CyberSec Research

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

This paper develops a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple $(H, G, \mathfrak{m})$: $H$ is a closed achronal set in a topological causal space, $G$ is a gauge function encoding affine parametrizations along null generators, and $\mathfrak{m}$ is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the synthetic null energy condition ($\mathsf{NC}^e(N)$), defined via the concavity of an entropy power functional along optimal transport, with parametrization given by the gauge $G$. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of $\mathsf{NC}^e(N)$ is the stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov--Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking's area theorem. Moreover, we obtain various sharpenings of the celebrated Penrose's singularity theorem: for smooth spacetimes we show that the incomplete null geodesic whose existence is guaranteed by Penrose's argument is actually maximizing; we extend Penrose's singularity theorem to continuous spacetimes; we prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic $\mathsf{NC}^e(N)$.

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Dru\c{t}u-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.

We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When restricted to bounded sequences in normed linear spaces, we prove that our notion of weak convergence agrees with the standard one.

Let $\mathcal{A}$ be the subdivision of $\mathbb{R}^d$ induced by $m$ convex polyhedra having $n$ facets in total. We prove that $\mathcal{A}$ has combinatorial complexity $O(m^{\lceil d/2 \rceil} n^{\lfloor d/2 \rfloor})$ and that this bound is tight. The bound is mentioned several times in the literature, but no proof for arbitrary dimension has been published before.

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichm{\"u}ller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichm{\"u}ller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schr\"odinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small.

In this paper, we revisit the notion of higher-order rigidity of a bar-and-joint framework. In particular, we provide a link between the rigidity properties of a framework, and the growth order of an energy function defined on that framework. Using our approach, we propose a general definition for the rigidity order of a framework, and we show that this definition does not depend on the details of the chosen energy function. Then we show how this order can be studied using higher order derivative tests. Doing so, we obtain a new proof that the lack of a second order flex implies rigidity. Our proof relies on our construction of a fourth derivative test, which may be applied to a critical point when the second derivative test fails. We also obtain a new proof that when the dimension of non-trivial first-order flexes equals $1$, then the lack of a $k$th order flex for some $k$ implies a framework is rigid. The higher order derivative tests that we study here may have applications beyond rigidity theory.

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds \`a la Lott-Villani-Sturm and Ohta in the metric measure case, and \`a la Cavalletti-Mondino in Lorentzian signature. Here, a generalized cone is a warped product of a one-dimensional base space, which will be positive or negative definite, over a fiber that is a metric space. We prove that Riemannian or Lorentzian generalized cones over $\mathsf{CD}$-spaces satisfy the (timelike) measure contraction property $\mathsf{(T)MCP}$ - a weaker version of a (timelike) curvature-dimension condition $\mathsf{(T)CD}$. Conversely, if the generalized cone is a $\mathsf{(T)CD}$-space, then the fiber is a $\mathsf{CD}$-space with the appropriate bounds on Ricci curvature and dimension. In proving these results we develop a novel and powerful two-dimensional localization technique, which we expect to be interesting in its own right and useful in other circumstances. We conclude by giving several applications including synthetic singularity and splitting theorems for generalized cones. The final application is that we propose a new definition for lower curvature bounds for metric and metric measure spaces via lower curvature bounds for generalized cones over the given space.

A long-standing conjecture of Lapidus asserts that, under certain conditions, a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in $\mathbb{R}^2$ that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results.

Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous problems in isotropic geometry. Isotropic geometry can be viewed as a structure-preserving simplification of Euclidean geometry. The solutions found in the isotropic case give insight and can initialize optimization algorithms to solve the original Euclidean problems. We illustrate this general approach with three examples: quad-mesh mechanisms, composite asymptotic-geodesic gridshells, and asymptotic gridshells with constant node angle.

After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \cite{Cs-Sz25}, we generalize the famous Menelaus' and Ceva's theorems for translation triangles in the mentioned spaces. The described method makes it possible to transfer further classical Euclidean theorems and notions to Thurston geometries with non-constant curvature. In our work we will use the projective models of Thurston geometries described by E. Moln\'ar in \cite{M97}.

We discuss the analogy between collapsing Conformal Field Theories and measured Gromov-Hausdorff limit of Riemannian manifolds with non-negative Ricci curvature. Motivated by this analogy we propose the notion of non-commutative (``quantum") Riemannian d-geometry. We explain how this structure is related to Connes's spectral triples in the case d=1. In the Appendix based on the unpublished joint work with Maxim Kontsevich we discuss deformation theory of Quantum Field Theories as well as an approach to QFTs in the case when the space-time is an arbitrary compact metric space.

For uniformly dicrete metric spaces without bounded geometry we suggest a modified version of property A based on metrics of bounded geometry greater than the given metric. We show that this version still implies coarse embeddability in Hilbert spaces, and that some examples of non-property A spaces of unbounded geometry satisfy this version. We also relate this version of property A to our version of uniform Roe algebras for spaces without bounded geometry and introduce an appropriate equivalence relation.

We establish geometric relationships between the average scale-invariant Cassinian metric and other hyperbolic type metrics. In addition, we study the local convexity properties of the scale-invariant metric balls in Euclidean once punctured spaces.

We show that under a lower Ricci curvature bound and an upper diameter bound, a torus admits a finite-sheeted covering space with volume bounded from below and diameter bounded from above. This partially recovers a result of Kloeckner and Sabourau, whose original proof contains a serious gap that currently lacks a resolution.

This paper establishes a metric framework for Spencer complexes based on the geometric theory of compatible pairs $(D,\lambda)$ in principal bundle constraint systems, solving fundamental technical problems in computing Spencer cohomology of constraint systems. We develop two complementary and geometrically natural metric schemes: a tensor metric based on constraint strength weighting and an induced metric arising from principal bundle curvature geometry, both maintaining deep compatibility with the strong transversality structure of compatible pairs. Through establishing the corresponding Spencer-Hodge decomposition theory, we rigorously prove that both metrics provide complete elliptic structures for Spencer complexes, thereby guaranteeing the existence, uniqueness and finite-dimensionality of Hodge decompositions. It reveals that the strong transversality condition of compatible pairs is not only a necessary property of constraint geometry, but also key to the elliptic regularity of Spencer operators, while the introduction of constraint strength functions and curvature weights provides natural weighting mechanisms for metric structures that coordinate with the intrinsic geometry of constraint systems. This theory tries to unify the differential geometric methods of constraint mechanics, cohomological analysis tools of gauge field theory, and classical techniques of Hodge theory in differential topology, establishing a mathematical foundation for understanding and computing topological invariants of complex constraint systems.