Classification of subdivision rules for geometric groups of low dimension

Subdivision rules create sequences of nested cell structures on CW-complexes, and they frequently arise from groups. In this paper, we develop several tools for classifying subdivision rules. We give a criterion for a subdivision rule to represent a Gromov hyperbolic space, and show that a subdivision rule for a hyperbolic group determines the Gromov boundary. We give a criterion for a subdivision rule to represent a Euclidean space of dimension less than 4. We also show that Nil and Sol geometries can not be modeled by subdivision rules. We use these tools and previous theorems to classify the geometry of subdivision rules for low-dimensional geometric groups by the combinatorial properties of their subdivision rules.Comment: To appear in Conformal Geometry and Dynamics, published by the American Mathematical Societ

Conflict-Free Coloring of String Graphs

Conflict-free coloring (in short, CF-coloring) of a graph $G = (V,E)$ is a coloring of $V$ such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane: (i) We provide a general upper bound of $O(\chi(G)^2 \log n)$ on the CF-chromatic number of any string graph $G$ with $n$ vertices in terms of the classical chromatic number $\chi(G)$. This result stands in contrast to general graphs where the CF-chromatic number can be $\Omega(\sqrt{n})$ already for bipartite graphs. (ii) For some central classes of string graphs, the CF-chromatic number is as large as $\Theta(\sqrt{n})$, which is the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of $\Theta(\log n)$ with respect to the notion of $k$-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most $k$ times), for a small constant $k$. (iii) We obtain a general upper bound on the $k$-CF-chromatic number of arbitrary hypergraphs: Any hypergraph with $m$ hyperedges can be $k$-CF colored with $\tilde{O}(m^{\frac{1}{k+1}})$ colors. This bound, which extends a bound of Pach and Tardos (2009), is tight for some string graphs, up to a logarithmic factor. (iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of $\Theta(\log n)$ on the CF-chromatic number of circle graphs, and an upper bound of $O(\log^{3} n)$ for a wider class that contains circle graphs, namely, intersection graphs of grounded L-shapes.Comment: 33 pages, 8 figures. A major extension of the previous version (that contained 19 pages), adding an upper bound on the CF-chromatic number of grounded L-shapes and other result

Embedding graphs into larger graphs: results, methods, and problems

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which are new breakthrough results, or which --- for some other reasons --- are very close to us. Some results discussed here got stronger emphasis, since they are connected to Lov\'asz (and sometimes to us).Comment: 153 pages, 15 figures, 3 tables. Almost final version of the survey for Building Bridges II (B\'ar\'any et al eds.), Laszlo Lovasz 70th Birthda

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an N-marginal optimal transport problem with N equal marginals supported on Rd and with cost of the form ∑i≠j|xi−xj|−s. In this setting we determine the second-order term in the N→∞ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents 0<s<d. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman–Gregg decomposition [20], [26], extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer [36], who dealt with the second-order term for the Coulomb case s=1,d=3

Packings and tilings in dense graphs

In this thesis we present results on selected problems from extremal graph theory, and discuss both known and new methods used to solve them. In Chapter 1, we give an introductory overview of the regularity method, the flag algebra framework, and some probabilistic tools, which we use to prove our results in subsequent chapters. In Chapter 2 we prove a new result on the packing density of triangles in graphs with given edge density. In doing so, we settle a few conjectures of Gyori and Tuza on decompositions and coverings of graphs with cliques of bounded size. In Chapter 3 we show that a famous conjecture on Hamilton decompositions of bipartite tournaments due to Jackson holds approximately, providing the first intermediate result towards a full proof of the conjecture. In Chapter 4, we introduce a novel absorbing paradigm for graph tilings, which we apply in a few different settings to obtain new results. Using this method, we are able to extend a result on triangle-tilings in graphs with high minimum degree and sublinear independence number to clique-tilings of arbitrary size. We also strengthen an existing result on tilings in randomly perturbed graphs. Finally, in Chapter 5, we consider a problem on quasi-randomness in permutations. We obtain simple density conditions for a sequence of permutations to be quasirandom, and give a full characterisation of all conditions of the same type that force quasi-randomness in the same way

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Covering Problems via Structural Approaches

The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible. In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide: - An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC. - A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure. - A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity. - Applications of the above to various capacitated covering problems via linear programming strengthening and rounding. In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability