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1. Volumes of line bundles on schemes

   https://doi.org/10.1090/proc/15526  

   Cutkosky, Steven; Núñez, Roberto (October 2021, Proceedings of the American Mathematical Society)

   Volumes of line bundles are known to exist as limits on generically reduced projective schemes. However, it is not known if they always exist as limits on more general projective schemes. We show that they do always exist as a limit on a codimension one subscheme of a nonsingular projective variety. 

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2. Nash blow-ups of jet schemes

   https://doi.org/10.5802/aif.3302  

   de Fernex, Tommaso; Docampo, Roi (January 2019, Annales de l'Institut Fourier)

   null (Ed.)
3. Deciding accuracy of differential privacy schemes

   https://doi.org/10.1145/3434289  

   Barthe, Gilles; Chadha, Rohit; Krogmeier, Paul; Sistla, A_Prasad; Viswanathan, Mahesh (January 2021, Proceedings of the ACM on Programming Languages)

   Differential privacy is a mathematical framework for developing statistical computations with provable guarantees of privacy and accuracy. In contrast to the privacy component of differential privacy, which has a clear mathematical and intuitive meaning, the accuracy component of differential privacy does not have a generally accepted definition; accuracy claims of differential privacy algorithms vary from algorithm to algorithm and are not instantiations of a general definition. We identify program discontinuity as a common theme in existingad hocdefinitions and introduce an alternative notion of accuracy parametrized by, what we call, — the of an inputxw.r.t.  a deterministic computationfand a distanced, is the minimal distanced(x,y) over allysuch thatf(y)≠f(x). We show that our notion of accuracy subsumes the definition used in theoretical computer science, and captures known accuracy claims for differential privacy algorithms. In fact, our general notion of accuracy helps us prove better claims in some cases. Next, we study the decidability of accuracy. We first show that accuracy is in general undecidable. Then, we define a non-trivial class of probabilistic computations for which accuracy is decidable (unconditionally, or assuming Schanuel’s conjecture). We implement our decision procedure and experimentally evaluate the effectiveness of our approach for generating proofs or counterexamples of accuracy for common algorithms from the literature. 

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4. On arc fibers of morphisms of schemes

   https://doi.org/10.4171/jems/1500  

   Chiu, Christopher; de_Fernex, Tommaso; Docampo, Roi (July 2024, Journal of the European Mathematical Society)

   Given a morphismf \colon X \to Yof schemes over a field, we prove several finiteness results about the fibers of the induced mapf_{\infty} \colon X_{\infty} \to Y_{\infty}on arc spaces. Assuming thatfis quasi-finite andXis separated and quasi-compact, our theorem states thatf_{\infty}has topologically finite fibers of bounded cardinality and its restriction toX_{\infty} \setminus R_{\infty}, whereRis the ramification locus off, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers off_{\infty}whenfis a finite morphism of varieties over an algebraically closed field, describe the ramification locus off_{\infty}, and prove a general criterion forf_{\infty}to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower bound on the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are also discussed. 

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5. Designing Fair Ranking Schemes

   https://doi.org/10.1145/3299869.3300079  

   Asudeh, Abolfazl; Jagadish, H. V.; Stoyanovich, Julia; Das, Gautam (July 2019, Proceedings of the ACM SIGMOD Intl Conference on Management of Data)

   Items from a database are often ranked based on a combination of criteria. The weight given to each criterion in the combination can greatly affect the fairness of the produced ranking, for example, preferring men over women. A user may have the flexibility to choose combinations that weigh these criteria differently, within limits. In this paper, we develop a system that helps users choose criterion weights that lead to greater fairness. We consider ranking functions that compute the score of each item as a weighted sum of (numeric) attribute values, and then sort items on their score. Each ranking function can be expressed as a point in a multidimensional space. For a broad range of fairness criteria, including proportionality, we show how to efficiently identify regions in this space that satisfy these criteria. Using this identification method, our system is able to tell users whether their proposed ranking function satisfies the desired fairness criteria and, if it does not, to suggest the smallest modification that does. Our extensive experiments on real datasets demonstrate that our methods are able to find solutions that satisfy fairness criteria effectively (usually with only small changes to proposed weight vectors) and efficiently (in interactive time, after some initial pre-processing). 

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