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Bourgain proves that every metric space with $n$ points embeds in Euclidean space with distortion $O(\log n)$, and constructs, from random graphs, metric spaces for which every embedding has distortion at least a constant multiple of $\log n/\log\log n$. The lower bound shows in particular that $O(\sqrt{\log n})$ is not achievable, and answers negatively a question of Johnson and Lindenstrauss (1984) about embeddings into normed spaces of dimension $\log n$. The embedding sends each point to its distances from random sets of points of every size up to $n$, and the analysis rests on a single estimate of how often such a set separates two given points. The theorem became a cornerstone of the theory of low-distortion embeddings of finite metric spaces. A decade later, Linial, London and Rabinovich (1995) made the construction algorithmic and proved that expander graphs require distortion of order $\log n$, so that Bourgain's bound is best possible; with the independent work of Aumann and Rabani (1998), this made the embedding a basic tool of approximation algorithms, beginning with the sparsest cut problem.

We calculate K_*(k[x]/x^e;Z_p) by evaluating the syntomic cohomology Z_p(i)(k[x]/x^e) introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case e=2 and p>2. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for e=2. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.

Turán's theorem is the founding result of extremal graph theory: among all graphs on $n$ vertices with no clique on $r+1$ vertices, the balanced complete multipartite graph has the most edges. We present the theorem through five classical proofs and follow one of them, the variational argument of Motzkin and Straus, upward into spectral graph theory, where Nikiforov's spectral strengthening returns and proves the original theorem in two lines. All mathematics herein is known; the contribution is the form of its presentation.

The rich club coefficient $\phi(k)$ quantifies whether high-degree nodes interconnect more densely than expected, but it is blind to the structure of those interconnections and offers no principled way to select the degree threshold $k$. We introduce the *cycle core* $H_k$, the maximal subgraph of vertices with degree at least $k$ that forms cycles among themselves. The nested sequence $H_1 \supseteq H_2 \supseteq \cdots$ reveals how the cycle core evolves as the degree threshold increases, capturing structure that $\phi(k)$ cannot see. We develop two spectral tools for this sequence. The first is structural: the *$k$-blocking operator* $\mathbf{M}_k$ generalizes the non-backtracking matrix by blocking $k - 1$ directions at each node. We prove that its spectrum is equivalent to that of a modified non-backtracking matrix weighted with a specific function of the nodes' degrees. Thus, $\mathbf{M}_k$ couples cycle structure with the degree sequence of the graph. We show the spectral radius of $\mathbf{M}_k$ peaks at an intrinsic threshold $k^*$, yielding the *optimal rich club* $H_{k^*}$ without null models or parameter tuning. The second tool is dynamical: the $k$-blocking walk defines a sub-stochastic Markov chain whose leading eigenvalue $\lambda_1(k)$ measures how long a process can persist on the cycle core before being expelled to the periphery. We evaluate the framework on fifteen networks spanning different domains, wherein $H_{k^*}$ closely recovers the published rich clubs.

Can graphs represent the dynamics of higher-order (multi-body) interactions, or are hypergraphs necessary? We decompose network dynamics into three components: the combinatorial structure (graph or hypergraph), per-edge coupling functions, and per-node aggregation functions. Within this framework, we prove that graphs can always reproduce the trajectories of any hypergraph dynamics by absorbing the coupling logic into the aggregation. This generalizes a recent result by Peixoto et al., who showed the same under the assumption of flat dynamics. However, the graph representation pays a price in parsimony: when the coupling and aggregation functions are polynomials of degree at most D, the graph aggregation requires Θ(k^D/D!) parameters relative to the hypergraph alternative, where k is the interaction size. Without symmetry, the comparison can reverse. The framework also clarifies two other recent contributions by Moutuou and by Lacasa, and identifies overlooked modeling degrees of freedom including partial interaction symmetries and the role of non-additive aggregation.

Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps (LE), which constructs a graph embedding based on the spectral properties of the Laplacian matrix of G. The intuition behind it, and many other embedding techniques, is that the embedding of a graph must respect node similarity: similar nodes must have embeddings that are close to one another. Here, we dispose of this distance-minimization assumption. Instead, we use the Laplacian matrix to find an embedding with geometric properties instead of spectral ones, by leveraging the so-called simplex geometry of G. We introduce a new approach, Geometric Laplacian Eigenmap Embedding, and demonstrate that it outperforms various other techniques (including LE) in the tasks of graph reconstruction and link prediction.