We propose a new notion of the formal tangent space to the Wasserstein space $\mathcal{P}(X)$ at a given measure. Modulo an integrability condition, we say that this tangent space is made of functions over $X$ which are valued in the probability measures over the tangent bundle to $X$. This generalization of previous concepts of tangent spaces allows us to define appropriate notions of parallel transport, $\mathcal{C}^{1,\alpha}$ regularity over $\mathcal{P}(X)$ and translation of a curve over $\mathcal{P}(X)$.

We introduce Magic Gems, a geometric representation of magic squares as three-dimensional polyhedra. By mapping an n x n magic square onto a centered coordinate grid with cell values as vertical displacements, we construct a point cloud whose convex hull defines the Magic Gem. This reveals a connection between magic square constraints and statistical structure: we prove that magic squares have vanishing covariances between position and value. We introduce a covariance energy functional -- the sum of squared covariances with row, column, and diagonal indicator variables -- and prove for n=3 (via exhaustive enumeration) that its zeros are precisely the magic squares. Large-scale sampling for n=4,5 (460+ million arrangements) provides strong numerical evidence that this characterization extends to larger orders. Perturbation analysis demonstrates that magic squares are isolated local minima. The representation is invariant under dihedral symmetry D_4, yielding canonical geometric objects for equivalence classes.

This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete and continuous classification of all finite clouds of unordered points under rigid motion in any Euclidean space. The key challenge was to avoid the exponential complexity arising from permutations of the given unordered points. For a fixed dimension of the ambient Euclidean space, the times of all algorithms for the resulting invariants and distance metrics depend polynomially on the number of points. The second part of the book advances a similar classification in the much more difficult case of periodic point sets, which model all periodic crystals at the atomic scale. The most significant result is the hierarchy of invariants from the ultra-fast to complete ones. The key challenge was to resolve the discontinuity of crystal representations that break down under almost any noise. Experimental validation on all major materials databases confirmed the Crystal Isometry Principle: any real periodic crystal has a unique location in a common moduli space of all periodic structures under rigid motion. The resulting moduli space contains all known and not yet discovered periodic crystals and hence continuously extends Mendeleev's table to the full crystal universe.

We analyze the gradient flow of a potential energy in the space of probability measures when we substitute the optimal transport geometry with a geometry based on Sinkhorn divergences, a debiased version of entropic optimal transport. This gradient flow appears formally as the limit of the minimizing movement scheme, a.k.a. JKO scheme, when the squared Wasserstein distance is substituted by the Sinkhorn divergence. We prove well-posedness and stability of the flow, and that, in the long term, the energy always converges to its minimal value. The analysis is based on a change of variable to study the flow in a Reproducing Kernel Hilbert Space, in which the evolution is no longer a gradient flow but described by a monotone operator. Under a restrictive assumption we prove the convergence of our modified JKO scheme towards this flow as the time step vanishes. We also provide numerical illustrations of the intriguing properties of this newly defined gradient flow.

Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative estimates between the sliced $1$-Wasserstein distance and the $1$-Wasserstein distance. We construct a concrete example to demonstrate the exponents in the estimate is sharp. We also provide a general analysis for the case where slicing involves projections onto $k$-planes and not just lines.

The concept of \emph{almost orthogonal vectors}, i.e.\ vectors whose cosine similarity is close to $0$, relates to topics both in pure mathematics and in coding theory under the guises of spherical packing and spherical codes. In recent years the rise of advanced language models in AI has created new interest in this concept as the models seem to store certain concepts as almost orthogonal directions in high-dimensional spaces. In this survey we represent some ideas regarding almost orthogonal vectors through three approaches: (1) the mathematical theory of almost orthogonality, (2) some observations from the embedding spaces of language models, and (3) generation of large sets of almost orthogonal vectors by simulations.

Nonnegative cross-curvature (NNCC) is a geometric property of a cost function defined on a product space that originates in optimal transportation and the Ma-Trudinger-Wang theory. Motivated by applications in optimization, gradient flows and mechanism design, we propose a variational formulation of nonnegative cross-curvature on c-convex domains applicable to infinite dimensions and nonsmooth settings. The resulting class of NNCC spaces is closed under Gromov-Hausdorff convergence and for this class, we extend many properties of classical nonnegative cross-curvature: stability under generalized Riemannian submersions, characterization in terms of the convexity of certain sets of c-concave functions, and in the metric case, it is a subclass of positively curved spaces in the sense of Alexandrov. One of our main results is that Wasserstein spaces of probability measures inherit the NNCC property from their base space. Additional examples of NNCC costs include the Bures-Wasserstein and Fisher-Rao squared distances, the Hellinger-Kantorovich squared distance (in some cases), the relative entropy on probability measures, and the 2-Gromov-Wasserstein squared distance on metric measure spaces.

The subject of this article is the proof that the Heron triangle with three integer medians does not exist. The article provides proofs of three lemmas. As a result, the infinite descent method proved that the Heron triangle with three integer medians does not exist. In the process of proving Lemma 3, the fact was used that there is no right-angled Heron triangle with three integer medians. And then it is proved that there is no Heron triangle with three natural medians. The relevance of this article lies in the fact that the problem under study is one of the unsolved problems of number theory.

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, non-flat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{soft tilings}. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.

Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.

A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra: for example, it is known that all 5 Platonic and 9 out of the 13 Archimedean solids admit Rupert's property. A commonly believed conjecture states that every convex polyhedron is Rupert. We prove that Rupert's problem is algorithmically decidable for polyhedra with algebraic coordinates. We also design a probabilistic algorithm which can efficiently prove that a given polyhedron is Rupert. Using this algorithm we not only confirm this property for the known Platonic and Archimedean solids, but also prove it for one of the remaining Archimedean polyhedra and many others. Moreover, we significantly improve on almost all known Nieuwland numbers and finally conjecture, based on statistical evidence, that the Rhombicosidodecahedron is in fact not Rupert.

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to calculate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

We introduce strings in metric spaces and define string complexes of metric spaces. We describe the class of 2-dimensional topological spaces which arise in this way from finite metric spaces.

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.

A theory of free spanning sets, free bases and their space group symmetric variants is developed for the first order flex spaces of infinite bar-joint frameworks. Such spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks. It is also shown that the existence of crystal flex bases is closely related to linear structure in the rigid unit mode (RUM) spectrum and a more general geometric flex spectrum.

A conjecture regarding the structure of expander graphs is discussed.