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1. Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands

   https://doi.org/10.1142/S0219199721500838  

   Àlvarez Montaner, Josep; Hernández, Daniel J.; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E. (December 2022, Communications in Contemporary Mathematics)

   This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct [Formula: see text]-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of [Formula: see text]-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. 

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2. Lim Ulrich sequences and Boij-Söderberg cones

   https://doi.org/10.1017/fms.2023.108  

   Iyengar, Srikanth B; Ma, Linquan; Walker, Mark E (January 2023, Forum of Mathematics, Sigma)

   Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings. 

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3. 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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4. K-Theory of Endomorphisms, the TR-Trace, and Zeta Functions

   https://doi.org/10.1007/s44007-025-00154-0  

   Campbell, Jonathan A; Lind, John A; Malkiewich, Cary; Ponto, Kate; Zakharevich, Inna (June 2025, La Matematica)

   We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the K-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy’s map from K-theory of endomorphisms to topological restriction homology, defining it for any Waldhausen category with a compatible enrichment in orthogonal spectra. In particular, this extends their construction from rings to ring spectra. We also give a revisionist treatment of the original Dennis trace map from K-theory to topological Hochschild homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories). 

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5. CATEGORICAL COMPLEXITY

   https://doi.org/10.1017/fms.2020.26  

   BASU, SAUGATA; ISIK, UMUT (January 2020, Forum of Mathematics, Sigma)

   We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes. 

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