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1. Asymptotics for the spectral function on Zoll manifolds

   https://doi.org/10.4171/JST/552  

   Canzani, Yaiza; Galkowski, Jeffrey; Keeler, Blake (June 2025, Journal of Spectral Theory)

   On a smooth, compact, Riemannian manifold without boundary(M,g), let\Delta_{g}be the Laplace–Beltrami operator. We define the orthogonal projection operator \Pi_{I_\lambda}\colon L^{2}(M)\to \bigoplus_{\mathclap{\lambda_j\in I_\lambda}}\ker(\Delta_{g}+\lambda_{j}^{2}) for an intervalI_{\lambda}centered around\lambda\in\Rof a small, fixed length. The Schwartz kernel,\Pi_{I_\lambda}(x,y), of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that\Pi_{I_\lambda}(x,y)has universal asymptotics as\lambda \to \inftyin a shrinking neighborhood of the diagonal inM\times M(providedI_{\lambda}is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points inMwith few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period. 

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2. Mean curvature flow with generic initial data

   https://doi.org/10.1007/s00222-024-01258-0  

   Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix (April 2024, Inventiones mathematicae)

   Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ $R^3$avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ $R^4$is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. 

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3. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

   https://doi.org/10.3934/dcdss.2022012  

   Antil, Harbir; Gal, Ciprian G.; Warma, Mahamadi (January 2022, Discrete and Continuous Dynamical Systems - S)

   We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$$ 0<\gamma <1 $$\end{document}) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$$\mathsf{first\;show}$$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$$\mathsf{backward\;(adjoint)}$$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$$\mathsf{controls }$$\end{document} and characterize the associated \begin{document}$$\mathsf{first\;order}$$\end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces. 

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4. A measure model for the spread of viral infections with mutations

   https://doi.org/10.3934/nhm.2022015  

   Gong, Xiaoqian; Piccoli, Benedetto (January 2022, Networks and Heterogeneous Media)

   Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $$\end{document} and removed \begin{document}$$ R $$\end{document} populations by ODEs and the infected \begin{document}$$ I $$\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$$ S $$\end{document} and \begin{document}$$ R $$\end{document} contains terms that are related to the measure \begin{document}$$ I $$\end{document}$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases. 

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5. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane; Demers, Mark F. (January 2022, Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $$\end{document} on the two-torus, we find \begin{document}$$ t_*>1 $$\end{document} such that for each \begin{document}$$ t\in (0,t_*) $$\end{document} there exists a unique equilibrium state \begin{document}$$ \mu_t $$\end{document} for \begin{document}$$ - t\log J^uT $$\end{document}, and \begin{document}$$ \mu_t $$\end{document} is \begin{document}$$ T $$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$$ - \log J^uT $$\end{document}.) We show that \begin{document}$$ \mu_t $$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $$\end{document} is analytic on \begin{document}$$ (0,t_*) $$\end{document}. In addition, \begin{document}$$ P(t) $$\end{document} is strictly convex if and only if \begin{document}$$ \log J^uT $$\end{document} is not \begin{document}$$ \mu_t $$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$$ t_a\ne t_b $$\end{document} with \begin{document}$$ \mu_{t_a} = \mu_{t_b} $$\end{document}, then \begin{document}$$ P(t) $$\end{document} is affine on \begin{document}$$ (0,t_*) $$\end{document}. An additional sparse recurrence condition gives \begin{document}$$ \lim_{t\downarrow 0} P(t) = P(0) $$\end{document}$. 

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