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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. 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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3. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

   https://doi.org/10.1090/tran/8976  

   Livshyts, Galyna (June 2023, Transactions of the American Mathematical Society)

   We show that for any even log-concave probability measure $\mathrm{\mu <\#comment/>}$on ${\mathbb{R}}^n$, any pair of symmetric convex sets $K$and $L$, and any $\mathrm{\lambda <\#comment/>}\in <\#comment/>[0,1]$, $\mathrm{\mu <\#comment/>}\left(\right(1-<\#comment/>\mathrm{\lambda <\#comment/>})K+\mathrm{\lambda <\#comment/>}L{)}^{c_n}\ge <\#comment/>(1-<\#comment/>\mathrm{\lambda <\#comment/>}\left)\mathrm{\mu <\#comment/>}\right(K{)}^{c_n}+\mathrm{\lambda <\#comment/>}\mathrm{\mu <\#comment/>}(L{)}^{c_n},$where $c_n\ge <\#comment/>n^{-<\#comment/>4-<\#comment/>o\left(1\right)}$. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. 

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4. The two-phase problem for harmonic measure in VMO and the chord-arc condition

   https://doi.org/10.1090/btran/197  

   Tolsa, Xavier; Toro, Tatiana (November 2024, Transactions of the American Mathematical Society, Series B)

   Let ${\mathrm{\Omega <\#comment/>}}^{+}\subset <\#comment/>{\mathbb{R}}^{n+1}$be a bounded $\mathrm{\delta <\#comment/>}$-Reifenberg flat domain, with $\mathrm{\delta <\#comment/>}>0$small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}={\mathbb{R}}^{n+1}\setminus <\#comment/>\stackrel{¯<\#comment/>}{{\mathrm{\Omega <\#comment/>}}^{+}}$is an NTA (non-tangentially accessible) domain as well and denote by ${\mathrm{\omega <\#comment/>}}^{+}$and ${\mathrm{\omega <\#comment/>}}^{-<\#comment/>}$the respective harmonic measures of ${\mathrm{\Omega <\#comment/>}}^{+}$and ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}$with poles $p^{\pm <\#comment/>}\in <\#comment/>{\mathrm{\Omega <\#comment/>}}^{\pm <\#comment/>}$. In this paper we show that the condition that $\log <\#comment/>\frac{d{\mathrm{\omega <\#comment/>}}^{-<\#comment/>}}{d{\mathrm{\omega <\#comment/>}}^{+}}\in <\#comment/>\mathrm{VMO}<\#comment/>\left({\mathrm{\omega <\#comment/>}}^{+}\right)$is equivalent to ${\mathrm{\Omega <\#comment/>}}^{+}$being a chord-arc domain with inner unit normal belonging to $\mathrm{VMO}<\#comment/>\left({\mathcal{H}}^n{|}_{\mathrm{\partial <\#comment/>}{\mathrm{\Omega <\#comment/>}}^{+}}\right)$. 

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5. Long strings of consecutive composite values of polynomials

   https://doi.org/10.1090/tran/9348  

   Ford, Kevin; Gabdullin, Mikhail (December 2024, Transactions of the American Mathematical Society)

   We show that for any polynomial $f:\mathbb{Z}\to \mathbb{Z}$with positive leading coefficient and irreducible over $\mathbb{Q}$, if $x$is large enough then there is a string of $(\log x)(\log \log x{)}^{1/835}$consecutive integers $n\in [1,x]$for which $f\left(n\right)$is composite. This improves the result by Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and Terence Tao [J. Eur. Math. Soc. (JEMS) 23 (2023), pp. 667–700], which has the exponent of $\log \log x$being a constant depending on $f$which can be exponentially small in the degree of $f$. 

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