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1. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}${\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m${A_{m}}with $m\le n${m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}${\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$\square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4${D_{4}}which do not arise from an embedding of $A_r${A_{r}}with $r\le 3${r\leq 3}in $D_4${D_{4}}. 

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2. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa; Teräväinen, Joni (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

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3. Record Warmth of 2023 and 2024 was Highly Predictable and Resulted From ENSO Transition and Northern Hemisphere Absorbed Shortwave Anomalies

   https://doi.org/10.1029/2025GL115614  

   Blanchard‐Wrigglesworth, Eduardo; Bilbao, Roberto; Donohoe, Aaron; Materia, Stefano (May 2025, Geophysical Research Letters)

   Abstract Global mean temperature rapidly warmed during 2023, making 2023 the second warmest year on record at 1.45°C above pre‐industrial climate, and 2024 became the first year on record to surpass 1.5°C. Here we explore the likelihood, mechanisms, and predictability of the rapid warming during 2023 with CMIP simulations and a fully‐coupled forecast ensemble initialized on 1 November 2022. The year‐to‐year (Y2Y) warming for the second half of 2023 of 0.49°C equaled the largest on record since 1850, and is simulated as a 1 in 6,000 years event. The forecast ensemble‐mean predicts about 75% of the observed warming during 2023. The remaining 25% of the warming lies within the forecast spread, with members that forecast a strong 2023 El Niño and positive absorbed shortwave anomalies more likely to forecast the entirety of the observed warming. The forecast ensemble succesfully predicts 2024 to be the first year on record above 1.5°C. 

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4. KMT-2023-BLG-0416, KMT-2023-BLG-1454, KMT-2023-BLG-1642: Microlensing planets identified from partially covered signals

   https://doi.org/10.1051/0004-6361/202348245  

   Han, Cheongho; Udalski, Andrzej; Lee, Chung-Uk; Zang, Weicheng; Albrow, Michael D; Chung, Sun-Ju; Gould, Andrew; Hwang, Kyu-Ha; Jung, Youn Kil; Ryu, Yoon-Hyun; et al (March 2024, Astronomy & Astrophysics)

   Aims. We investigate the 2023 season data from high-cadence microlensing surveys with the aim of detecting partially covered shortterm signals and revealing their underlying astrophysical origins. Through this analysis, we ascertain that the signals observed in the lensing events KMT-2023-BLG-0416, KMT-2023-BLG-1454, and KMT-2023-BLG-1642 are of planetary origin. Methods. Considering the potential degeneracy caused by the partial coverage of signals, we thoroughly investigate the lensing-parameter plane. In the case of KMT-2023-BLG-0416, we have identified two solution sets, one with a planet-to-host mass ratio ofq~ 10⁻²and the other withq~ 6 × 10⁻⁵, within each of which there are two local solutions emerging due to the inner-outer degeneracy. For KMT-2023-BLG-1454, we discern four local solutions featuring mass ratios ofq~ (1.7−4.3) × 10⁻³. When it comes to KMT-2023-BLG-1642, we identified two locals withq~ (6 − 10) × 10⁻³resulting from the inner-outer degeneracy. Results. We estimate the physical lens parameters by conducting Bayesian analyses based on the event time scale and Einstein radius. For KMT-2023-BLG-0416L, the host mass is ~0.6M_⊙, and the planet mass is ~(6.1−6.7)M_Jaccording to one set of solutions and ~0.04M_Jaccording to the other set of solutions. KMT-2023-BLG-1454Lb has a mass roughly half that of Jupiter, while KMT-2023-BLG-1646Lb has a mass in the range of between 1.1 to 1.3 times that of Jupiter, classifying them both as giant planets orbiting mid M-dwarf host stars with masses ranging from 0.13 to 0.17 solar masses. 

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5. Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

   https://doi.org/10.1515/crelle-2023-0043  

   Argüz, Hülya; Bousseau, Pierrick (July 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Cluster varieties come in pairs: for any $\mathcal{𝒳}${\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{𝒳}${\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties. 

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