Convex and Combinatorial Tropical Geometry

Mathematical research in Hungary started with geometry: with the work of the two Bolyais early in the 19th century. The father, Farkas Bolyai, showed that equal area polygons are equidecomposable. The son, Janos Bolyai, laid down the foundations of non-Euclidean geometry. The study of geometric objects has been continuing ever since. The present chapter of this book is devoted to describing what investigations took place in Hungary in the 20th century in the field of convex and combinatorial geometry. This includes incidence geometries, finite geometries, and stochastic geometry as well. The selection of the material is, of course, a personal one, and some omissions are inevitable (though most likely unjustified). The choice is made difficult by the wide variety of topics that were to be included.

This survey is devoted to some results in the area of combinatorial and convex geometry, from classical theorems up to the latest contemporary results, mainly those results whose proofs make essential use of the methods of algebraic topology. Various generalizations of the Borsuk-Ulam theorem for a -action are explained in detail, along with applications to Knaster's problem about levels of a function on a sphere, and applications are discussed to the Lyusternik-Shnirel'man theory for estimating the number of critical points of a smooth function. An overview is given of the topological methods for estimating the chromatic number of graphs and hypergraphs, in theorems of Tverberg and van Kampen-Flores type. The author's results on the `dual' analogues of the central point theorem and Tverberg's theorem are described. Results are considered on the existence of inscribed and circumscribed polytopes of special form for convex bodies and on the existence of billiard trajectories in a convex body. Results on partition of measures by hyperplanes and other partitions of Euclidean space are presented. For theorems of Helly type a brief overview is given of topological approaches connected with the nerve of a family of convex sets in Euclidean space. Also surveyed are theorems of Helly type for common flat transversals, and results using the topology of the Grassmann manifold and of the canonical vector bundle over it are considered in detail.

The past 30 years have not only seen substantial progress and lively activity in various areas within convex geometry, e.g., in asymptotic geometric analysis, valuation theory, the L_p -Brunn-Minkowski theory and stochastic geometry, but also an increasing amount and variety of applications of convex geometry to other branches of mathematics (and beyond), e.g. to PDEs, statistics, discrete geometry, optimization, or geometric algorithms in computer science. Thus convex geometry is a flourishing and attractive field, which is also reflected by the considerable number of talented young mathematicians at this meeting.

In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.

Motivated by Barman [6], we initiate a systematic study of the 'no-dimensional' analogues of some basic theorems in combinatorial and convex geometry, including the colorful Caratheodory's theorem, Tverberg's theorem, Helly's theorem as well as their fractional and colorful extensions.

The workshop Convex and Algebraic Geometry was organized by Klaus Altmann (Berlin), Victor Batyrev (Tübingen), and Bernard Teissier (Paris). Both title subjects meet primarily in the theory of toric varieties. These constitute the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Thus, toric geometry and its several generalizations provide a kind of section from polyhedral into algebraic geometry. While this reflects only a thin slice of algebraic geometry, it is general enough to display many important phenomena, techniques, and methods. It serves as a wonderful testing ground for general theories such as the celebrated mirror symmetry in its different flavours. In particular, much of the popularity of toric geometry originates in mathematical physics. The meeting was attended by almost 50 participants from many European countries, Canada, the USA, and Japan. The program consisted of talks by 23 speakers, among them many young researchers. Most subjects fit more or less into the following main areas: One of the major discussions during the meeting concerned the existence of strongly exceptional sequences on toric varieties which consist of line bundles. A full exceptional sequence provides a kind of “basis” for the derived category. While Hille and Perling presented an example that does not carry such a sequence of full length, Bondal suggested a method to link this question to sheaves on the dual real torus that are constructible with respect to a certain stratification. In general, one expects to gain exceptional sequences from the universal bundles on moduli spaces. Using this method, Craw constructs those sequences on smooth toric Fano threefolds. In this context, Maclagan and Ueda consider the case of three-dimensional abelian quotient singularities. Ueda investigates the Fukaya category of the corresponding potential on the dual torus explicitly. Using mirror symmetry, Horja establishes a connection between the orbifold K -theory of toric Deligne-Mumford stacks and solutions to GKZ-hypergeometric D -modules. Gross and Siebert have developed a program to understand mirror symmetry as the duality of certain degeneration data. The special fibers split into toric components, and the degeneration is encoded in a topological manifold B with an affine and a polytopal structure. Duality is now inherited from discrete geometry, and the topology of B reflects the topology of the general fiber. In particular, if B is a ( \mathbb Q -homology) \mathbb P^n_{\mathbb C} , then this construction might lead to (compact) Hyperkähler varieties. Considering, in a special case, a certain contraction of the total space of these families leads to a description of torus actions on algebraic varieties via divisors on their Chow quotients. These divisors carry polytopes or even polyhedral complexes as their coefficients, compare the talks of Hausen, Süss, and Vollmert. In a similar setting, but with an explicit manipulation of Pfaffians, Brown and Reid construct smoothings of certain non-isolated singularities giving rise to four-dimensional flips. The most rigorous degeneration of a variety is the tropical one. Here, everything takes place over the so-called tropical semiring, and one ends up with piecewise linear spaces. In fact, Siebert's degeneration data mentioned above correspond to these objects. Itenberg and Shustin use this approach to calculate the Welschinger invariants, which are a kind of real version of Gromov–Witten invariants. Along the lines of the method of Gathmann and Markwig, there is a recursive formula for theses invariants. In the case of del Pezzo surfaces, it turns out that both invariants are (log-) asymptotically equivalent. A generalization of toric varieties in a different direction from the torus actions mentioned above is given by the notion of spherical varieties. Pasquier considers horospherical Fano varieties and comes up with an adapted notion of (generalized, coloured) reflexive polytopes. Bruns, Haase, and Hering deal with ordinary polytopes and their relations to syzygies of toric varieties. For an integral matrix A one obtains a semigroup algebra \mathbb C[\mathbb N A] (leading to the usual affine toric variety) and a GKZ-hypergeometric system of differential equations. The latter depends on a parameter \beta , and Miller has reported on a result that relates the set of \beta where the rank of the system jumps to the set of those multidegrees where the semigroup algebra \mathbb C[\mathbb N A] carries local cohomology. In particular, the Cohen–Macaulay property is equivalent to the constant rank condition, answering an old question of Sturmfels. One of the nighttime discussions gave rise to the suggestion to not include normality in the definition of a toric variety, thus overcoming the cumbersome term of a “not necessarily normal toric variety”. The workshop was closed on Friday night by an informal piano recital by Benjamin Nill and Milena Hering featuring Strawinsky, Liszt, and Chopin.

A geometric method of sector decomposition

It is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in ℝ n {\mathbb{R}^{n}} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n ↗ ∞ {n\nearrow\infty} . Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

This paper attempts to look at the interconnections existing between metrics, convexity, and integral geometry from the point of view of combinatorial integral geometry. Along with general expository material, some new concepts and results are presented, in particular the sin2-representations of breadth functions, translative versions of mean curvature integral, and the notion of 2-zonoids. The main aim is to apply these new ideas for a better understanding of the nature of zonoids.

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean valued elds. To achieve this, we develop a theory of toric schemes over valuation rings of rank 1. As a basic tool, we use techniques from non-archimedean analysis. MSC2010: 14T05, 14M25, 32P05 X) of polyhedra in R n . This process is called tropicaliza- tion and it can be used to transform a problem from algebraic geometry into a corresponding problem in convex geometry which is usually easier. If the toric co- ordinates are well suited to the problem, it is sometimes possible to use a solution of the convex problem to solve the original algebraic problem. Another strategy is to vary the ambient torus to compensate the loss of information due to the tropi- calization process. Tropicalization originates from a paper of Bergman (Berg) on logarithmic limit sets. The convex structure of the tropical variety Trop(X) was worked out by Bieri{Groves (BG) with applications to geometric group theory in mind. Sturmfels (Stu) pointed out that Trop(X) is a subcomplex of the Grobner complex. In fact, the polyhedral complex Trop(X) has some natural weights satisfying a balancing condition which appears rst in Speyer's thesis (Spe). This relies on the description

We continue our study of 'no-dimension' analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper \cite{adiprasito2019theorems} and prove no-dimension versions of colorful Tverberg's theorem, selection lemma and the weak $\epsilon$-net theorem in Banach spaces of type $p > 1.$ To prove this results we use the original ideas of \cite{adiprasito2019theorems} for the Euclidean case and our slightly modified version of the celebrated Maurey lemma.

During 1996–7 MSRI held a full academic year program on Combinatorics, with special emphasis on the connections with other branches of mathematics, such as algebraic geometry, topology, commutative algebra, representation theory, and convex geometry. The rich combinatorial problems arising from the study of various algebraic structures are the subject of this book, which represents work done or presented at seminars during the program. It contains contributions on matroid bundles, combinatorial representation theory, lattice points in polyhedra, bilinear forms, combinatorial differential topology and geometry, Macdonald polynomials and geometry, enumeration of matchings, the generalized Baues problem, and Littlewood–Richardson semigroups. These expository articles, written by some of the most respected researchers in the field, will continue to be of use to graduate students and researchers in combinatorics as well as algebra, geometry, and topology.

A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space intersecting any convex body of volume one. A suitable weakening of the volume constraint leads one to the (much more recent) problem of constructing dense forests . These are discrete point sets becoming uniformly close to long enough line segments. Progress towards these problems has so far involved a wide range of ideas surrounding areas as varied as combinatorial and computation geometry, convex geometry, Diophantine approximation, discrepancy theory, the theory of dynamical systems, the theory of exponential sums, Fourier analysis, homogeneous dynamics, the mathematical theory of quasicrystals and probability theory. The goal of this paper is to survey the known results related to the Danzer problem and to the construction of dense forests, to generalise some of them and to state a number of open problems to make further progress towards this longstanding question.

Combinatorial Optimization is an area of mathematics that thrives from a continual influx of new questions and problems from practice. Attacking these problems has required the development and combination of ideas and techniques from different mathematical areas including graph theory, matroids and combinatorics, convex and nonlinear optimization, discrete and convex geometry, algebraic and topological methods. We continued a tradition of triannual Oberwolfach workshops, bringing together the best international researchers with younger talent to discover new connections with a particular emphasis on emerging breakthrough areas.

Quantitative versions of Helly's and Steinitz' theorems were first introduced by Bárány, Katchalski and Pach in 1982, and have grown into a well-studied field within discrete and convex geometry in the last decade. This note is an invitation to the field in the form of an incomplete collection of open problems.