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1. Invariant theory for the commuting scheme of symplectic Lie algebras

   Chen, Tsao-Hsien; Ngo, Bao Chau (October 2024, Tata Institute of Fundamental Research Publications)

   We prove the Chevalley restriction theorem for the commuting scheme of symplectic Lie algebras. The key step is the construction of the inverse map of the Chevalley restriction map called the spectral data map. Along the way, we establish a certain multiplicative property of the Pfaffian which is of independent interest. 

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2. Lipschitz Stability of Travel Time Data

   https://doi.org/10.1007/s12220-025-02084-3  

   Ilmavirta, Joonas; Kykkänen, Antti; Lassas, Matti; Saksala, Teemu; Shedlock, Andrew (June 2025, The Journal of Geometric Analysis)

   Abstract We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel’fand’s inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees. 

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3. Uniqueness theorems for meromorphic inner functions and canonical systems

   https://doi.org/10.4153/S0008439524000614  

   Hatinoğlu, Burak (March 2025, Canadian Mathematical Bulletin)

   Abstract We consider uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems, we consider spectral data depending partially or fully on the spectrum, derivative values at the spectrum, Clark measure, or the spectrum of the negative of a meromorphic inner function. Moreover, we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of the Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions. 

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4. Untilting Line Bundles on Perfectoid Spaces

   https://doi.org/10.1093/imrn/rnab314  

   Dorfsman-Hopkins, Gabriel (November 2021, International Mathematics Research Notices)

   Abstract Let $$X$$ be a perfectoid space with tilt $$X^\flat $$. We build a natural map $$\theta :\Pic X^\flat \to \lim \Pic X$$ where the (inverse) limit is taken over the $$p$$-power map and show that $$\theta $$ is an isomorphism if $$R = \Gamma (X,\sO _X)$$ is a perfectoid ring. As a consequence, we obtain a characterization of when the Picard groups of $$X$$ and $$X^\flat $$ agree in terms of the $$p$$-divisibility of $$\Pic X$$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence. 

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5. Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method

   Dang, T D; Le, C V; Luu, K D; Nguyen, L H (June 2025, preprint arXiv:2506.13000)

   We introduce a time-dimensional reduction method for the inverse source problem in linear elasticity, where the goal is to reconstruct the initial displacement and velocity fields from partial boundary measurements of elastic wave propagation. The key idea is to employ a novel spectral representation in time, using an orthonormal basis composed of Legendre polynomials weighted by exponential functions. This Legendre polynomial-exponential basis enables a stable and accurate decomposition in the time variable, effectively reducing the original space-time inverse problem to a sequence of coupled spatial elasticity systems that no longer depend on time. These resulting systems are solved using the quasi-reversibility method. On the theoretical side, we establish a convergence theorem ensuring the stability and consistency of the regularized solution obtained by the quasi-reversibility method as the noise level tends to zero. On the computational side, two-dimensional numerical experiments confirm the theory and demonstrate the method's ability to accurately reconstruct both the geometry and amplitude of the initial data, even under substantial measurement noise. The results highlight the effectiveness of the proposed framework as a robust and computationally efficient strategy for inverse elastic source problems. 

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