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1. Fukaya categories of surfaces, spherical objects and mapping class groups

   https://doi.org/10.1017/fms.2021.21  

   Auroux, Denis; Smith, Ivan (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $$2$$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $$1$$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included. 

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2. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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3. Polynomial Representations of the Witt Lie Algebra

   https://doi.org/10.1093/imrn/rnae139  

   Sam, Steven_V; Snowden, Andrew; Tosteson, Philip (June 2024, International Mathematics Research Notices)

   Abstract The Witt algebra  $${\mathfrak{W}}_{n}$$ is the Lie algebra of all derivations of the $$n$$-variable polynomial ring $$\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$$ (or of algebraic vector fields on $$\textbf{A}^{n}$$). A representation of $${\mathfrak{W}}_{n}$$ is polynomial if it arises as a subquotient of a sum of tensor powers of $$\textbf{V}_{n}$$. Our main theorems assert that finitely generated polynomial representations of $${\mathfrak{W}}_{n}$$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $$\textbf{Fin}^{\textrm{op}}$$, where $$\textbf{Fin}$$ is the category of finite sets. We also show that polynomial representations of $${\mathfrak{W}}_{n}$$ are equivalent to polynomial representations of the endomorphism monoid of $$\textbf{A}^{n}$$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish. 

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4. Upper Tail Large Deviations for Arithmetic Progressions in a Random Set

   https://doi.org/10.1093/imrn/rny022  

   Bhattacharya, Bhaswar B; Ganguly, Shirshendu; Shao, Xuancheng; Zhao, Yufei (March 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract Let Xk denote the number of k-term arithmetic progressions in a random subset of $$\mathbb{Z}/N\mathbb{Z}$$ or $$\{1, \dots , N\}$$ where every element is included independently with probability p. We determine the asymptotics of $$\log \mathbb{P}\big (X_{k} \ge \big (1+\delta \big ) \mathbb{E} X_{k}\big )$$ (also known as the large deviation rate) where p → 0 with $$p \ge N^{-c_{k}}$$ for some constant ck > 0, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of p, the large deviation rate up to a constant factor. 

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5. Sectorial descent for wrapped Fukaya categories

   https://doi.org/10.1090/jams/1035  

   Ganatra, Sheel; Pardon, John; Shende, Vivek (October 2023, Journal of the American Mathematical Society)

   We develop a set of tools for doing computations in and of (partially) wrapped Fukaya categories. In particular, we prove (1) a descent (cosheaf) property for the wrapped Fukaya category with respect to so-called Weinstein sectorial coverings and (2) that the partially wrapped Fukaya category of a Weinstein manifold with respect to a mostly Legendrian stop is generated by the cocores of the critical handles and the linking disks to the stop. We also prove (3) a ‘stop removal equals localization’ result, and (4) that the Fukaya–Seidel category of a Lefschetz fibration with Liouville fiber is generated by the Lefschetz thimbles. These results are derived from three main ingredients, also of independent use: (5) a Künneth formula (6) an exact triangle in the Fukaya category associated to wrapping a Lagrangian through a Legendrian stop at infinity and (7) a geometric criterion for when a pushforward functor between wrapped Fukaya categories of Liouville sectors is fully faithful. 

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