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We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope - corresponding to the triangulations of A - are very…

We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a…

Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence…

In this paper we investigate the existence of ``partially'' isometric immersions. These are maps f:M->R^q which, for a given Riemannian manifold M, are isometries on some sub-bundle H of TM. The concept of free maps, which is essential in…

Let X --> B be a proper submersion with a Riemannian structure. Given a differential K-theory class on X, we define its analytic and topological indices as differential K-theory classes on B. We prove that the two indices are the same.

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We…

An Independent Parallelism Theorem is proven in the theory of adhesive HLR categories. It shows the bijective correspondence between sequential independent and parallel independent direct derivations in the Weak Double-Pushout framework,…

We classify immersions $f$ of $S^1$ in a $2$-manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the number of double points of a self-transverse $C^1$-approximation of $f$, and the turning number $T(e\bar f)$ of the…

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature…

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and…

We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider Hermitian 2-forms on the tensor product $H=K\otimes L$, where $K,L$ are finite dimensional complex spaces. Inspired by quantum…

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a…

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some…

We show how to find a complete set of necessary and sufficient conditions that solve the fixed-parameter local congruence problem of immersions in $G$-spaces, whether homogeneous or not, provided that a certain $k^{\rm th}$ order jet bundle…

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and…

We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic)…

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a…

The universal order 1 invariant f^U of immersions of a closed orientable surface into R^3, whose existence has been established in [N3], takes values in the group G_U = K \oplus Z/2 \oplus Z/2 where K is a countably generated free Abelian…