Let $M^{({k})}_{d}(n)$ be the manifold of n-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-k-equal coordinates. We show that, for $d\geq2$, $M^{({3})}_{d}(n)$ is rationally formal if and only if $n\leq6$. This stands in sharp contrast with the fact that all classical configuration spaces $M^{({2})}_d(n)=\text{Conf}(\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non-formality of $M^{({3})}_{d}(n)$ for $n \gt 6$ is established via detection of non-trivial triple Massey products, which are assessed geometrically through Poincaré duality.

We define a generic rigidity matroid for k-volumes of a simplicial complex in $\mathbb {R}^d$ and prove that for $2\leq k \leq d-1$ it has the same rank as the classical generic d-rigidity matroid on the same vertex set (namely, the case $k=1$). This is in contrast with the $k=d$ case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of d-rigidity of the $1$-skeleton of the complex and a combinatorial Hall condition on incidences of edges in k-faces.

A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behavior with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

A finite point set in $\mathbb{R}^d$ is in general position if no $d + 1$ points lie on a common hyperplane. Let $\alpha _d(N)$ be the largest integer such that any set of $N$ points in $\mathbb{R}^d$, with no $d + 2$ members on a common hyperplane, contains a subset of size $\alpha _d(N)$ in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that $\alpha _2(N) \lt N^{5/6 + o(1)}$. In this paper, we also use the container method to obtain new upper bounds for $\alpha _d(N)$ when $d \geq 3$. More precisely, we show that if $d$ is odd, then $\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{2d} + o(1)}$, and if $d$ is even, we have $\alpha _d(N) \lt N^{\frac {1}{2} + \frac {1}{d-1} + o(1)}$. We also study the classical problem of determining $a(d,k,n)$, the maximum number of points selected from the grid $[n]^d$ such that no $k + 2$ members lie on a $k$-flat, and improve the previously best known bound for $a(d,k,n)$, due to Lefmann in 2008, by a polynomial factor when $k$ = 2 or 3 (mod 4).

For each closed subtorus T of $(\mathbb{R}/\mathbb{Z})^n$, let D(T) denote the (infimal) $L^\infty$-distance from T to the point $(1/2,\ldots, 1/2)$. The nth Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by D(T) as T ranges over the 1-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture directly, we study the qualitative structure of the sets $\mathcal{S}(n)$ via their accumulation points. This project brings into the picture the analogues of $\mathcal{S}(n)$ where 1-dimensional subtori are replaced by k-dimensional subtori or k-dimensional subgroups.

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to $\binom {n+1}{2}^{n-1} \le n^{2n}$ are enough.

We prove the same finite-checking result, with the same bound, for the more general shifted Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the Lonely Vector Problem (LVP), about sumsets of n rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with n generators, a class of lattice zonotopes that we introduce.

In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to $195$.

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda $ can be defined by a finite set of forbidden induced subgraphs if and only if $\lambda < \lambda ^*$, where $\lambda ^* = \rho ^{1/2} + \rho ^{-1/2} \approx 2.01980$, and $\rho $ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman’s work on those limit points in $[-2, \infty )$.

We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{\alpha , \beta }(d)$ the maximum number of unit vectors in $\mathbb {R}^d$ where all pairwise inner products lie in $\{\alpha , \beta \}$ with $-1 \le \beta < 0 \le \alpha < 1$. Very recently Jiang, Tidor, Yao, Zhang, and Zhao determined the limit of $N_{\alpha , \beta }(d)/d$ as $d\to \infty $ when $\alpha + 2\beta < 0$ or $(1-\alpha )/(\alpha -\beta ) \in \{1,\sqrt 2,\sqrt 3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-\alpha )/(\alpha - \beta ) < \lambda ^*$.

We consider a new family $(\mathcal {T}_n)_{n\geq 1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $\Omega _n$ of valid configurations $\mathbb {Z}^2\to \mathcal {T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari–Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $\Omega _n$ are the codings of a $\mathbb {Z}^2$-action on a 2-dimensional torus like the Jeandel–Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $\Omega _n$ involves the positive root $\beta $ of the polynomial $x^2-nx-1$, also known as the n-th metallic mean. We show the existence of an almost one-to-one factor map $\Omega _n\to \mathbb {T}^2$ which commutes the shift action on $\Omega _n$ with horizontal and vertical translations by $\beta $ on $\mathbb {T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $\Omega _n$ (proved in a previous article) and a polygonal partition of $\mathbb {T}^2$ which we show is a Markov partition for the toral $\mathbb {Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation.

In this article, we will provide a new proof of the fact that for any convex body $K\subseteq \mathbb {R}^n$,

$$ \begin{align*}\frac{{{2n}\choose{n}}}{n^n}n\int_0^\infty r^{n-1}\mathrm{vol}_n(K\cap(re_n+K))dr\leq\frac{(\mathrm{vol}_n(K))^{n+1}}{(\mathrm{vol}_{n-1}(P_{e_n^\perp}(K)))^n}, \end{align*} $$

$(e_i)_{i=1}^n$

$\mathbb {R}^n$

$P_{e_n^\perp }(K)$

$e_n$

$\mathrm {vol}_k$

We show that for every non-spherical set X in $\mathbb {E}^d$, there exists a natural number m and a red/blue-coloring of $\mathbb {E}^n$ for every n such that there is no red copy of X and no blue progression of length m with each consecutive point at distance $1$. This verifies a conjecture of Wu and the first author.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

For every positive integer n, we introduce a set ${\mathcal {T}}_n$ made of $(n+3)^2$ Wang tiles (unit squares with labeled edges). We represent a tiling by translates of these tiles as a configuration $\mathbb {Z}^2\to {\mathcal {T}}_n$. A configuration is valid if the common edge of adjacent tiles has the same label. For every $n\geq 1$, we show that the Wang shift ${\Omega }_n$, defined as the set of valid configurations over the tiles ${\mathcal {T}}_n$, is self-similar, aperiodic and minimal for the shift action. We say that $\{{\Omega }_n\}_{n\geq 1}$ is a family of metallic mean Wang shifts, since the inflation factor of the self-similarity of $\Omega _n$ is the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the n-th metallic mean, and in particular, the golden mean when $n=1$, and the silver mean when $n=2$. When $n=1$, the set of Wang tiles ${\mathcal {T}}_1$ is equivalent to the Ammann aperiodic set of 16 Wang tiles.

Factorization structures occur in toric differential and discrete geometry and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such or as an apex generalization of cyclic polytopes featuring a generalized Gale’s evenness condition. This article presents a comprehensive study of this new concept called factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains a construction of polytopes and cones compatible with a given factorization structure and exemplifies it for the product Segre–Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalized Gale’s evenness condition for compatible cones, polytopes, and their duals and explicitly describes faces of these. Factorization structures naturally provide generalized Vandermonde identities, which relate normals of any compatible polytope, and which are used to find examples of Delzant and rational Delzant polytopes compatible with the Veronese factorization structure. The article offers a myriad of factorization structure examples, which are later characterized to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.

Let $A\ \mathrm{and}\ B$ be subsets of $(\mathbb {Z}/p^r\mathbb {Z})^2$. In this note, we provide conditions on the densities of A and B such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb {Z}/p^r\mathbb {Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.

In [5], a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call hyperpolygonal arrangements ${\mathscr H}_n$. In this note, we study these arrangements and investigate their properties systematically. Remarkably, the arrangements ${\mathscr H}_n$ discriminate between essentially all local properties of arrangements. In addition, we show that hyperpolygonal arrangements are projectively unique and combinatorially formal.

We note that the arrangement ${\mathscr H}_5$ is the famous counterexample of Edelman and Reiner [17] of Orlik’s conjecture that the restriction of a free arrangement is again free.

Orbit separation dimension ($\mathrm {OSD}$), previously introduced as amorphic complexity, is a powerful complexity measure for topological dynamical systems with pure-point spectrum. Here, we develop methods and tools for it that allow a systematic application to translation dynamical systems of tiling spaces that are generated by primitive inflation rules. These systems share many nice properties that permit the explicit computation of the $\mathrm {OSD}$, thus providing a rich class of examples with non-trivial $\mathrm {OSD}$.

Let $H_n$ be the minimal number such that any n-dimensional convex body can be covered by $H_n$ translates of the interior of that body. Similarly $H_n^s$ is the corresponding quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger’s covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note, we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions n. Our main idea is to cover the body by translates of John’s ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John’s position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger n, we describe how the general asymptotic bounds $H_n\le \binom {2n}{n}n(\ln n+\ln \ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln \ln n+5)$ due to Rogers, Shephard and Roger, Zong, respectively, can be improved for specific values of n.

We give an affirmative answer to a long-standing conjecture of Thomassen, stating that every sufficiently highly connected graph has a k-vertex-connected orientation. We prove that a connectivity of order $O(k^2)$ suffices. As a key tool, we show that for every pair of positive integers d and t, every $(t \cdot h(d))$-connected graph contains t edge-disjoint d-rigid (in particular, d-connected) spanning subgraphs, where $h(d) = 10d(d+1)$. This also implies a positive answer to the conjecture of Kriesell that every sufficiently highly connected graph G contains a spanning tree T such that $G-E(T)$ is k-connected.

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension:

• ◦ We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them, we find five empty ones of width $11$ and none of larger width.

• ◦ Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension d and width growing asymptotically as $d/\operatorname {\mathrm {arcsinh}}(1) \sim 1.1346\,d$.

The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology $\check H^1(\Omega )$ gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of $\check H^1(\Omega )$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of $\check H^1(\Omega )$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of $\check H^1(\Omega )$.