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1. Some sharp inequalities of Mizohata–Takeuchi-type

   https://doi.org/10.4171/RMI/1463  

   Carbery, Anthony; Iliopoulou, Marina; Wang, Hong (June 2024, Revista Matemática Iberoamericana)

   Let\Sigmabe a strictly convex, compact patch of aC^{2}hypersurface in\mathbb{R}^{n}, with non-vanishing Gaussian curvature and surface measured\sigmainduced by the Lebesgue measure in\mathbb{R}^{n}. The Mizohata–Takeuchi conjecture states that \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for allg\in L^{2}(\Sigma)and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), whereXdenotes theX-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every\varepsilon>0, there exists a positive constantC_{\varepsilon}, which depends only on\Sigmaand\varepsilon, such that for allR \geq 1and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, whereTranges over the family of tubes in\mathbb{R}^{n}of dimensionsR^{1/2}\times \cdots \times R^{1/2}\times R. From this we deduce the Mizohata–Takeuchi conjecture with anR^{(n-1)/(n+1)}-loss; i.e., that \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ballB_{R}of radiusRand any\varepsilon>0. The power(n-1)/(n+1)here cannot be replaced by anything smaller unless properties of\widehat{g d\sigma}beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds. 

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2. Asymptotic second moment of Dirichlet L -functions along a thin coset

   https://doi.org/10.1017/fms.2025.44  

   Garcia, Bradford; Young, Matthew P (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove an asymptotic formula for the second moment of central values of DirichletL-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulodinside the full group of characters moduloq. Suppose that$$\nu _p(d) \geq \nu _p(q)/2$$for all primespdividingq. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size$$q^{1/2}$$here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smallerd, with$$\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$$, with a power-saving error term fordlarger than$$q^{2/5}$$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughlyd, which is only slightly smaller than the diagonal main term. 

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3. The eleventh cohomology group of $$\overline {\mathcal {M}}_{g,n}$$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the rational cohomology group$$H^{11}(\overline {\mathcal {M}}_{g,n})$$vanishes unless$$g = 1$$and$$n \geq 11$$. We show furthermore that$$H^k(\overline {\mathcal {M}}_{g,n})$$is pure Hodge–Tate for all even$$k \leq 12$$and deduce that$$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$$is surprisingly well approximated by a polynomial inq. In addition, we use$$H^{11}(\overline {\mathcal {M}}_{1,11})$$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology. 

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4. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin; Harper, Alicia; Landsberg, J.M. (January 2023, Forum of Mathematics, Pi)

   Abstract Let$$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$$denote the matrix multiplication tensor (and write$$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$$), and let$$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$$denote the determinant polynomial considered as a tensor. For a tensorT, let$$\underline {\mathbf {R}}(T)$$denote its border rank. We (i) give the first hand-checkable algebraic proof that$$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$$, (ii) prove$$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$$and$$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$$M_{\langle 2\rangle }$$, (iii) prove$$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$$, (iv) prove$$\underline {\mathbf {R}}(\operatorname {det}_3)=17$$, improving the previous lower bound of$$12$$, (v) prove$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$$for all$$\mathbf {n}\geq 25$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$$was known, as well as lower bounds for$$4\leq \mathbf {n}\leq 25$$, and (vi) prove$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$$for all$$\mathbf {n} \ge 18$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 

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5. Nonexistence of isoperimetric sets in spaces of positive curvature

   https://doi.org/10.1515/crelle-2024-0032  

   Antonelli, Gioacchino; Glaudo, Federico (May 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract For every $d\ge 3$d\geq 3, we construct a noncompact smooth 𝑑-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below 1.We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in ${\mathbb{R}}^d$\mathbb{R}^{d}.The examples we construct have nondegenerate asymptotic cone.The dimensional constraint $d\ge 3$d\geq 3is sharp.Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed, in nonnegatively curved spaces with nondegenerate asymptotic cones, isoperimetric sets with large volumes always exist.This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets. 

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