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1. Epsilon multiplicity and analytic spread of filtrations

   https://doi.org/10.1215/00192082-11081310  

   Cutkosky, Steven Dale; Sarkar, Parangama (April 2024, Illinois Journal of Mathematics)

   We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties. 

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2. ANALYTIC SPREAD OF FILTRATIONS ON TWO-DIMENSIONAL NORMAL LOCAL RINGS

   https://doi.org/10.1017/nmj.2022.35  

   CUTKOSKY, STEVEN DALE (March 2023, Nagoya Mathematical Journal)

   Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R , and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R . The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations. 

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3. Convexity of multiplicities of filtrations on local rings

   https://doi.org/10.1112/S0010437X23007972  

   Blum, Harold; Liu, Yuchen; Qi, Lu (April 2024, Compositio Mathematica)

   We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions. 

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4. Perfect even modules and the even filtration

   https://doi.org/10.4171/JEMS/1669  

   Pstrągowski, Piotr (July 2025, Journal of the European Mathematical Society)

   Inspired by the work of Hahn–Raksit–Wilson, we introduce a variant of the even filtration which is naturally defined on\mathbf{E}_{1}-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn–Raksit–Wilson filtration on ring spectra of arithmetic interest, showing that various “motivic” filtrations are in fact invariants of the\mathbf{E}_{1}-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective\mathbf{E}_{1}-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights. 

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5. Efficient Approximation of Multiparameter Persistence Modules

   David Loiseaux, Mathieu Carrière (June 2022, ArXivorg)

   Topological Data Analysis is a growing area of data science, which aims at computing and characterizing the geometry and topology of data sets, in order to produce useful descriptors for subsequent statistical and machine learning tasks. Its main computational tool is persistent homology, which amounts to track the topological changes in growing families of subsets of the data set itself, called filtrations, and encode them in an algebraic object, called persistence module. Even though algorithms and theoretical properties of modules are now well-known in the single-parameter case, that is, when there is only one filtration to study, much less is known in the multi-parameter case, where several filtrations are given at once. Though more complicated, the resulting persistence modules are usually richer and encode more information, making them better descriptors for data science. In this article, we present the first approximation scheme, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence, for computing and decomposing general multi-parameter persistence modules. Our algorithm has controlled complexity and running time, and works in arbitrary dimension, i.e., with an arbitrary number of filtrations. Moreover, when restricting to specific classes of multi-parameter persistence modules, namely the ones that can be decomposed into intervals, we establish theoretical results about the approximation error between our estimate and the true module in terms of interleaving distance. Finally, we present empirical evidence validating output quality and speed-up on several data sets. 

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