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Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{s_1, s_2, s_3, t_1, t_2, t_3} \subseteq VG$. The goal is to find a collection $\calP$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$…
Listed as No. 53 among the one hundred famous unsolved problems in [J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008] is Steinberg's conjecture, which states that every planar graph without 4- and 5-cycles is 3-colorable.…
Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of…
We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle…
The $p$-rank of a Steiner triple system $B$ is the dimension of the linear span of the set of characteristic vectors of blocks of $B$, over GF$(p)$. We derive a formula for the number of different Steiner triple systems of order $v$ and…
A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are…
Our work explores the hardness of $3$SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving $3$SUM on a size-$n$ integer set that avoids solutions to $a+b=c+d$ for $\{a,…
We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find…
An ordered set $S$ of vertices of a graph $G$ is a resolving set for $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of G is the minimum cardinality of a resolving set. In…
We consider real polynomial systems $f=g=0$ in two variables where $f$ has $t\geq 3$ monomial terms and $g$ has $3$ monomials terms. We prove that the number of positive isolated solutions of such a system does not exceed $3\cdot 2^{t-2} -…
We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…
One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be…
We construct a new solution $(R,K)$ to the three-dimensional reflection equation, a boundary analogue of the tetrahedron equation. The $R$-operator is the one obtained by Sun, Terashima, Yagi, and the authors in 2024, involving four quantum…
In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of…
Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the…
A set of vertices of a graph $G$ such that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$. If additionally, the set of vertices induces a connected subgraph of $G$ then the…
We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite…
Three-body and n-body problems in celestial mechanics are age-old and challenging puzzles. In recent years, several breakthroughs are made in finding periodic orbits for three-body problem. And Bohua Sun proposed a conjecture on Kepler's…
The tree spanner problem for a graph $G$ is as follows: For a given integer $k$, is there a spanning tree $T$ of $G$ (called a tree $k$-spanner) such that the distance in $T$ between every pair of vertices is at most $k$ times their…
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…