Related papers: Mostow Rigidity Made Easier
Methods for measuring convexity defects of compacts in R^n abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global…
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for…
We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…
We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several…
Manifolds have uses throughout and beyond Mathematics and it is not surprising that topologists have expended a huge effort in trying to understand them. In this article we are particularly interested in the question: `when is a manifold…
Possible models of modified gravity are being extensively studied now, with most phenomenological motivations coming from puzzles and tensions in cosmology due to a natural desire to better fit the known and newly coming data. At the same…
This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution…
Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis…
We announce here that Fermat's Last theorem was solved, but there is an easy proof of it on the basis of elemetary undergraduate mathematics. We shall disclose such an easy proof.
This paper aims to give an elementary proof for Toponogov's theorem in Alexandrov geometry with lower curvature bound. The idea of the proof comes from the fact that, in Riemannian geometry, sectional curvature can be embodied in the second…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the…
We prove a formality theorem for algebraic objects internal to smooth complex varieties that are not compact but whose mixed Hodge structure has a certain purity property.
In this note we prove a lower bound for the rank of 2-dimensional generic rigidity matroid for regular graphs of degree four and five. Also, we give examples to show the order of the bound we give is sharp.
The aim of this brief note is to provide a quick and elementary proof of the following known fact: on a metric measure space whose Sobolev space is separable, there exists a test plan that is sufficient to identify the minimal weak upper…
In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
A smooth compactification of Donaldson moduli spaces is given. As an application, we use this new space to study the wall-crossing formula and prove the Kotschick-Morgan conjecture.
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle…