Recent work on higher-dimensional type theory has explored connections between Martin-Löf type theory, higher-dimensional category theory, and homotopy theory. These connections suggest a generalization of dependent type theory to account for computationally relevant proofs of propositional equality—for example, taking IdSet A B to be the isomorphisms between A and B. The crucial observation is...

We present results concerning geometric topology of 2-dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3-manifolds (and thickenings), the problem of resolving arbitrary 2-polyhedra by fake surfaces (including an application to reduce the classical Whitehead asphericity conjecture to special polyhedra) and existence of nonhomeomorphic 3-manifolds with eq...

In these notes we investigate a particular partial differential equation arising from the quantum mechanical description of a hydrogen atom. For simplicity we consider the atom in a 2-dimensional instead of a 3-dimensional universe. For convenience we put all fundamental constants of nature equal to 1. The problem. In quantum mechanics the electron is described by a so called wave function. A w...

In this paper we classify the possible degenerate fibers which can occur in a semistable degeneration of two-dimensional tori under the assumption that the canonical bundle of the total space of the family is trivial. 1. Preliminaries Let π : X → ∆ be a proper map of a Kähler manifold X onto the unit disk ∆ = {t ∈ C :| t |< 1}, such that the fibers Xt are nonsingular compact complex manifolds f...

Polygraphs generalize to n-categories the usual notion of equational theory, thus allowing one to describe a category by the means of generators and relations. When the relations are oriented, such a presentation can be considered as a rewriting system and one might wonder whether the rewriting system is confluent and terminating in order to provide a notion of canonical representative of morph...

This paper extends the problem of palindrome searching into a higher dimension, addressing two definitions of 2D palindromes. The first definition implies a square, while the second definition (also known as a centrosymmetric factor), can be any rectangular shape. We describe two algorithms for searching a 2D text for maximal palindromes, one for each type of 2D palindrome. The first algorithm ...

In this note, we reformulateHyland-Ong-Nickau pointer game semantics for PCF, as a more concurrent style of games, called 2-dimensional games. The main novelty of these games is that they are played on concurrent graphs, instead of trees. This enables to permute moves, and to decompose the innocence constraint on strategies, as local permutation properties. 1 Step one: reformulate pointer games...