Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝ d of cardinality (d+1)t, partitioned into t-point subsets C 1,C 2, ,C d+1 (which we think of as color classes; e.g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint sets R 1,R 2, ,R r ⊆C that are rainbow, meaning that |R i ∩C j |≤1 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audienc

Regular systems of paths and families of convex sets in convex position

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies have at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies have at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Toth, and generalizes a theorem of Bisztriczky and Fejes Toth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions $f:[0,1] \to \mathbb{R}$. On our way towards proving the Pach-Toth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdos and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest

Adjacency Graphs of Polyhedral Surfaces

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $\mathbb{R}^3$. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $K_5$, $K_{5,81}$, or any nonplanar $3$-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $K_{4,4}$, and $K_{3,5}$ can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable $n$-vertex graphs is in $\Omega(n \log n)$. From the non-realizability of $K_{5,81}$, we obtain that any realizable $n$-vertex graph has $O(n^{9/5})$ edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202

Colouring Polygon Visibility Graphs and Their Generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time

Claude Ambrose Rogers. 1 November 1920 — 5 December 2005

Claude Ambrose Rogers and his identical twin brother, Stephen Clifford, were born in Cambridge in 1920 and came from a long scientific heritage. Their great-great-grandfather, Davies Gilbert, was President of the Royal Society from 1827 to 1830; their father was a Fellow of the Society and distinguished for his work in tropical medicine. After attending boarding school at Berkhamsted with his twin brother from the age of 8 years, Ambrose, who had developed very different scientific interests from those of his father, entered University College London in 1938 to study mathematics. He completed the course in 1940 and graduated in 1941 with first-class honours, by which time the UK had been at war with Germany for two years. He joined the Applied Ballistics Branch of the Ministry of Supply in 1940, where he worked until 1945, apparently on calculations using radar data to direct anti-aircraft fire. However, this did not lead to research interests in applied mathematics, but rather to several areas of pure mathematics. Ambrose's PhD research was at Birkbeck College, London, under the supervision of L. S. Bosanquet and R. G. Cooke, his first paper being on the subject of geometry of numbers. Later, Rogers became known for his very wide interests in mathematics, including not only geometry of numbers but also Hausdorff measures, convexity and analytic sets, as described in this memoir. Ambrose was married in 1952 to Joan North, and they had two daughters, Jane and Petra, to form a happy family

Lattice point inequalities and face numbers of polytopes in view of central symmetry

Magdeburg, Univ., Fak. für Mathematik, Diss., 2012von Matthias Henz

TORIC VARIETIES AND COBORDISM

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950\u27s, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension eight. In addition, the role of smooth projective toric varieties in the polynomial ring structure of complex cobordism is examined. More specifically, smooth projective toric varieties are constructed as polynomial ring generators in most dimensions, and evidence is presented suggesting that a smooth projective toric variety can be chosen as a polynomial generator in every dimension. Finally, toric varieties with an additional fiber bundle structure are used to study some manifolds in oriented cobordism. In particular, manifolds with certain fiber bundle structures are shown to all be cobordant to zero in the oriented cobordism ring