Related papers: Recent developments in toric geometry

This survey presents recent Helly-type geometric theorems published since the appearance of the last comprehensive survey, more than ten years ago. We discuss how such theorems continue to be influential in computational geometry and in…

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable…

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…

This is a survey article, based on the author's lectures in the 2015 AMS Summer Research Institute in Algebraic Geometry, and to appear in the Proceedings.

This is the author's Habilitation which took place at University of Essen on July 11, 1993. The manuscript contains two parts. The first one is devoted to the author's combinatorial construction of mirrors of Calabi-Yau hypersurfaces in…

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we…

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated,…

This is a survey on coarse geometry with an emphasis on coarse homology theories.

This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.

In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be…

In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…

This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be…

We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a…

This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider…

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we…

We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and…

This book is devoted to review two of the most relevant approaches to the study of classical field theories of first order, say k-symplectic and k-cosymplectic. In the last part, we relate the k-symplectic and k-cosymplectic manifolds with…

The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…