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1. Proper mappings between indefinite hyperbolic spaces and type I classical domains

   https://doi.org/10.1090/tran/8618  

   Huang, Xiaojun; Lu, Jin; Tang, Xiaomin; Xiao, Ming (December 2022, Transactions of the American Mathematical Society)

   In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. 

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2. An exponential bound on the number of non-isotopic commutative semifields

   https://doi.org/10.1090/tran/8785  

   Göloğlu, Faruk; Kölsch, Lukas (March 2023, Transactions of the American Mathematical Society)

   We show that the number of non-isotopic commutative semifields of odd order p n p^{n} is exponential in n n when n = 4 t n = 4t and t t is not a power of 2 2 . We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in n n and given by Zhou and Pott [Adv. Math. 234 (2013), pp. 43–60]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003), pp. 96–114] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004), pp. 895–938]. 

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3. Matrix Schubert varieties, binomial ideals, and reduced Gröbner bases

   https://doi.org/10.1090/proc/17175  

   Stelzer, Ada (July 2025, Proceedings of the American Mathematical Society)

   We prove a sharp lower bound on the number of terms in an element of the reduced Gröbner basis of a Schubert determinantal ideal $I_w$under the term order of Knutson–Miller [Ann. of Math. (2) 161 (2005), pp. 1245–1318]. We give three applications. First, we give a pattern-avoidance characterization of the matrix Schubert varieties whose defining ideals are binomial. This complements a result of Escobar–Mészáros [Proc. Amer. Math. Soc. 144 (2016), pp. 5081–5096] on matrix Schubert varieties that are toric with respect to their natural torus action. Second, we give a combinatorial proof that the recent formulas of Rajchgot–Robichaux–Weigandt [J. Algebra 617 (2023), pp. 160–191] and Almousa–Dochtermann–Smith [Preprint, arXiv:2209.09851, 2022] computing the Castelnuovo–Mumford regularity of vexillary $I_w$and toric edge ideals of bipartite graphs respectively agree for binomial $I_w$. Third, we demonstrate that the Gröbner basis for $I_w$given by the minimal generators of Gao–Yong [J. Commut. Algebra 16 (2024), pp. 267–273] is reduced if and only if the defining permutation $w$is vexillary. 

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4. On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

   https://doi.org/10.1090/cams/32  

   Lührmann, Jonas; Schlag, Wilhelm (February 2024, Communications of the American Mathematical Society)

   We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]). The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172]. 

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5. Model theoretic characterizations of large cardinals revisited

   https://doi.org/10.1090/tran/9067  

   Boney, Will; Dimopoulos, Stamatis; Gitman, Victoria; Magidor, Menachem (July 2024, Transactions of the American Mathematical Society)

   In Boney [Israel J. Math. 236 (2020), pp. 133–181], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals. 

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