We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.

We consider a class of two-dimensional Schrdinger operator with a singular interaction of the type and a fixed strength supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an AharonovBohm flux in the center. It is shown that if , there is a critical value such that the discrete spectrum has an accumulation point when , while for the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed and small enough.

The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.

A large class of symmetry-protected topological phases (SPT) in boson / spin systems have been recently predicted by the group cohomology theory. In this work, we consider SPT states at least with charge symmetry (U(1) or ZN) or spin Sz rotation symmetry (U(1) or ZN) in 2D, 3D, and the surface of 3D. If both are U(1), we apply external electromagnetic field / `spin gauge field' to study the charge / spin response. For the SPT examples we consider (i.e. Uc(1)⋊ZT2, Us(1)×ZT2, Uc(1)×[Us(1)⋊Z2]; subscripts c and s are short for charge and spin; ZT2 and Z2 are time-reversal symmetry and π-rotation about Sy, respectively), many variants of Witten effect in the 3D SPT bulk and various versions of anomalous surface quantum Hall effect are defined and systematically investigated. If charge or spin symmetry reduces to ZN by considering charge-N or spin-N condensate, instead of the linear response approach, we gauge the charge/spin symmetry, leading to a dynamical gauge theory with some remaining global symmetry. The 3D dynamical gauge theory describes a symmetry-enriched topological phase (SET), i.e. a topologically ordered state with global symmetry which admits nontrivial ground state degeneracy depending on spatial manifold topology. For the SPT examples we consider, the corresponding SET states are described by dynamical topological gauge theory with topological BF term and axionic Θ-term in 3D bulk. And the surface of SET is described by the chiral boson theory with quantum anomaly.

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