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We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

In this paper, we aim at developing scalable neural network-type learning systems. Motivated by the idea of constructive neural networks in approximation theory, we focus on constructing rather than training feed-forward neural networks (FNNs) for learning, and propose a novel FNNs learning system called the constructive FNN (CFN). Theoretically, we prove that the proposed method not only overcomes the classical saturation problem for constructive FNN approximation, but also reaches the optimal learning rate when the regression function is smooth, while the state-of-the-art learning rates established for traditional FNNs are only near optimal (up to a logarithmic factor). A series of numerical simulations are provided to show the efficiency and feasibility of CFN.

Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory.

In this paper, a structure-preserving algorithm is developed for the computation of a semi-stabilizing solution of a Generalized Algebraic Riccati Equation (GARE). The semi-stabilizing solution of GAREs has been used to characterize the solvability of the (J, J ′ )-spectral factorization problem in control theory for general rational matrices which may have poles and zeros on the extended imaginary axis. The main difficulty in solving such a GARE lies in the fact that its associated Hamiltonian/skew-Hamiltonian pencil has eigenvalues on the extended imaginary axis. Consequently, it is not clear which eigenspace of the associated Hamiltonian/skew-Hamiltonian pencil can characterize the desired semi-stabilizing solution. That is, it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute. Hence, the well-known generalized eigenspace approach for the classical algebraic Riccati equations cannot be employed directly. The proposed algorithm consists of a structure-preserving doubling algorithm (SDA) and a postprocessing procedure to determine the desired eigenvectors and principal vectors corresponding to the purely imaginary and infinite eigenvalues. Under mild assumptions, linear convergence of rate 1/2 for the SDA is proved. Numerical experiments illustrate that the proposed algorithm performs efficiently and reliably.