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1. A Trace Map on Higher Scissors Congruence Groups

   https://doi.org/10.1093/imrn/rnae153  

   Bohmann, Anna_Marie; Gerhardt, Teena; Malkiewich, Cary; Merling, Mona; Zakharevich, Inna (August 2024, International Mathematics Research Notices)

   Abstract Cut-and-paste $$K$$-theory has recently emerged as an important variant of higher algebraic $$K$$-theory. However, many of the powerful tools used to study classical higher algebraic $$K$$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $$K$$-theory. In this paper we address the particular case of the $$K$$-theory of polyhedra, also called scissors congruence $$K$$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $$K$$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $$K$$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $$K$$-theory. 

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2. Distributions on partitions arising from Hilbert schemes and hook lengths

   https://doi.org/10.1017/fms.2022.45  

   Bringmann, Kathrin; Craig, William; Males, Joshua; Ono, Ken (January 2022, Forum of Mathematics, Sigma)

   Abstract Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on $${\mathbb {C}}^2$$ . For the Hilbert schemes, we prove that homology is equidistributed as $$n\to \infty $$ . For t -hooks, we prove distributions that are often not equidistributed. The cases where $$t\in \{2, 3\}$$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$ 

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3. Chemical scissor–mediated structural editing of layered transition metal carbides

   https://doi.org/10.1126/science.add5901  

   Ding, Haoming; Li, Youbing; Li, Mian; Chen, Ke; Liang, Kun; Chen, Guoxin; Lu, Jun; Palisaitis, Justinas; Persson, Per O.; Eklund, Per; et al (March 2023, Science)

   Intercalated layered materials offer distinctive properties and serve as precursors for important two-dimensional (2D) materials. However, intercalation of non–van der Waals structures, which can expand the family of 2D materials, is difficult. We report a structural editing protocol for layered carbides (MAX phases) and their 2D derivatives (MXenes). Gap-opening and species-intercalating stages were respectively mediated by chemical scissors and intercalants, which created a large family of MAX phases with unconventional elements and structures, as well as MXenes with versatile terminals. The removal of terminals in MXenes with metal scissors and then the stitching of 2D carbide nanosheets with atom intercalation leads to the reconstruction of MAX phases and a family of metal-intercalated 2D carbides, both of which may drive advances in fields ranging from energy to printed electronics. 

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4. Prospective High School Teachers’ Understanding and Application of the Connection Between Congruence and Transformation in Congruence Proofs

   St. Goar, J.; Lai, Y.; Funk, R. (January 2019, Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   Undergraduate mathematics instructors are called by recent standards to promote prospective teachers’ learning of a transformation approach in geometry and its proofs. The novelty of this situation means it is unclear what is involved in prospective teachers’ learning of geometry from a transformation perspective, particularly if they learned geometry from an approach based on the Elements; hence undergraduate instructors may need support in this area. To begin to approach this problem, we analyze the prospective teachers’ use of the conceptual link between congruence and transformation in the context of congruence. We identify several key actions involved in using the definition of congruence in congruence proofs, and we look at ways in which several of these actions are independent of each other, hence pointing to concepts and actions that may need to be specifically addressed in instruction. 

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5. Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties

   https://doi.org/10.46298/lmcs-19(1:19)2023  

   Kapur, Deepak (March 2023, Logical Methods in Computer Science)

   Algorithms for computing congruence closure of ground equations overuninterpreted symbols and interpreted symbols satisfying associativity andcommutativity (AC) properties are proposed. The algorithms are based on aframework for computing a congruence closure by abstracting nonflat terms byconstants as proposed first in Kapur's congruence closure algorithm (RTA97).The framework is general, flexible, and has been extended also to developcongruence closure algorithms for the cases when associative-commutativefunction symbols can have additional properties including idempotency,nilpotency, identities, cancellativity and group properties as well as theirvarious combinations. Algorithms are modular; their correctness and terminationproofs are simple, exploiting modularity. Unlike earlier algorithms, theproposed algorithms neither rely on complex AC compatible well-foundedorderings on nonvariable terms nor need to use the associative-commutativeunification and extension rules in completion for generating canonical rewritesystems for congruence closures. They are particularly suited for integratinginto the Satisfiability modulo Theories (SMT) solvers. A new way to viewGroebner basis algorithm for polynomial ideals with integer coefficients as acombination of the congruence closures over the AC symbol * with the identity 1and the congruence closure over an Abelian group with + is outlined. 

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