Related papers: Th{\'e}orie de l'homotopie quantitative
Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold…
Topoi are categories which have enough structure to interpret higher order logic. They admit two notions of morphism: logical morphisms which preserve all of the structure and therefore the interpretation of higher order logic, and…
Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. In this paper, we study sequential parametrized topological…
Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, - topology and number theory, - was built only during the last fifty years. This bridge is the theory of spectra in stable homotopy theory.…
Parametrized topological complexity is a homotopy invariant that represents the degree of instability of motion planning problem that involves external constraints. We consider the parametrized topological complexity in the case of…
Fields exhibit a variety of topological properties, like different topological charges, when field space in the continuum is composed by more than one topological sector. Lattice treatments usually encounter difficulties describing those…
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…
For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3-manifolds, including the ``piecewise geometric'' ones in the sense of…
This book is an account of certain topics in general and algebraic topology. Instead of laying out a synopsis of each chapter, here is a sample of some of what is taken up: 1) Nilpotency and its role in homotopy theory. 2) Bousfield's…
We survey some topics in ${\mathbb A}^1$-homotopy theory. Our main goal is to highlight the interplay between ${\mathbb A}^1$-homotopy theory and affine algebraic geometry, focusing on the varieties that are "contractible" from various…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
We investigate various homotopy invariant formulations of commutative algebra in the context of rational homotopy theory. The main subject is the complete intersection condition, where we show that a growth condition implies a structure…
Given an algebraic theory $\ct$, a homotopy $\ct$-algebra is a simplicial set where all equations from $\ct$ hold up to homotopy. All homotopy $\ct$-algebras form a homotopy variety. We give a characterization of homotopy varieties…
This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…
This expository article discusses some connections between the geometry of a hyperbolic 3-manifold homotopy-equivalent to a surface, and the combinatorial properties of its end invariants. In particular a necessary and sufficient condition…
Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological…
We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be…
We investigate a special kind of contraction of symmetric spaces (respectively, of Lie triple systems), called homotopy. In this first part of a series of two papers we construct such contractions for classical symmetric spaces in an…
I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects (e.g., spacetimes) living in infinite-dimensional…
In this paper, we are concerned with studying the existence of invariant complex manifolds of two-dimensional holomorphic systems. From the geometric singular perturbation theory we know that if a slow-fast system has associated a normally…