely follow my talk in the Lectures on Modern Mathematics Series at the Mathematical Sciences Center in Tsinghua Un...
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".
The qualitative nature of singularities in Ricci flow of dimension 3 was significantly studied in Perelman’s three renowned papers [4–6]. The arguments in [4–6] were detailedly addressed in the exposi...
The eld of Topology was born out of the realisation that in some fundamentalsense, a sphere and an ellipsoid resemble each other but di er from a torus { thesurface of a rubber tube (or a doughnut). A...