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1. The Hilbert scheme of infinite affine space and algebraic K-theory

   https://doi.org/10.4171/jems/1340  

   Hoyois, Marc; Jelisiejew, Joachim; Nardin, Denis; Totaro, Burt; Yakerson, Maria (April 2025, Journal of the European Mathematical Society)

   We study the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})from an\mathbf{A}^{1}-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})is\mathbf{A}^{1}-equivalent to the Grassmannian of(d-1)-planes in\mathbf{A}^{\infty}. We then describe the\mathbf{A}^{1}-homotopy type of\mathrm{Hilb}_{d}(\mathbf{A}^{n})in a certain range, fornlarge compared tod. For example, we compute the integral cohomology of\mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C})in a range. We also deduce that the forgetful map\mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect}from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an\mathbf{A}^{1}-equivalence after group completion. This implies that the moduli stack\mathcal{FF}\mathrm{lat}, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum\mathrm{kgl}representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the\mathrm{kgl}-homology of smooth proper schemes over a perfect field. 

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2. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

   https://doi.org/10.1017/fms.2019.49  

   ARANA-HERRERA, FRANCISCO; ATHREYA, JAYADEV S. (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 

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3. Rationality of twists of the Siegel modular variety of genus 2 and level 3

   https://doi.org/10.1090/proc/15854  

   Calegari, Frank; Chidambaram, Shiva (February 2022, Proceedings of the American Mathematical Society)

   Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 

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4. 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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5. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar; Kalhan, Sanchit; Makarychev, Konstantin; Makarychev, Yury (January 2021, Proceedings of Machine Learning Research)

   Meila, Marina; Zhang, Tong (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph $$G$$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $$\mathcal{C}$$ of graph $$G$$, a similar edge is in disagreement with $$\mathcal{C}$$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $$\mathcal{C}$$ if its endpoints belong to the same cluster. The disagreements vector, $$\mathbf{disagree}$$, is a vector indexed by the vertices of $$G$$ such that the $$v$$-th coordinate $$\mathbf{disagree}_v$$ equals the weight of all disagreeing edges incident on $$v$$. The goal is to produce a clustering that minimizes the $$\ell_p$$ norm of the disagreements vector for $$p\geq 1$$. We study the $$\ell_p$$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $$[\alpha\mathbf{w},\mathbf{w}]$$ and every dissimilar edge has weight at least $$\alpha\mathbf{w}$$ (where $$\alpha \leq 1$$ and $$\mathbf{w}>0$$ is a scaling parameter). We give an $$O\left((\frac{1}{\alpha})^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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