Related papers: Updates on Hirzebruch's 1954 Problem List

Branched covers between Riemann surfaces are associated with certain combinatorial data, and Hurwitz existence problem asks whether given data satisfying those combinatorial constraints can be realized by some branched cover. We connect…

We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.

In this paper, we try to generalize to the case of compact Riemannian orbifolds $Q$ some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds $M$. We shall also consider the problem of…

This is an overview of recent developments regarding the complexity of matrix multiplication, with an emphasis on the uses of algebraic geometry and representation theory in complexity theory.

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.

In this note, we discuss the interactions between differential topology and isoparametric foliations, surveying some recent progress and open problems.

We provide a list of (mainly unsolved) problems in ordered and orderable groups. These were originally compiled 10 years ago by the last two authors. New problems have been added to the list. Progress on some of these is noted and…

In this paper, a survey about recent progress on problems solved using graph amalgamations is presented, along with some new results with complete proofs, and some related open problems.

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…

We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.

The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this…

In this paper, we focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an…

We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.

Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…

This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after…

Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in…

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final…

We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in…

This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory…