Related papers: Shape Theory
The Hilbert-Einstein equations are insufficient to describe the geometry of the Universe, as they only constrain a local geometrical property: curvature. A global knowledge of the geometry of space, if possible, would require measurement of…
The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is ``what is the physical content of spacetime…
Shape dynamics is a completely background-independent universal framework of dynamical theories from which all absolute elements have been eliminated. For particles, only the variables that describe the shapes of the instantaneous particle…
This is an expository article written for the Notices of the AMS in which we discuss the weak Lefschetz Principle in birational geometry. Our departing point is the influential work of Solomon Lefschetz started in 1924. Indeed, we look at…
This is the first expression of my thoughts and my experiments with Nature about the mathematical description of the Universe. The theories about our surrounding Nature became popular from our ancient civilizations and may be from the…
In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was…
The notion of a topological Ramsey space was introduced by Carlson some 30 years ago. Studying the topological Ramsey space of variable words, Carlson was able to derive many classical combinatorial results in a unifying manner. For the…
There have been three geometrizations in history. The first one is historically due to the Pythagorean school and Plato, the second one comes from Galileo, Kepler, Descartes and Newton, and the third geometrization of nature begins with…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant…
In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive…
Shape dynamics is classical theory of gravity which agrees with general relativity in many important cases, but possesses different gauge symmetries and constraints. Rather than spacetime diffeomorphism invariance, shape dynamics takes…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
This is a guided tour through some selected topics in geometric analysis. We have chosen to illustrate many of the basic ideas as they apply to the theory of minimal surfaces. This is, in part, because minimal surfaces is, if not the…
An old branch of mathematics, Topology, has opened the road to the discovery of new phases of matter. A hidden topology in the energy spectrum is the key for novel conducting/insulating properties of topological matter.
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…
The structural analysis of shape boundaries leads to the characterization of objects as well as to the understanding of shape properties. The literature on graphs and networks have contributed to the structural characterization of shapes…
Regions-based theories of space aim -- among others -- to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed,…
This article provides a historical overview of Geometry of Numbers. 1. Figures, 2. The circuit problem and its relatives, 3. Minkowski lattice point set, 4. The young Hermann Minkowski, 5. The geometry of numbers develops, 6. Minkowski…
Starting from topological quantum field theory, we derive space-time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large N matrix model.…