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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. Uniqueness of some cylindrical tangent cones to special Lagrangians

   https://doi.org/10.1007/s00039-023-00634-x  

   Collins, Tristan C.; Li, Yang (April 2023, Geometric and Functional Analysis)

   Abstract We show that if an exact special Lagrangian $$N\subset {\mathbb {C}}^n$$ N ⊂ C n has a multiplicity one, cylindrical tangent cone of the form $${\mathbb {R}}^{k}\times {\textbf{C}}$$ R k × C where $${\textbf{C}}$$ C is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $${\textbf{C}}$$ C satisfies an integrability condition. This applies, for example, when $${\textbf{C}}= {\textbf{C}}_{HL}^{m}$$ C = C HL m is the Harvey-Lawson $$T^{m-1}$$ T m - 1 cone for $$m\ne 8,9$$ m ≠ 8 , 9 . 

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3. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

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4. Uniformization of Cantor sets with bounded geometry

   https://doi.org/10.1090/ecgd/360  

   Vellis, Vyron (August 2021, Conformal Geometry and Dynamics of the American Mathematical Society)

   In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} . 

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5. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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