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Conformally St{\"a}ckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G($\nabla$u, $\nabla$u) = 0 for null geodesics and the Laplace equation --$\Delta$…

We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi…

J.H.C. Whitehead introduced the concept of crossed modules in the early 20th century. These crossed modules are crucial for algebraic models of 2-type homotopy, which involve connected spaces with no higher than second-degree homotopy…

In this paper, we discuss two well-known open problems in the regularity theory for nonlinear, conformally invariant elliptic systems in dimensions $n\ge 3$, with a critical nonlinearity: $H$-systems (equations of hypersurfaces of…

In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…

Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing…

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex…

We prove a global uniqueness result for the Calder\'{o}n inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension $n\ge 3$. Performing higher order linearizations of…

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits…

The Lusternik-Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik-Schnirelmann category and…

Let $M$ be an analytic complete finite volume pseudo-Riemannian manifold and $\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R})$ a connected semisimple Lie group such that its Lie algebra is…

We determine the Euclidean distance degrees of the three most common manifolds arising in manifold optimization: flag, Grassmann, and Stiefel manifolds. For the Grassmannian, we will also determine the Euclidean distance degree of an…

We define an infinite sequence of new invariants, delta_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold…

J. Milnor introduced a specific class of codimension-$1$ submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides…

The three-manifold topological invariants $\hat Z$ capture the half-index of the three-dimensional theory with ${\mathcal{N}}=2$ supersymmetry obtained by compactifying the M5 brane theory on the closed three-manifold. In 2019, surprising…

In this paper, we prove various results on the topology of the Grassmannian of oriented 3-planes in Euclidean 6-space and compute its cohomology ring. We give self-contained proofs. These spaces come up when studying submanifolds of…

Let $\widetilde{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of…

For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K_3(C) Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold.…

In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…