# Dr James Grant

## About

### Biography

- 1989: B.Sc. (Hons.) in Mathematical Physics, University of Edinburgh.
- 1990: Certificate of Advanced Study in Mathematics (with distinction), University of Cambridge.
- 1993: Ph.D. Mathematics, University of Cambridge.

Numerous postdoctoral positions. Joined Department of Mathematics, University of Surrey as a Reader in January 2013.

For more details, please visit http://jdegrant.wordpress.com.

### University roles and responsibilities

- 2018-Present: Chair of Examinations.
- 2013-18: Coordinator of the literature reviews and projects for B.Sc. and MMath students.

## Research

### Research interests

My work now centres on the application of tools from analysis and the theory of partial differential equations to problems in geometry and mathematical physics. I am particularly interested in problems related to low-regularity geometrical structures. I also have long-term interests in mathematical relativity, spectral theory and the theory of integrable systems.

## Teaching

- 2013 - Present: Real Analysis 2 (MAT2004).
- 2014 - 2019: General Topology (MATM042).
- 2019 - Present: Manifolds and Topology (MAT3009).

## Publications

We show differentiability of a class of Geroch's volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally anti-Lipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably causal spacetimes Hawking's time function can be uniformly approximated by smooth time functions with timelike gradient.

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by space-like hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse function of the foliation and sup-norm bounds on the deformation tensors of the foliation. Our proof is inspired by techniques from Riemannian geometry, and it should be noted that we impose no restriction on the size of the bound satisfied by the curvature or deformation tensors, and allow for metrics that are “far” from the Minkowski one. The relevance of our estimate is illustrated with a class of plane-symmetric spacetimes which satisfy our assumptions but admit no uniform lower bound on the curvature not even in the L2 norm. The conditions we put forward, therefore, lead to a uniform control of the spacetime geometry and should be useful in the context of general relativity.

We present recent developments concerning Lorentzian geometry in algebras of generalized functions. These have, in particular, raised a new interest in refined regularity theory for the wave equation on singular space-times.

We give a spinorial set of Hamiltonian variables for General Relativity in any dimension greater than 2. This approach involves a study of the algebraic properties of spinors in higher dimension, and of the elimination of second-class constraints from the Hamiltonian theory. In four dimensions, when restricted to the positive spin-bundle, these variables reduce to the standard Ashtekar variables. In higher dimensions, the theory can either be reduced to a spinorial version of the ADM formalism, or can be left in a more general form which seems useful for the investigation of some spinorial problems such as Riemannian manifolds with reduced holonomy group. In dimensions $0 ____pmod 4$, the theory may be recast solely in terms of structures on the positive spin-bundle $____mathbb{V}^+$, but such a reduction does not seem possible in dimensions $2 ____pmod 4$, due to algebraic properties of spinors in these dimensions.

We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space $W^{2, n/2}_{____mathrm{loc}}(M)$. We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvature fails to be non-negative, where the negative part has compact support and sufficiently small $L^{n/2}$ norm. We show that a Riemannian metric in $W^{2, p}_{loc}$ for some $p > ____frac{n}{2}$ with non-negative scalar curvature in the distributional sense can be approximated locally uniformly by smooth metrics with non-negative scalar curvature. For continuous metrics in $W^{2, n/2}_{loc}$, there exist smooth approximating metrics with non-negative scalar curvature that converge in $L^p_{loc}$ for all $p < ____infty$.

We show that the Hawking–Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of C1 , 1-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for C1 , 1-metrics, and of C0-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.

A generalised canonical formulation of gravity is devised for foliations of spacetime with codimension $n_ge1$. The new formalism retains n-dimensional covariance and is especially suited to 2+2 decompositions of spacetime. It is also possible to use the generalised formalism to obtain boundary contributions to the 3+1 Hamiltonian.

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open and may differ from the corresponding sets defined via piecewise C¹ -curves. By refining the notion of a causal bubble from Chruściel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (push-up). The phenomena described here are, in particular, relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.

We consider some natural connections which arise between right-flat (p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form, with a "Lax p-tuple" of linear differential operators, depending a spectral parameter which lives in (q-1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives with respect to the spectral parameter.

We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in [KS17]. To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on recent analytic work in this direction and, for the first time, relate low-regularity inextendibility to (synthetic) curvature blow-up.

We conjecture an interpretation in terms of multipole moments of the obstructions to smoothness at infinity found for time-symmetric, conformally-flat initial data by Kroon (Commun Math Phys 244(1):133–156, 2004).

We apply the methods of value distribution theory to the spectral asymptotics of Schrodinger operators with L^2-sparse potentials.