Geometric realizations of Tamari interval lattices via cubic coordinates

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable

Hypertiling -- a high performance Python library for the generation and visualization of hyperbolic lattices

Hypertiling is a high-performance Python library for the generation and visualization of regular hyperbolic lattices embedded in the Poincar\'e disk model. Using highly optimized, efficient algorithms, hyperbolic tilings with millions of vertices can be created in a matter of minutes on a single workstation computer. Facilities including computation of adjacent vertices, dynamic lattice manipulation, refinements, as well as powerful plotting and animation capabilities are provided to support advanced uses of hyperbolic graphs. In this manuscript, we present a comprehensive exploration of the package, encompassing its mathematical foundations, usage examples, applications, and a detailed description of its implementation.Comment: 52 pages, 20 figure

Partition functions:Zeros, unstable dynamics and complexity

This thesis considers the complexity of approximating the partition functions of the ferromagnetic Ising model and of the hard-core model (independence polynomial) within the class of bounded degree graphs. It is known that the absence of zeros essentially implies that approximation is easy. In this thesis the inverse is proved: the presence of zeros implies that approximation is #P hard. The most important step of the proof is relating both the "zero parameters" and the "#P hard parameters" to the set of parameters around which a related set of functions, namely the occupation ratios, behaves chaotically. The first two chapters contain the proof of the main theorem for the ferromagnetic Ising model and the independence polynomial respectively. Chapters 3 and 4 concern the set of zeros of the independence polynomial for bounded degree graphs. In Chapter 3 it is shown that zeros of Cayley trees are not extremal within the set of zeros of all bounded degree graphs, something that was previously conjectured. In Chapter 4 a very precise description of the set of zeros is given as the degree bound goes to infinity

The Potts model and the independence polynomial:Uniqueness of the Gibbs measure and distributions of complex zeros

Part 1 of this dissertation studies the antiferromagnetic Potts model, which originates in statistical physics. In particular the transition from multiple Gibbs measures to a unique Gibbs measure for the antiferromagnetic Potts model on the infinite regular tree is studied. This is called a uniqueness phase transition. A folklore conjecture about the parameter at which the uniqueness phase transition occurs is partly confirmed. The proof uses a geometric condition, which comes from analysing an associated dynamical system.Part 2 of this dissertation concerns zeros of the independence polynomial. The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the independence polynomial is related to phase transitions in terms of the analycity of the free energy and plays an important role in the design of efficient algorithms to approximately compute evaluations of the independence polynomial. Chapter 5 directly relates the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. This is done by moreover relating the set of zeros of the independence polynomial to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios. Chapter 6 studies boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. It is shown that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori

Domänen parallele Maschinen

A computational model is introduced, which abstracts and idealizes computers with access to fragment shaders. While the set of functions computable by this model remains the same, the running times can be drastically reduced through parallelization compared to conventional models. Some of the algorithms designed for the model can be approximated using fragment shaders. With an automatic transcompilation scheme, fragment shader programs can be generated automatically from a description in a high-level language.In dieser Arbeit wird ein Rechenmodell, das Computer mit Zugriff zu Fragment Shader abstrahiert und idealisiert, eingeführt. Zwar bleibt der Umfang der durch dieses Modell berechenbarer Funktionen gleich, jedoch können die Laufzeiten durch Parallelisierung im Vergleich zu herkömmlichen Modellen drastisch verkürzt werden. Einige der für das Modell entworfenen Algorithmen lassen sich mithilfe von Fragment Shadern approximieren. In einer Hochsprache beschriebene Algorithmen werden automatisiert in Fragment Shader Programme übersetzt

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry