These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein’s time. I shall explain these difficulties and progresses that have been made. In particular, I shall introduce new definitions of center of mass and angular momentum at both the quasi-local and total levels, which are derived from first principles in general relativity and by the method of geometric analysis. With these new definitions, the classical formula $p = mv$ is shown to be consistent with Einstein’s field equation for the first time. This paper is based on joint work with Po-Ning Chen and Shing-Tung Yau.
We describe recent progress on rationality and stable rationality questions. We discuss the cohomological or Chow decomposition of the diagonal, a very strong stably birationally invariant property which controls many of the previously defined stable birational invariants. On the other hand, it behaves very well under specialization and desingularization of mild singularities.
The Katz–Klemm–Vafa conjecture expresses the Gromov–Witten theory of K3 surfaces (and K3-fibred 3-folds in fibre classes) in terms of modular forms. Its recent proof gives the first non-toric geometry in dimension greater than 1 where Gromov–Witten theory is exactly solved in all genera.
We survey the various steps in the proof. The MNOP correspondence and a new Pairs/Noether–Lefschetz correspondence for K3-fibred 3-folds transform the Gromov–Witten problem into a calculation of the full stable pairs theory of a local K3-fibred 3-fold. The stable pairs calculation is then carried out via degeneration, localisation, vanishing results, and new multiple cover formulae.
In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.
In this article we survey what is known about the existence of minimal varieties of dimension $l \geq 2$ in compact Riemannian manifolds. We describe how min-max methods can be used in conjunction with the nontrivial topology of the space of cycles. In the final section, we propose some open questions in the subject.
By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere, we encounter the notion of a folded hyperkähler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmüller space.
This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel. While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality. This suggestion originates with the work of the late Andrei Tyurin. We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties.
This is an expository article, discussing problems and developments in the theory of singularity formation for sequences of Kähler–Einstein metrics. In particular we consider the algebro-geometric significance of the metric tangent cones of a limit of Kähler–Einstein manifolds.
A well-known monograph of Almgren proves that the singular set of a general $n$-dimensional area-minimizing integral currents has dimension at most $n-2$, which is an optimal bound when the dimension of the ambient manifold is larger than $n+1$. Almgren’s original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren’s theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry.
I shall discuss the Chen–Wang–Yau quasilocal angular momentum, which is defined based on the theory of optimal isometric embedding and quasilocal mass of Wang–Yau, and the limits of which at spatial and null infinity of an isolated gravitating system. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.