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1. On the maximal spectral type of nilsystems

   https://doi.org/10.1090/bproc/229  

   Ackelsberg, Ethan; Richter, Florian; Shalom, Or (September 2024, Proceedings of the American Mathematical Society, Series B)

   Let $(G/\mathrm{\Gamma <\#comment/>},R_a)$be an ergodic $k$-step nilsystem for $k\ge <\#comment/>2$. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that $L^2\left(G/\mathrm{\Gamma <\#comment/>}\right)$decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124(2014), pp. 261–295] for $2$-step nilsystems and a result by Stepin [Uspehi Mat. Nauk24(1969), pp. 241–242] for nilsystems $G/\mathrm{\Gamma <\#comment/>}$with connected, simply connected $G$. 

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2. Characterization of positive definite, radial functions on free groups

   https://doi.org/10.1090/bproc/158  

   Chuah, Chian Yeong; Liu, Zhenchuan; Mei, Tao (April 2023, Proceedings of the American Mathematical Society, Series B)

   This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby [Proc. Amer. Math. Soc. 143 (2015), pp. 1477–1489]. We obtain characterizations of radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^2$length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^p$length on the free real line with infinite generators for $0>p\le <\#comment/>2$. We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well. 

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3. Total power operations in spectral sequences

   https://doi.org/10.1090/tran/9073  

   Balderrama, William (May 2024, Transactions of the American Mathematical Society)

   We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the $C_2$-equivariant Adams spectral sequence, to compute norms on ${\mathrm{\pi <\#comment/>}}_0$of the equivariant $KU$-local sphere, and to compute power operations for the $K\left(1\right)$-local sphere. An appendix contains material on equivariant Bousfield localizations which may be of independent interest. 

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4. Randomized Runge-Kutta-Nyström methods for unadjusted Hamiltonian and kinetic Langevin Monte Carlo

   https://doi.org/10.1090/mcom/4061  

   Bou-Rabee, Nawaf; Kleppe, Tore (February 2025, Mathematics of Computation)

   We introduce $5/2$- and $7/2$-order $L^2$-accurate randomized Runge-Kutta-Nyström methods, tailored for approximating Hamiltonian flows within non-reversible Markov chain Monte Carlo samplers, such as unadjusted Hamiltonian Monte Carlo and unadjusted kinetic Langevin Monte Carlo. We establish quantitative $5/2$-order $L^2$-accuracy upper bounds under gradient and Hessian Lipschitz assumptions on the potential energy function. The numerical experiments demonstrate the superior efficiency of the proposed unadjusted samplers on a variety of well-behaved, high-dimensional target distributions. 

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5. Unavoidable structures in infinite tournaments

   https://doi.org/10.1090/proc/16940  

   Benford, Alistair; DeBiasio, Louis; Larson, Paul (October 2024, Proceedings of the American Mathematical Society)

   We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$such that $G⊈K$, or (ii) every countably-infinite tournament contains aspanningcopy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $\mathrm{\kappa <\#comment/>}$; however, we are only able to provide a concise characterization in the case when $\mathrm{\kappa <\#comment/>}={\mathrm{\aleph <\#comment/>}}_1$. 

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