Related papers: New examples of locally algebraically integrable b…

We study some topological properties of maximal ideal spaces of certain algebras of almost periodic functions. Our main result is that such spaces are contractible. We present certain corollaries of this result.

A uniform space is said to be non-Archimedean if it is generated by equivalence relations. If $\lambda$ is a cardinal, then a non-Archimedean uniform space $(X,\mathcal{U})$ is $\lambda$-totally bounded if each equivalence relation in…

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…

We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…

Let $K$ be a convex body in Euclidean space ${\mathbb R}^d$, and let a translation invariant, locally finite Borel measure on the space of hyperplanes in ${\mathbb R}^d$ be given. For $\delta\ge 0$, we consider the set of all points $x$ for…

Under natural conditions (such as split property and geometric modular action of wedge algebras) it is shown that the unitary equivalence class of the net of local (von Neumann) algebras in the vacuum sector associated to double cones with…

We introduce a notion of volume for an l-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions…

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a…

Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…

We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…

For $n \ge 2$, we prove that a finite volume complex hyperbolic $n$-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two is arithmetic, paralleling our previous work for real…

We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, using the $n$-dimensional area of the sections cut off by hyperplanes and the $(n+1)$-dimensional…

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…

We conjecture that quantum Gaudin models in affine types admit families of local higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated…

We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the…

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any…

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost…

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They…

This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational…

This is a note on the graphs of two smooth real-valued functions in the plane with no intersection and the natural map onto the region surrounded by them with the canonical projection to the line composed, yielding its Reeb space. The Reeb…