Related papers: Combinatorics and their evolution in resolution of…
We describe the embedded resolution of a quasi-ordinary surface singularity (V,p) which results from applying the canonical resolution of Bierstone-Milman to (V,p). We show that this process depends solely on the characteristic pairs of…
Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically…
This paper introduces a new shape-matching methodology, combinative matching, to combine interlocking parts for geometric shape assembly. Previous methods for geometric assembly typically rely on aligning parts by finding identical surfaces…
We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every…
Surface plasmons dominate the optical response of metal surfaces, and their nature is controlled by surface geometry. Here we study metasurfaces containing singularities in the form of sharp edges and characterized by three quantum numbers…
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. In this paper, we introduce combinatorial…
Directed assembly of block polymers is rapidly becoming a viable strategy for lithographic patterning of nanoscopic features. One of the key attributes of directed assembly is that an underlying chemical or topographic substrate pattern…
The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a…
Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…
Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
The structural complexity of metamaterials is limitless, although in practice, most designs comprise periodic architectures which lead to materials with spatially homogeneous features. More advanced tasks, arising in e.g. soft robotics,…
Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter…
We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (at the local level) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework.…
For any unbranched double covering of compact Riemann surfaces, we study the associated character varieties that are unitary in the global sense, which we call $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-character varieties. We introduce $k>0$…
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
A new formula is obtained in algebraic topology, in terms of Betti numbers, and a new method, called the spinal method, is suggested and developed for generating quadrangulations of closed orientable surfaces. Those surfaces arise as the…