In this paper, we discuss elliptic genera of (2,2) and (0,2) supersymmetric Landau–Ginzburg models over nontrivial spaces, i.e., nonlinear sigma models on nontrivial noncompact manifolds with superpotential, generalizing old computations in Landau–Ginzburg models over (orbifolds of) vector spaces. For Landau–Ginzburg models in the same universality class as nonlinear sigma models, we explicitly check that the elliptic genera of the Landau–Ginzburg models match that of the nonlinear sigma models, via a Thom class computation of a form analogous to that appearing in recent studies of other properties of Landau–Ginzburg models on nontrivial spaces.

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald-type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present paper, we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in general relativity, cosmology and astrophysics. In particular, we investigate the Jacobi stability of the general relativistic static fluid sphere with a linear barotropic equation of state, of the vacuum in the brane world models, of a dynamical dark energy model, and of the Lane–Emden equation, respectively. It is shown that the Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second-order differential equations.

We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce precisely to the common objects of Lorentzian spin geometry, up to higher-order curvature corrections.

We exhaustively analyze the toric symmetries of $\mathbb{CP}^3$ and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov–Witten (GW) theory and Donaldson–Thomas (DT) theory. We identify all non-trivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of GW Theory and DT theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.

We define an infinite class of unitary transformations between position and momentum fractional spaces, thus generalizing the Fourier transform to a special class of fractal geometries. Each transform diagonalizes a unique Laplacian operator. We also introduce a new version of fractional spaces, where coordinates and momenta span the whole real line. In one topological dimension, these results are extended to more general measures.