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1. Local decoding and testing of polynomials over grids

   https://doi.org/10.1002/rsa.20933  

   Bafna, Mitali; Srinivasan, Srikanth; Sudan, Madhu (June 2020, Random Structures & Algorithms)

   We study the local decodability and (tolerant) local testability of low‐degreen‐variate polynomials over arbitrary fields, evaluated over the domain {0,1}ⁿ. We show that for every field there is a tolerant local test whose query complexity depends only on the degree. In contrast we show that decodability is possible over fields of positive characteristic, but not over the reals. 

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2. Limit models in strictly stable abstract elementary classes

   https://doi.org/10.1002/malq.202200075  

   Boney, Will; VanDieren, Monica_M (December 2024, Mathematical Logic Quarterly)

   Abstract In this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfyingthe joint embedding and amalgamation properties with no maximal model of cardinality ,stability in ,,continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed. 

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3. Base Change and Iwasawa Main Conjectures for GL2

   https://doi.org/10.1093/imrn/rnaf082  

   Burungale, Ashay; Castella, Francesc; Skinner, Christopher (April 2025, International Mathematics Research Notices)

   Abstract Let $$E$$ be an elliptic curve defined over $${\mathbb{Q}}$$ of conductor $$N$$, $$p$$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $$K$$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the $${\mathbb{Z}}_{p}$$-cyclotomic and $${\mathbb{Z}}_{p}$$-anticyclotomic deformations of $$E$$ over $${\mathbb{Q}}$$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $$E$$ over $$K$$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $$E$$ over a quartic CM field containing $$K$$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $$E$$ over $$K$$. The aforementioned one-variable main conjectures imply the $$p$$-part of the conjectural Birch and Swinnerton-Dyer formula for $$E$$ if $$\operatorname{ord}_{s=1}L(E,s)\leq 1$$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1]. 

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4. Notes on solutions of KZ equations modulo $p^s$ and $$p$$-adic limit $$s\to\infty$$

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ equations over C in the case, when the hypergeometric solutions are hyperelliptic integrals of genus g. Then the space of solutions is a 2g-dimensional complex vector space. We also consider the same equations modulo ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case g=1 of elliptic integrals. The p-adic limit of Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. We describe the Frobenius transformations of solutions of the KZ equations for g=1 and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

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5. Witnessing “De‐Socialization”: Investigating unexpected longitudinal trends in engineering doctoral socialization

   https://doi.org/10.1002/jee.70040  

   Jwa, Kyeonghun; Berdanier, Catherine GP (January 2026, Journal of Engineering Education)

   Abstract BackgroundGraduate‐level education is gaining attention in engineering education scholarship. While “socialization” is a key term in doctoral literature, little is known about how socialization occurs over time. One common assumption asserts that socialization increases over time, encompassing factors such as belongingness, research ability, and advisor relationship as students acclimate to the norms and values of their advisors, departments, universities, and disciplines. We investigate engineering doctoral student socialization trends: students likely to complete their degrees and those who have questioned whether to persist in their programs. Understanding these trends is essential, as many students consider leaving their programs. Purpose/HypothesisThis paper aims to understand how socialization processes occur over several years in engineering students who questioned leaving their PhD programs. Design/MethodWe present longitudinal survey data collected from two cohorts (N_A = 113 andN_B = 355) of engineering doctoral students at R1 universities in the United States. Data were collected over 2 years through SMS surveys with participants receiving text messages three times per week. We analyzed data using descriptive and time series analysis methods. ResultsBoth cohorts showed lower levels of belongingness over time, reported declining advisor relationships, and experienced higher levels of stress. Students later in their programs also reported deteriorating overall social relationships. These findings contradict canonical socialization theory, which expects socialization to naturally improve over time. ConclusionWhile many assume socialization occurs passively and students acculturate into their department and research team over time, our results show students who question whether to persist are de‐socializing from graduate school. 

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