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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. Random walks on direct products of groups

   https://doi.org/10.4171/JEMS/1550  

   Golsefidy, Alireza Salehi; Srinivas, Srivatsa (November 2024, Journal of the European Mathematical Society)

   Suppose\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})is generated by a symmetric setSof cardinalitynwherepis a prime number. Suppose the Cheeger constants of the Cayley graphs of\operatorname{SL}_{2}(\mathbb{F}_{p})with respect to\pi_{L}(S)and\pi_{R}(S)are at leastc_{0}, where\pi_{L}and\pi_{R}are the projections to the left and the right components of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}), respectively. Then the Cheeger constant of the Cayley graph of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})with respect toSis at leastcwherecis a positive number which only depends onnandc_{0}. This gives an affirmative answer to a question of Lindenstrauss and Varjú. 

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3. Sharp superlevel set estimates for small cap decouplings of the parabola

   https://doi.org/10.4171/rmi/1393  

   Fu, Yuqiu; Guth, Larry; Maldague, Dominique (June 2023, Revista Matemática Iberoamericana)

   We prove sharp bounds for the size of superlevel sets\{x \in \mathbb{R}^2\colon |f(x)| >\alpha\}, where\alpha>0andf\colon\mathbb{R}^2\to\mathbb{C}is a Schwartz function with Fourier transform supported in anR^{-1}-neighborhood of the truncated parabola\mathbb{P}^1. These estimates imply the small cap decoupling theorem for\mathbb{P}^1of Demeter, Guth, and Wang (2020) and the canonical decoupling theorem for\mathbb{P}^1of Bourgain and Demeter (2015). New(\ell^q,L^p)small cap decoupling inequalities also follow from our sharp level set estimates. 

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4. Fixed points of parking functions

   https://doi.org/10.1090/tran/8994  

   McCammond, Jon; Thomas, Hugh; Williams, Nathan (January 2024, Transactions of the American Mathematical Society)

   We define an action of words in $[m{]}^n$on ${\mathbb{R}}^m$to give a new characterization of rational parking functions—they are exactly those words whose action has a fixed point. We use this viewpoint to give a simple definition of Gorsky, Mazin, and Vazirani’s zeta map on rational parking functions when $m$and $n$are coprime [Trans. Amer. Math. Soc. 368 (2016), pp. 8403–8445], and prove that this zeta map is invertible. A specialization recovers Loehr and Warrington’s sweep map on rational Dyck paths (see D. Armstrong, N. A. Loehr, and G. S. Warrington [Adv. Math. 284 (2015), pp. 159–185; E. Gorsky, M. Mazin, and M. Vazirani [Electron. J. Combin. 24 (2017), p. 29; H. Thomas and N. Williams, Selecta Math. (N.S.) 24 (2018), pp. 2003–2034]). 

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5. 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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