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1. Planar Algebras in Braided Tensor Categories

   https://doi.org/10.1090/memo/1392  

   Henriques, André; Penneys, David; Tener, James (February 2023, Memoirs of the American Mathematical Society)

   We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when $\mathcal{C}$is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$and pivotal module tensor categories over $\mathcal{C}$equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras. 

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2. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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3. Global well-posedness for 2D generalized parabolic Anderson model via paracontrolled calculus

   https://doi.org/10.1007/s40072-025-00358-z  

   Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan (May 2025, Stochastics and Partial Differential Equations: Analysis and Computations)

   Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ $R^{+}\times T^2$within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ $\begin{array}{c}({\partial }_t-\Delta )u=F\left(u\right)\eta \end{array}$where$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$with$$1/6>\kappa >0$$ $1/6>\kappa >0$, and$$F\in C_b^2(\mathbb {R})$$ $F\in C_b^2\left(R\right)$. Assume that$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 

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4. Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations

   https://doi.org/10.1007/s00220-025-05257-x  

   Chae, Dongho; Jeong, In-Jee; Na, Jungkyoung; Oh, Sung-Jin (April 2025, Communications in Mathematical Physics)

   Abstract We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,$$\begin{aligned} \begin{aligned} \partial _t \theta - \nabla ^\perp \log (10+(-\Delta )^{\frac{1}{2}})\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta -{\nabla }^{\perp }log(10+{(-\Delta )}^{\frac{1}{2}})\theta ·\nabla \theta =0,\end{array}\end{array}$and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities,arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the$$\delta $$ $\delta$-SQG equations, defined by$$\begin{aligned} \begin{aligned} \partial _t \theta + \nabla ^\perp (10+(-\Delta )^{\frac{1}{2}})^{-\delta }\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta +{\nabla }^{\perp }{(10+{(-\Delta )}^{\frac{1}{2}})}^{-\delta }\theta ·\nabla \theta =0,\end{array}\end{array}$for all sufficiently small$$\delta >0$$ $\delta >0$depending on the size of the initial data. For the same range of$$\delta $$ $\delta$, we establish global well-posedness of smooth solutions to the dissipative SQG equations. 

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5. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy; Feneuil, Joseph; Mayboroda, Svitlana (March 2022, Mathematische Annalen)

   Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ $L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$ $\Gamma$of dimension$$d < n-1$$ $d<n-1$, the now usual distance to the boundary$$D = D_\beta $$ $D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ $D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$ $X\in \Omega$, where$$\beta >0$$ $\beta >0$and$$\gamma \in (-1,1)$$ $\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$ $L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$ $D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ $|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$ $\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear). 

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