An Efficient Triangulation of $\mathbb{R}P^5$

ArXiv.org · 2026-03-08

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.

Conformal Rigidity and Spectral Embeddings of Graphs

Annals of Combinatorics · 2026-05-16

Distance Equilibrium Measures and Curvature in Metric Spaces

arXiv (Cornell University) · 2026-01-01

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $μ$ with the property that $$ \int_{X} d(x, y) dμ(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist, encode a `curvature-type' quantity. We investigate this in the special case where $X$ is a closed, convex curve in $\mathbb{R}^2$ and $d = \| \cdot \|_2$ is the Euclidean distance: even a single point with small curvature implies non-existence of such a measure. Conversely, such a measure $μ$ exists for all curves whose curvature is sufficiently close to constant. Curvature is usually defined by second derivatives; this one is defined via an integral equation which makes sense in much rougher spaces. Connections to curvature on graphs, the Gross-Stadje Theorem and magnitude are discussed.

A Stability Version of the Jones Opaque Set Inequality

Discrete & Computational Geometry · 2026-05-06

Slow dispersion in Floquet-Dirac Hamiltonians

arXiv (Cornell University) · 2026-03-30

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of $t^{-1/5}$. It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than $t^{-1/10}$. Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, $t^{-\varepsilon}$ for every $\varepsilon > 0$, should be achievable.

Hyperintense FLAIR signal in the anterior cranial fossa

Nature Communications · 2026-05-06

Slow dispersion in Floquet-Dirac Hamiltonians

arXiv (Cornell University) · 2026-03-30

We study dispersive decay for non-autonomous Hamiltonian systems. While the general theory for dispersion in such non-autonomous systems is largely open, it was shown \cite{kraisler2025time} that there exists a time-periodically forced one-dimensional Dirac equation with unusually slow dispersive decay rate of $t^{-1/5}$. It is to be expected that such behavior is not generic and requires a very particular forcing term; we provide a more general ansatz and systematic procedure to construct such an equation with a dispersive decay rate no faster than $t^{-1/10}$. Our limitations are purely algebraic and it stands to reason that arbitrarily slow decay, $t^{-\varepsilon}$ for every $\varepsilon > 0$, should be achievable.

Distance Equilibrium Measures and Curvature in Metric Spaces

arXiv (Cornell University) · 2026-02-22

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $μ$ with the property that $$ \int_{X} d(x, y) dμ(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist, encode a `curvature-type' quantity. We investigate this in the special case where $X$ is a closed, convex curve in $\mathbb{R}^2$ and $d = \| \cdot \|_2$ is the Euclidean distance: even a single point with small curvature implies non-existence of such a measure. Conversely, such a measure $μ$ exists for all curves whose curvature is sufficiently close to constant. Curvature is usually defined by second derivatives; this one is defined via an integral equation which makes sense in much rougher spaces. Connections to curvature on graphs, the Gross-Stadje Theorem and magnitude are discussed.

An Efficient Triangulation of $\mathbb{R}P^5$

Open MIND · 2026-03-08

We present a $6$-dimensional centrally symmetric simplicial polytope for which the antipodal quotient of its boundary forms a $24$-vertex triangulation of the $5$-dimensional real projective space. This $6$-polytope is highly symmetric with an automorphism group of order $192$, and is of independent interest. We conjecture that our construction uses the fewest number of vertices among all triangulations of $\mathbb{R}P^5$. Our method also produces two triangulations of $\mathbb{R}P^6$ on $45$ and $49$ vertices; both improve the previously best known construction in dimension $6$ that used $53$ vertices.

An effective variant of the Hartigan $k$-means algorithm

arXiv (Cornell University) · 2026-04-23

The k-means problem is perhaps the classical clustering problem and often synonymous with Lloyd's algorithm (1957). It has become clear that Hartigan's algorithm (1975) gives better results in almost all cases, Telgarsky-Vattani note a typical improvement of $5\%$ -- $10\%$. We point out that a very minor variation of Hartigan's method leads to another $2\%$ -- $5\%$ improvement; the improvement tends to become larger when either dimension or $k$ increase.