Mastodawn

Today my paper "Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds" got published in the Journal of Geometry and Physics!

The paper defines: A globally hyperbolic spacetime $(X,g)$ is observer-refocusing if there exists a point $p$ and a timelike curve $\gamma$ in $X$ so that all lightrays emitted from $p$ intersect $\gamma$. The paper proves that spacetimes $(X,g)$ with $\dim(X)\geq 3$ which are observer-refocusing with respect to a compact timelike curve have compact Cauchy surfaces with finite fundamental group. This extends known results on strongly refocusing spacetimes, which are spacetimes with points $p,q$ so that all lightrays through $p$ go through $q$. Further, observer-refocusing spacetimes of dimension at least $3$ with an analytic metric are strongly refocusing.

These results lead to immediate corollaries in Riemannian geometry: Let $(M,h)$ be a connected, complete Riemannian manifold and let $x \in M$. We call $(M,h)$ a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We show that if $\dim(M) \geq 2$ and if the return time of unit-speed geodesics starting at $x$ is uniformly bounded, then $M$ is compact with finite fundamental group. Further, if the metric of a $Z^x$ manifold is analytic, then all unit-speed geodesics starting at $x$ return to $x$ at a common time. This resolves the question "Are all $Z^x$ manifolds $Y^x_l$ manifolds for some $l>0$?" posed in Besse's book "Manifolds all of whose geodesics are closed" affirmatively for analytic manifolds.

https://doi.org/10.1016/j.geomphys.2026.105834
https://arxiv.org/abs/2503.23565

#DifferentialGeometry #MathematicalPhysics #GeneralRelativity

Show thread

katch wreck Mar 11

#theoreticalPhysics #quantumMechanics #decoherence #wavefunction #wavefunctionCollapse #measurement #mathematicalPhysics #functionalAnalysis #equation

Her Maths Story Feb 25

“Looking back, my path has not been linear, and I changed direction more than once; however, what has stayed constant is curiosity, even when the topics and the places were changing. This is also what I like most about mathematics: there is room for many different trajectories, as long as you keep following questions that genuinely interest you.” - Mikaela Iacobelli

➡️ https://hermathsstory.eu/mikaela-iacobelli

#Academia #PhD #AssociateProfessor #MathematicalPhysics #PDEs #KineticTheory #MathsJourney #WomenInSTEM #WomenInMaths #HerMathsStory