Related papers: On Rigid, Hard and Soft Problems and Results in Ar…
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
An important, if relatively less well known aspect of the singularity theorems in Lorentzian Geometry is to understand how their conclusions fare upon weakening or suppression of one or more of their hypotheses. Then, theorems with modified…
Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u \i\ curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from…
GR and other metric theories of gravity are formulated with an arbitrary auxiliary curved background in a Lagrangian formalism. A new sketch of how to include spinor fields is included. Conserved quantities are obtained using Noether's…
We develop a rigidity theory for frameworks in $\mathbb{R}^3$ which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered.…
We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class.…
[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…
We introduce a general notion of solution for a Noetherian differential $k$-algebra and study its relationship with simplicity, where k is an algebraically closed field; then we analyze conditions under which such solutions may exist and be…
We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.
We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local…
We prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. In the route, we also discover new characterizations of…
We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a point or orthogonal to an hypersurface, one of these metrics being a warped product if the dimension $n$ is greater than or equal to 3.…
We obtain sharp volume bounds on the boundaries of Alexandrov spaces with given lower curvature bound, dimension, and radius. We also completely classify the rigidity case and analyze almost rigidity. Our results are new even for smooth…
We overview some of the foundations of the so-called henselian rigid geometry, and show that henselian rigid geometry has many aspects, useful in applications, that one cannot expect in the usual rigid geometry. This is done by announcing a…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i…
Recent studies showed that hardness, a complex property, can be calculated using very simple approaches or even analytical formulae. These form the basis for evaluating controversial experimental results (as we illustrate for…
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…
The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the…
We consider the global rigidity problem for bar-joint frameworks where each vertex is constrained to lie on a particular line in $\mathbb R^d$. In our setting we allow multiple vertices to be constrained to the same line. Under a mild…