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Stephen Lynch

Publications and source records attributed to Stephen Lynch.

17 recordsLinked to original sources
arXiv2026

Time-to-Injury Forecasting in Elite Female Football: A DeepHit Survival Approach

Injury occurrence in football poses significant challenges for athletes and teams, carrying personal, competitive, and financial consequences. While machine learning has been applied to injury prediction before, existing approaches often rely on static pre-season data and binary outcomes, limiting their real-world utility. This study investigates the feasibility of using a DeepHit neural network to forecast time-to-injury from longitudinal athlete monitoring data, while providing interpretable predictions. The analysis utilised the publicly available SoccerMon dataset, containing two seasons of training, match, and wellness records from elite female footballers. Data was pre-processed through cleaning, feature engineering, and the application of three imputation strategies. Baseline models (Random Forest, XGBoost, Logistic Regression) were optimised via grid search for benchmarking, while the DeepHit model, implemented with a multilayer perceptron backbone, was evaluated using chronological and leave-one-player-out (LOPO) validation. DeepHit achieved a concordance index of 0.762, outperforming baseline models and delivering individualised, time-varying risk estimates. Shapley Additive Explanations (SHAP) identified clinically relevant predictors consistent with established risk factors, enhancing interpretability. Overall, this study provides a novel proof of concept: survival modelling with DeepHit shows strong potential to advance injury forecasting in football, offering accurate, explainable, and actionable insights for injury prevention across competitive levels.

cs.LG↗
arXiv2025

Translators Asymptotic to Planes

We prove that a vertical plane is the only complete translator, properly immersed in $\mathbb{R}^3$ and having finite topology, whose ends are asymptotic to vertical planes.

math.DG↗
arXiv2024

Canonical foliation of bubblesheets

We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC.

math.DG↗
arXiv2023

A differential Harnack inequality for noncompact evolving hypersurfaces

We prove a differential Harnack inequality for noncompact convex hypersurfaces flowing with normal speed equal to a symmetric function of their principal curvatures. This extends a result of Andrews for compact hypersurfaces. We assume that the speed of motion is one-homogeneous, uniformly elliptic, and suitably 'uniformly' inverse-concave as a function of the principal curvatures. In addition, we assume the hypersurfaces satisfy pointwise scaling-invariant gradient estimates for the second fundamental form. For many natural flows all of these hypotheses are met by any ancient solution which arises as a blow-up of a singularity.

math.DG↗
arXiv2023

Plateau's problem via the Allen--Cahn functional

Let $\Gamma$ be a compact codimension-two submanifold of $\mathbb{R}^n$, and let $L$ be a nontrivial real line bundle over $X = \mathbb{R}^n \setminus \Gamma$. We study the Allen--Cahn functional, \[E_\varepsilon(u) = \int_X \varepsilon \frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon}\,dx,\] on the space of sections $u$ of $L$. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to $\Gamma$. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as $\varepsilon \to 0$, the associated family of energy measures converges to an integer rectifiable $(n-1)$-varifold $V$. Moreover, $V$ is stationary with respect to any variation which leaves $\Gamma$ fixed. Away from $\Gamma$, this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary $\Gamma$. Under additional hypotheses, we can say more about $V$. When $V$ arises as a limit of critical sections with uniformly bounded Morse index, $\Sigma := \operatorname{supp} \|V\|$ is a minimal hypersurface, smooth away from $\Gamma$ and a singular set of Hausdorff dimension at most $n-8$. If the sections are globally energy minimizing and $n = 3$, then $\Sigma$ is a smooth surface with boundary, $\partial \Sigma = \Gamma$ (at least if $L$ is chosen correctly), and $\Sigma$ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fr\"{o}hlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in $\mathbb{R}^3$. This also works if $4 \leq n\leq 7$ and $\Gamma$ is assumed to lie in a strictly convex hypersurface.

math.DG↗
arXiv2022

Ancient solutions of Ricci flow with Type I curvature growth

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type~I curvature growth. We show that a $\kappa$-noncollapsed Type~I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean factor. It follows that a $\kappa$-noncollapsed Type~I ancient solution which is weakly PIC2 is a locally symmetric space.

math.DG↗
arXiv2021

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell}$, the round $(n+\ell)$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2}|H|^{2} + 4K$ when $n\geq 8$, $|A|^{2} < \frac{4}{3n}|H|^{2}+\frac{n}{2}K$ when $n=7$, and $|A|^2<\frac{3(n+1)}{2n(n+2)}|H|^2+\frac{2n(n-1)}{3(n+1)}K$ when $n=5$ or $6$. This condition is related to a theorem of Li and Li [Arch. Math., 58:582--594, 1992] which states that the only $n$-dimensional minimal submanifolds of $S_K^{n+\ell}$ satisfying $|A|^2<\frac{2n}{3}K$ are the totally geodesic $n$-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of $S_K^{n+1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^n$ or to the connected sum of a finite number of handles $S^1\times S^{n-1}$. The results are sharp when $n\geq 8$ due to hypersurface counterexamples.

math.DG↗
arXiv2021

Collapsing and noncollapsing in convex ancient mean curvature flow

We provide several characterisations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in $(n-1)$-convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that $(n-1)$-convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.

math.DG↗
arXiv2021

Uniqueness of convex ancient solutions to hypersurface flows

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

math.DG↗
arXiv2020

Convexity estimates for hypersurfaces moving by concave curvature functions

We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

math.DG↗
arXiv2020

Convexity Estimates for High Codimension Mean Curvature Flow

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.

math.DG↗
arXiv2020

High Codimension Mean Curvature Flow with Surgery

We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions $n \geq 8$. Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to $\mathbb{S}^n$ or to a finite connected sum of $\mathbb{S}^{n-1}$-bundles over $\mathbb{S}^1$.

math.DG↗
arXiv2019

Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov-Arnold-Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic.

physics.atom-ph↗
arXiv2017

Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.

math.DG↗
arXiv2017

Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates') for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of the shrinking sphere amongst ancient solutions of these flows.

math.DG↗
arXiv2016

Bifurcation and Stability Analysis of Bistable Neuromodules

This paper presents a stability analysis of simple neuromodules displaying fold bifurcations (leading to hysteresis), flip bifurcations (period doubling and undoubling to and from chaos) and Neimark-Sacker bifurcations (quasiperiodic and periodic bifurcations). For the first time, bifurcation diagrams are plotted using a feedback mechanism. It is shown that the stability curves and bifurcation diagrams must be dealt with simultaneously in order to fully understand the dynamics of the systems involved. Synaptic weights, biases and gradients of transfer functions are varied and the system is shown to be history dependent. The work can be applied to artificial neural networks and developing brains and gives a very important generalization of previous work in this field.

math.DS↗