I will report on a joint work in progress with Egor Morozov proving that for generic elements in several families of Laplace-type operators invariant under a finite group action, all eigenspaces are irreducible representations. In particular, for the case of Laplace-Beltrami operators, this provides a natural generalization of Uhlenbeck's result on the generic simplicity of the spectrum to the equivariant setting. Moreover, this extends previous work of Zelditch and solves the finite group case of a well-known question raised by Guillemin and Yau. For Schrödinger operators, our results rigorously underpin the notion of accidental degeneracy for certain quantum-mechanical systems with finite symmetry. Our approach involves modern methods of equivariant transversality which we extend to higher dimensions.

We discuss the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian N X N non-Hermitian random band matrices with a bandwidth W. Given W,N → ∞, we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at W ∼√N: it coincides with those for Ginibre ensemble for W ≫√N, and factorized as 1 ≪ W ≪√N. The behavior of the correlation function near the threshold (W/√N →C) will be also discussed.

Compactifying topological actions using only de Rham forms fails to capture torsion sectors encoded in integral cohomology. Differential cohomology remedies this by combining integral characteristic classes, differential-form curvatures, and holonomy data into a single framework. In the context of deriving SymTFTs from M-theory, such a refinement is crucial for capturing background gauge fields for discrete 1-form global symmetries in the physical theory. In this talk, we will review the construction of differential cohomology and, time permitting, show how a refined Kaluza-Klein compactification leads to background gauge fields that encode these higher-form symmetries.