K-stability of constant scalar curvature K\"ahler manifolds

We show that a polarised manifold with a constant scalar curvature K\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.Comment: 14 page

The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R(X)$ to the existence of conical Kahler-Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth simple divisor and the Mabuchi $K$-energy is bounded below, then there exists a unique conical Kahler-Einstein metric satisfying $Ric(g) = \beta g + (1-\beta) [D]$ for any $\beta \in (0,1)$. We also construct unique smooth conical toric Kahler-Einstein metrics with $\beta=R(X)$ and a unique effective Q-divisor $D\in [-K_X]$ for all toric Fano manifolds. Finally we prove a Miyaoka-Yau type inequality for Fano manifolds with $R(X)=1$

A Simple Geometric Representative for μ\mu of a Point

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $\mu$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p)$ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.Comment: Updated 2018 to published version. 8 pages, AmS-TeX, no figure

CRUK Basic and Clinical Research DMP Assessment Rubric v2.0

This rubric is intended to assist in the assessment of CRUK Basic and Clinical Research data management plans, against the criteria required by the funder. It is not intended to be used as a template for researchers to follow when writing a data management plan. The rubric has been divided into 'performance criteria' (on the left hand side) which cover the information the funder expects to be covered by the data management plan. Each performance criteria is followed by three descriptions of how it might be addressed, each indicating a different level of response. The descriptions are intended as examples of how the performance criteria might be addressed and are not considered to be exhaustive. The rubric also lists the documents and resources on which it was based

Reducible Connections in Massless Topological QCD and 4-manifolds

A role of reducible connections in Non-Abelian Seiberg-Witten invariants is analyzed with massless Topological QCD where monopole is extended to non-Abelian groups version. By giving small external fields, we found that vacuum expectation value can be separated into a part from Donaldson theory, a part from Abelian Monopole theory and a part from non-Abelian monopole theory. As a by-product, we find identities of U(1) topological invariants. In our proof, the duality relation and Higgs mechanism are not necessary.Comment: 23 pages, Latex, Some mistakes and typographical errors are revised in it. Especially, results written in section 4 are change