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1. Bernstein-Sato theory for singular rings in positive characteristic

   https://doi.org/10.1090/tran/8917  

   Jeffries, Jack; Núñez-Betancourt, Luis; Quinlan-Gallego, Eamon (July 2023, Transactions of the American Mathematical Society)

   The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case. 

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2. Bernstein’s Inequality and Holonomicity for Certain Singular Rings

   https://doi.org/10.1093/imrn/rnad121  

   Montaner, Josep Àlvarez; Hernández, Daniel J; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E (June 2023, International Mathematics Research Notices)

   Abstract In this manuscript, we prove the Bernstein inequality and develop the theory of holonomic $$D$$-modules for rings of invariants of finite groups in characteristic zero, and for strongly $$F$$-regular finitely generated graded algebras with finite $$F$$-representation type in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic $$D$$-modules, in this context, have finite length, and we prove the existence of Bernstein–Sato polynomials in characteristic zero. We obtain these results using a more general version of Bernstein filtrations. 

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3. Games with filters I

   https://doi.org/10.1142/S021906132450003X  

   Foreman, Matthew; Magidor, Menachem; Zeman, Martin (August 2023, Journal of Mathematical Logic)

   Chong, Chita; Feng, Qi; Slaman, Theodore_A; Woodin, W_Hugh (Ed.)

   This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. 

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4. Relative injective modules, superstability and noetherian categories

   https://doi.org/10.1142/S0219061324500272  

   Mazari-Armida, Marcos; Rosický, Jiří (April 2026, Journal of Mathematical Logic)

   We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. 

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5. The stationary Boussinesq problem under singular forcing

   https://doi.org/10.1142/S0218202521500196  

   Allendes, Alejandro; Otárola, Enrique; Salgado, Abner J. (April 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to [Formula: see text], where [Formula: see text] is a weight in the Muckenhoupt class [Formula: see text] that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and [Formula: see text], show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates. 

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