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1. Counterexamples to the tilting and (p,r)-filtration conjectures

   https://doi.org/10.1515/crelle-2019-0036  

   Bendel, Christopher P.; Nakano, Daniel K.; Pillen, Cornelius; Sobaje, Paul (November 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p -Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the {(p,r)} -Filtration Conjecture. 

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2. A counterexample to the HK-conjecture that is principal

   https://doi.org/10.1017/etds.2022.25  

   DEELEY, ROBIN J. (June 2023, Ergodic Theory and Dynamical Systems)

   Abstract Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture. 

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3. Counterexample Classification

   https://doi.org/10.1007/978-3-030-92124-8_18  

   Vick, Cole; Kang, Eunsuk; Tripakis, Stavros (December 2021, International Conference on Software Engineering and Formal Methods)

   Calinescu, R.; Păsăreanu, C.S. (Ed.)

   In model checking, when a given model fails to satisfy the desired specification, a typical model checker provides a counterexample that illustrates how the violation occurs. In general, there exist many diverse counterexamples that exhibit distinct violating behaviors, which the user may wish to examine before deciding how to repair the model. Unfortunately, obtaining this information is challenging in existing model checkers since (1) the number of counterexamples may be too large to enumerate one by one, and (2) many of these counterexamples are redundant, in that they describe the same type of violating behavior. In this paper, we propose a technique called counterexample classification. The goal of classification is to partition the space of all counterexamples into a finite set of counterexample classes, each of which describes a distinct type of violating behavior for the given specification. These classes are then presented as a summary of possible violating behaviors in the system, freeing the user from manually having to inspect or analyze numerous counterexamples to extract the same information. We have implemented a prototype of our technique on top of an existing formal modeling and verification tool, the Alloy Analyzer, and evaluated the effectiveness of the technique on case studies involving the well-known Needham-Schroeder protocol with promising results. 

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4. On Learning Invariant Representations for Domain Adaptation

   Han Zhao, Remi Tachet (January 2019, Proceedings of the 36th International Conference on Machine Learning)

   Due to the ability of deep neural nets to learn rich representations, recent advances in unsupervised domain adaptation have focused on learning domain-invariant features that achieve a small error on the source domain. The hope is that the learnt representation, together with the hypothesis learnt from the source domain, can generalize to the target domain. In this paper, we first construct a simple counterexample showing that, contrary to common belief, the above conditions are not sufficient to guarantee successful domain adaptation. In particular, the counterexample exhibits conditional shift: the class-conditional distributions of input features change between source and target domains. To give a sufficient condition for domain adaptation, we propose a natural and interpretable generalization upper bound that explicitly takes into account the aforementioned shift.Moreover, we shed new light on the problem by proving an information-theoretic lower bound on the joint error of any domain adaptation method that attempts to learn invariant representations.Our result characterizes a fundamental tradeoff between learning invariant representations and achieving small joint error on both domains when the marginal label distributions differ from source to target. Finally, we conduct experiments on real-world datasets that corroborate our theoretical findings. We believe these insights are helpful in guiding the future design of domain adaptation and representation learning algorithms. 

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5. On Cm solutions to systems of linear inequalities

   O'Neill, Kevin; Luli, Garving K (April 2023, Advances in mathematics)

   Recent work of C. Fefferman and the first author [8] has demonstrated that the linear system of equations has a solution if and only if satisfy a certain finite collection of partial differential equations. Here, the are fixed semialgebraic functions. In this paper, we consider the analogous problem for systems of linear inequalities: Our main result is a negative one, demonstrated by counterexample: the existence of a solution F may not, in general, be determined via an analogous finite set of partial differential inequalities in . 

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