A duality transform for realizing convex polytopes with small integer coordinates

Wir entwickeln eine Dualitätstransformation für Polyeder, die eine Einbettung auf dem polynomiellen Gitter berechnet, wenn das ursprüngliche Polyeder auf einem polynomiellen Gitter gegeben ist. Die Konstruktion erfordert einen beschränkten Knotengrad des Polytop-Graphen, funktioniert aber im allgemeinen Fall für die Klasse der Stapelpolytope. Als Konsequenz können wir zeigen, dass sich die "Truncated Polytopes" auf einem polynomiellen Gitter realisieren lassen. Dieses Ergebnis gilt für jede (feste) Dimension.We study realizations of convex polytopes with small integer coordinates. We develop an efficient duality transform, that allows us to go from an efficient realization of a convex polytope to an efficient realization of its dual.Our methods prove to be especially efficient for realizing the class of polytopes dual to stacked polytopes, known as truncated polytopes. We show that every 3d truncated polytope with n vertices can be realized on an integer grid of size O(n^(9lg(6)+1)), and in R^d the required grid size is n^(O(d^2*lg(d))). The class of truncated polytopes is only the second nontrivial class of polytopes, the first being the class of stacked polytopes, for which realizations on a polynomial size integer grid are known to exists

Embedding Stacked Polytopes on a Polynomial-Size Grid

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking operations. We show that for a fixed $d$ every stacked $d$-polytope with $n$ vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure

Geometry-based structural analysis and design via discrete stress functions

This PhD thesis proposes a direct and unified method for generating global static equilibrium for 2D and 3D reciprocal form and force diagrams based on reciprocal discrete stress functions. This research combines and reinterprets knowledge from Maxwell’s 19th century graphic statics, projective geometry and rigidity theory to provide an interactive design and analysis framework through which information about designed structural performance can be geometrically encoded in the form of the characteristics of the stress function. This method results in novel, intuitive design and analysis freedoms. In contrast to contemporary computational frameworks, this method is direct and analytical. In this way, there is no need for iteration, the designer operates by default within the equilibrium space and the mathematically elegant nature of this framework results in its wide applicability as well as in added educational value. Moreover, it provides the designers with the agility to start from any one of the four interlinked reciprocal objects (form diagram, force diagram, corresponding stress functions). This method has the potential to be applied in a wide range of case studies and fields. Specifically, it leads to the design, analysis and load-path optimisation of tension-and compression 2D and 3D trusses, tensegrities, the exoskeletons of towers, and in conjunction with force density, to tension-and-compression grid-shells, shells and vaults. Moreover, the abstract nature of this method leads to wide cross-disciplinary applicability, such as 2D and 3D discrete stress fields in structural concrete and to a geometrical interpretation of yield line theory

Discrete Geometry

[no abstract available

New directions in bipartite field theories

We perform a detailed investigation of Bipartite Field Theories (BFTs), a general class of 4d N = 1 gauge theories which are defined by bipartite graphs. This class of theories is considerably expanded by identifying a new way of assigning gauge symmetries to graphs. A new procedure is introduced in order to determine the toric Calabi-Yau moduli spaces of BFTs. For graphs on a disk, we show that the matroid polytope for the corresponding cell in the Grassmannian coincides with the toric diagram of the BFT moduli space. A systematic BFT prescription for determining graph reductions is presented. We illustrate our ideas in infinite classes of BFTs and introduce various operations for generating new theories from existing ones. Particular emphasis is given to theories associated to non-planar graphs

On vanishing of Kronecker coefficients

We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P=NP. We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples $(\lambda, \mu, \pi)$ such that the Kronecker coefficient $k^\lambda_{\mu, \pi} = 0$ but the Kronecker coefficient $k^{l \lambda}_{l \mu, l \pi} > 0$ for some integer $l>1$. Such "holes" are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any $0<\epsilon\leq1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda_{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|=|\mu| \le m^3$. The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur

Polytopes and Loop Quantum Gravity

The main aim of this thesis is to give a geometrical interpretation of ``spacetime grains'' at Planck scales in the framework of Loop Quantum Gravity. My work consisted in analyzing the details of the interpretation of the quanta of space in terms of polytopes. The main results I obtained are the following: We clarified details on the relation between polytopes and interwiners, and concluded that an intertwiner can be seen unambiguously as the state of a \emph{quantum polytope}. Next we analyzed the properties of these polytopes: studying how to reconstruct the solid figure from LQG variables, the possible shapes and the volume. We adapted existing algorithms to express the geometry of the polytopes in terms of the holonomy-fluxes variables of LQG, thus providing an explicit bridge between the original variables and the interpretation in terms of polytopes of the phase space. Finally we present some direct application of this geometrical picture. We defined a volume operator such as in the large spin limit it reproduce the geometrical volume of a polytope, we computed numerically his spectrum for some elementary cases and we pointed out some asymptotic property of his spectrum. We discuss applications of the picture in terms of polytopes to the study of the semiclassical limit of LQG, in particular commenting a connection between the quantum dynamics and a generalization of Regge calculus on polytopes

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version