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1. On realizations of the subalgebra 𝒜^{ℝ}(1) of the ℝ-motivic Steenrod algebra

   https://doi.org/10.1090/btran/114  

   Bhattacharya, P.; Guillou, B.; Li, A. (July 2022, Transactions of the American Mathematical Society, Series B)

   In this paper, we show that the finite subalgebra A R ( 1 ) \mathcal {A}^\mathbb {R}(1) , generated by S q 1 \mathrm {Sq}^1 and S q 2 \mathrm {Sq}^2 , of the R \mathbb {R} -motivic Steenrod algebra A R \mathcal {A}^\mathbb {R} can be given 128 different A R \mathcal {A}^\mathbb {R} -module structures. We also show that all of these A \mathcal {A} -modules can be realized as the cohomology of a 2 2 -local finite R \mathbb {R} -motivic spectrum. The realization results are obtained using an R \mathbb {R} -motivic analogue of the Toda realization theorem. We notice that each realization of A R ( 1 ) \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an R \mathbb {R} -motivic v 1 v_1 -self-map. The C 2 {\mathrm {C}_2} -equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the R O ( C 2 ) \mathrm {RO}({\mathrm {C}_2}) -graded Steenrod operations on a C 2 {\mathrm {C}_2} -equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C 2 {\mathrm {C}_2} -equivariant realizations of A C 2 ( 1 ) \mathcal {A}^{\mathrm {C}_2}(1) . We find another application of the R \mathbb {R} -motivic Toda realization theorem: we produce an R \mathbb {R} -motivic, and consequently a C 2 {\mathrm {C}_2} -equivariant, analogue of the Bhattacharya-Egger spectrum Z \mathcal {Z} , which could be of independent interest. 

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2. New Variants of Arithmetic Quantum Ergodicity

   https://doi.org/10.1007/s00220-024-05203-3  

   Humphries, Peter; Thorner, Jesse (March 2025, Communications in Mathematical Physics)

   Abstract We establish two new variants of arithmetic quantum ergodicity. The first is for self-dual$$\textrm{GL}_2$$ ${\text{GL}}_2$Hecke–Maaß newforms over$$\mathbb {Q}$$ $Q$as the level and Laplace eigenvalue vary jointly. The second is a nonsplit analogue wherein almost all restrictions of Hilbert (respectively Bianchi) Hecke–Maaß cusp forms to the modular surface dissipate as their Laplace eigenvalues grow. 

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3. Torsion Invariants of complexes of groups

   https://doi.org/10.1215/00127094-2023-0024  

   Okun, Boris; Schreve, K (February 2024, Duke)

   Hain, Richard (Ed.)

   Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal Z-subgroups. Let dQ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of dQ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual L^2-torsion of G in terms of the torsion of stabilizers and topology of dQ. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem. 

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4. On the Okounkov–Olshanski formula for standard tableaux of skew shapes

   Morales, A. H.; Zhu, D. (July 2020, Séminaire lotharingien de combinatoire)

   The classical hook-length formula counts the number of standard tableaux of straight shapes, but there is no known product formula for skew shapes. Okounkov– Olshanski (1996) and Naruse (2014) found new positive formulas for the number of standard Young tableaux of a skew shape. We prove various properties of the Okounkov– Olshanski formula: a reformulation similar to the Naruse formula, determinantal formulas for the number of terms, and a q-analogue extending the formula to reverse plane partitions, which complements work by Chen and Stanley for semistandard tableaux. 

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5. Irregular loci in the Emerton–Gee stack for GL ₂

   https://doi.org/10.1515/crelle-2024-0045  

   Bellovin, Rebecca; Borade, Neelima; Hilado, Anton; Kansal, Kalyani; Lee, Heejong; Levin, Brandon; Savitt, David; Wiersema, Hanneke (July 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $K/{\mathbf{Q}}_p$K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack ${\mathcal{X}}_2$\mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of $\mathrm{Gal}\left(\bar{K}/K\right)$\mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci. 

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