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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. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung; Wang, Yilong; Wilson, Samuel (January 2023, Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. The linear representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of ${\mathrm{SL}}_2<\#comment/>\left(\mathbb{Z}\right)$that are subrepresentations of a symmetric one. 

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3. Intersections of dual 𝑆𝐿₃-webs

   https://doi.org/10.1090/tran/9349  

   Shen, Linhui; Sun, Zhe; Weng, Daping (August 2025, Transactions of the American Mathematical Society)

   We introduce a topological intersection number for an ordered pair of ${\mathrm{SL}}_3$-webs on a decorated surface. Using this intersection pairing between reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs and a collection of $({\mathrm{SL}}_3,\mathcal{X})$-webs associated with the Fock–Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced $({\mathrm{SL}}_3,\mathcal{A})$-webs to the tropical set ${\mathcal{A}}_{{\mathrm{PGL}}_3,\stackrel{^<\#comment/>}{S}}^{+}\left({\mathbb{Z}}^t\right)$, as established by Douglas and Sun in [Forum Math. Sigma 12 (2024), p. e5, 55]. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock–Goncharov duality conjecture of higher Teichmüller spaces for ${\mathrm{SL}}_3$. 

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4. Almost commuting matrices and stability for product groups

   https://doi.org/10.4171/JEMS/1483  

   Ioana, Adrian (June 2024, Journal of the European Mathematical Society)

   We prove that any product of two non-abelian free groups,\Gamma=\mathbb{F}_{m}\times\mathbb{F}_{k}, form,k\geq 2, is not Hilbert–Schmidt stable. This means that there exist asymptotic representations\pi_{n}\colon \Gamma\rightarrow \mathrm{U}({d_n})with respect to the normalized Hilbert–Schmidt norm which are not close to actual representations. As a consequence, we prove the existence of contraction matricesA,Bsuch thatAalmost commutes withBandB^{*}, with respect to the normalized Hilbert–Schmidt norm, butA,Bare not close to any matricesA',B'such thatA'commutes withB'andB'^{*}. This settles in the negative a natural version of a question concerning almost commuting matrices posed by Rosenthal in 1969. 

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5. On a conjectural symmetric version of Ehrhard’s inequality

   https://doi.org/10.1090/tran/9177  

   Livshyts, Galyna (May 2024, Transactions of the American Mathematical Society)

   We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-<\#comment/>1}\left(s\right)={\int <\#comment/>}_0^s{t}^{k-<\#comment/>1}{e}^{-<\#comment/>\frac{t^2}{2}}dt$and $c_{k-<\#comment/>1}=J_{k-<\#comment/>1}(+\mathrm{\infty <\#comment/>})$, we conjecture that the function $F:[0,1]\to <\#comment/>\mathbb{R}$, given by $F\left(a\right)=\underset{k=1}{\overset{n}{\sum <\#comment/>}}{1}_{a\in <\#comment/>E_k}\cdot <\#comment/>\left({\mathrm{\beta <\#comment/>}}_k{J}_{k-<\#comment/>1}^{-<\#comment/>1}\right(c_{k-<\#comment/>1}a)+{\mathrm{\alpha <\#comment/>}}_k)$(with an appropriate choice of a decomposition $[0,1]={\cup <\#comment/>}_i{E}_i$and coefficients ${\mathrm{\alpha <\#comment/>}}_i,{\mathrm{\beta <\#comment/>}}_i$) satisfies, for all symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $F\left(\mathrm{\gamma <\#comment/>}\right(\mathrm{\lambda <\#comment/>}K+(1-<\#comment/>\mathrm{\lambda <\#comment/>})L\left)\right)\ge <\#comment/>\mathrm{\lambda <\#comment/>}F\left(\mathrm{\gamma <\#comment/>}\right(K\left)\right)+(1-<\#comment/>\mathrm{\lambda <\#comment/>})F\left(\mathrm{\gamma <\#comment/>}\right(L\left)\right).$We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set $K$, itsGaussian concavity power $p_s(K,\mathrm{\gamma <\#comment/>})$is greater than or equal to $p_s(R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k},\mathrm{\gamma <\#comment/>})$, for some $k\in <\#comment/>\{1,\dots <\#comment/>,n\}$. We call the sets $R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$round $k$-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set $K$in ${\mathbb{R}}^n$, $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>\underset{F\in <\#comment/>L^2(K,\mathrm{\gamma <\#comment/>})\cap <\#comment/>Lip\left(K\right):\int <\#comment/>F=1}{sup}\left(2T_{\mathrm{\gamma <\#comment/>}}^F\right(K)-<\#comment/>Var(F\left)\right)+\frac{1}{n-<\#comment/>\mathbb{E}{X}^2},$where $T_{\mathrm{\gamma <\#comment/>}}^F\left(K\right)$is the $F-<\#comment/>$torsional rigidity of $K$with respect to the Gaussian measure.Moreover, the equality holds if and only if $K=R{B}_2^k\times <\#comment/>{\mathbb{R}}^{n-<\#comment/>k}$for some $R>0$and $k=1,\dots <\#comment/>,n$.As a consequence, we get $p_s(K,\mathrm{\gamma <\#comment/>})\ge <\#comment/>Q(\mathbb{E}|X{|}^2,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^4,\mathbb{E}\Vert <\#comment/>X{\Vert <\#comment/>}_K^2,r(K\left)\right),$where $Q$is a certain rational function of degree $2$, the expectation is taken with respect to the restriction of the Gaussian measure onto $K$, $\Vert <\#comment/>\cdot <\#comment/>{\Vert <\#comment/>}_K$is the Minkowski functional of $K$, and $r\left(K\right)$is the in-radius of $K$. The result follows via a combination of some novel estimates, the $L2$method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185;Geometric aspects of functional analysis, Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function $F$whose composition with $\mathrm{\gamma <\#comment/>}\left(K\right)$is concave in the Minkowski sense for all symmetric convex sets. 

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