Abstract We show that the cohomology of the Regge complex in three dimensions is isomorphic to $$ \mathcal {H}^{{\scriptscriptstyle \bullet }}_{dR}(\varOmega )\otimes \mathcal{R}\mathcal{M}$$ H dR ∙ ( Ω ) ⊗ R M , the de Rham cohomology of differential forms with values in infinitesimal rigid body motions. Based on an observation that the twisted de Rham complex extends the elasticity complex (based on Riemannian deformation) to the linearized version of coframes, connection 1-forms, curvature and Cartan’s torsion, we construct a discrete version of linearized Riemann-Cartan geometry on any triangulation and determine its cohomology.

Abstract We study convergence rates of the Trotter splitting $$\begin{aligned} e^{A+L} = \lim _{n \rightarrow \infty } \Big (e^{L/n} e^{A/n}\Big )^n \end{aligned}$$ e A + L = lim n → ∞ ( e L / n e A / n ) n in the strong operator topology. In the first part, we use complex interpolation theory to treat generators L and A of contraction semigroups on Banach spaces, with L relatively A-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schrödinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $$d=3$$ d = 3 . Using the Brezis-Mironescu inequality, we derive convergence rates for the Schrödinger operator with $$V(x)=\pm |x|^{-a}$$ V ( x ) = ± | x | - a potential. In each case, our conditions are fully explicit.

Abstract Semi-algebraic priors are ubiquitous in signal processing and machine learning. Prevalent examples include a) linear models where the signal lies in a low-dimensional subspace; b) sparse models where the signal can be represented by only a few coefficients under a suitable basis; and c) a large family of neural network generative models. In this paper, we prove a transversality theorem for semi-algebraic sets in orthogonal or unitary representations of groups: with a suitable dimension bound, a generic translate of any semi-algebraic set is transverse to the orbits of the group action. This, in turn, implies that if a signal lies in a low-dimensional semi-algebraic set, then it can be recovered uniquely from measurements that separate orbits. As an application, we consider the implications of the transversality theorem to the problem of recovering signals that are translated by random group actions from their second moment. As a special case, we discuss cryo-EM. This is a leading technology to constitute the spatial structure of biological molecules, and serves as our prime motivation. In particular, we derive explicit bounds for recovering a molecular structure from the second moment under a semi-algebraic prior and deduce information-theoretic implications. We also obtain information-theoretic bounds for three additional applications: factoring Gram matrices, multi-reference alignment, and phase retrieval. Finally, we deduce bounds for designing permutation invariant separators in machine learning.