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1. Dens, nests and the Loehr-Warrington conjecture

   https://doi.org/10.1090/jams/1057  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G (June 2025, Journal of the American Mathematical Society)

   We prove and extend the longest-standing conjecture in ‘ $q,t$-Catalan combinatorics,’ namely, the combinatorial formula for ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$conjectured by Loehr and Warrington, where $s_{\mathrm{\mu <\#comment/>}}$is a Schur function and $\mathrm{\nabla <\#comment/>}$is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of $G{L}_l$characters involvingSchur Catalanimals; these were recently shown by the authors to represent Schur functions $s_{\mathrm{\mu <\#comment/>}}[-<\#comment/>M{X}^{m,n}]$in subalgebras $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)\subset <\#comment/>\mathcal{E}$isomorphic to the algebra of symmetric functions $\mathrm{\Lambda <\#comment/>}$over $\mathbb{Q}(q,t)$, where $\mathcal{E}$is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths callednests, having endpoints and bounding constraints controlled by data called aden. The special case for $\mathrm{\Lambda <\#comment/>}\left(X^{m,1}\right)$proves the Loehr-Warrington conjecture, giving ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)$our formula implies a new $(m,n)$version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$Loehr-Warrington formula generalize the $(km,kn)$shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations. 

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2. 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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3. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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4. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$be analytic on $[0,1]$with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$for some constants $A$and $\mathrm{\alpha <\#comment/>}>2$and all $k⩾<\#comment/>1$. We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$under random linear scrambling with $n=2^m$points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$random linearly scrambled estimates, when $k/m$is bounded away from zero. When $f$has a $p$’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$-Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$for any $\mathrm{ϵ<\#comment/>}>0$, if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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5. On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

   https://doi.org/10.1090/tran/9165  

   Dong, Hongjie; Li, Haigang; Xu, Longjuan (September 2024, Transactions of the American Mathematical Society)

   In this paper, we consider higher regularity of a weak solution $(\mathbf{u},p)$to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\mathrm{\delta <\#comment/>}}$in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mathrm{\mu <\#comment/>}}$, where $s$is a positive integer, $\mathrm{\delta <\#comment/>}\in <\#comment/>(0,1)$, and $\mathrm{\mu <\#comment/>}\in <\#comment/>(0,1]$, we show that $D\mathbf{u}$and $p$are piecewise $C^{s,{\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}}$, where ${\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}=min\{\frac{1}{2},\mathrm{\mu <\#comment/>},\mathrm{\delta <\#comment/>}\}$. Our result is new even in the 2D case with piecewise constant coefficients. 

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