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1. Extensions of tautological rings and motivic structures in the cohomology of $${\overline {\mathcal {M}}}_{g,n}$$

   https://doi.org/10.1017/fmp.2024.24  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2024, Forum of Mathematics, Pi)

   Abstract We study collections of subrings of$$H^*({\overline {\mathcal {M}}}_{g,n})$$that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the$$\ell $$-adic Galois representations and Hodge structures that appear in$$H^k({\overline {\mathcal {M}}}_{g,n})$$for$$k = 13$$,$$14$$and$$15$$. We also show that$$H^4({\overline {\mathcal {M}}}_{g,n})$$is generated by tautological classes for allgandn, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final base cases needed for the inductive proofs of our main results, we use Mukai’s construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of$${\mathcal {M}}_{7,n}$$is generated by algebraic cycle classes, for$$n \leq 3$$. 

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2. The eleventh cohomology group of $$\overline {\mathcal {M}}_{g,n}$$

   https://doi.org/10.1017/fms.2023.59  

   Canning, Samir; Larson, Hannah; Payne, Sam (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that the rational cohomology group$$H^{11}(\overline {\mathcal {M}}_{g,n})$$vanishes unless$$g = 1$$and$$n \geq 11$$. We show furthermore that$$H^k(\overline {\mathcal {M}}_{g,n})$$is pure Hodge–Tate for all even$$k \leq 12$$and deduce that$$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$$is surprisingly well approximated by a polynomial inq. In addition, we use$$H^{11}(\overline {\mathcal {M}}_{1,11})$$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology. 

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3. Multi-linear forms, graphs, and $L^p$-improving measures in $${\Bbb F}_q^d$$

   https://doi.org/10.4153/S0008414X2300086X  

   Bhowmik, Pablo; Iosevich, Alex; Koh, Doowon; Pham, Thang (February 2025, Canadian journal of mathematics)

   Abstract The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$where$$f: X \to {\Bbb R}$$,Xa set, finite or infinite, andKand$$\mu $$denote a suitable kernel and a measure, respectively. Given a connected ordered graphGonnvertices, consider the multi-linear form$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$where$${\mathcal E}(G)$$is the edge set ofG. Define$$\Lambda _G(p_1, \ldots , p_n)$$as the smallest constant$$C>0$$such that the inequality(0.1)$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$holds for all nonnegative real-valued functions$$f_i$$,$$1\le i\le n$$, onX. The basic question is, how does the structure ofGand the mapping properties of the operator$$T_K$$influence the sharp exponents in (0.1). In this paper, this question is investigated mainly in the case$$X={\Bbb F}_q^d$$, thed-dimensional vector space over the field withqelements,$$K(x^i,x^j)$$is the indicator function of the sphere evaluated at$$x^i-x^j$$, and connected graphsGwith at most four vertices. 

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4. On the ergodic theory of the real Rel foliation

   https://doi.org/10.1017/fmp.2024.6  

   Chaika, Jon; Weiss, Barak (January 2024, Forum of Mathematics, Pi)

   Abstract Let$${{\mathcal {H}}}$$be a stratum of translation surfaces with at least two singularities, let$$m_{{{\mathcal {H}}}}$$denote the Masur-Veech measure on$${{\mathcal {H}}}$$, and let$$Z_0$$be a flow on$$({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$$obtained by integrating a Rel vector field. We prove that$$Z_0$$is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces$$({\mathcal L}, m_{{\mathcal L}})$$, where$${\mathcal L} \subset {{\mathcal {H}}}$$is an orbit-closure for the action of$$G = \operatorname {SL}_2({\mathbb {R}})$$(i.e., an affine invariant subvariety) and$$m_{{\mathcal L}}$$is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of$$Z_0$$with respect to any of the measures$$m_{{{\mathcal L}}}$$is zero. 

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5. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao; Longbrake, Sean; Ma, Jie (February 2023, Combinatorics, Probability and Computing)

   Abstract Given a family$$\mathcal{F}$$of bipartite graphs, theZarankiewicz number$$z(m,n,\mathcal{F})$$is the maximum number of edges in an$$m$$by$$n$$bipartite graph$$G$$that does not contain any member of$$\mathcal{F}$$as a subgraph (such$$G$$is called$$\mathcal{F}$$-free). For$$1\leq \beta \lt \alpha \lt 2$$, a family$$\mathcal{F}$$of bipartite graphs is$$(\alpha,\beta )$$-smoothif for some$$\rho \gt 0$$and every$$m\leq n$$,$$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$$(\alpha,\beta )$$-smooth family$$\mathcal{F}$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$$is bipartite. In this paper, we strengthen their result by showing that for every real$$\delta \gt 0$$, there exists$$k_0$$such that for all odd$$k\geq k_0$$and sufficiently large$$n$$, any$$n$$-vertex$$\mathcal{F}\cup \{C_k\}$$-free graph with minimum degree at least$$\delta n^{\alpha -1}$$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$$\mathcal{F}$$consisting of the single graph$$K_{s,t}$$when$$t\gg s$$. We also prove an analogous result for$$C_{2\ell }$$-free graphs for every$$\ell \geq 2$$, which complements a result of Keevash, Sudakov and Verstraëte. 

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