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1. Fixed point sets and the fundamental group I: semi-free actions on G -CW-complexes

   https://doi.org/10.1017/prm.2023.63  

   Cappell, Sylvain; Weinberger, Shmuel; Yan, Min (August 2023, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Smith theory says that the fixed point set of a semi-free action of a group$$G$$on a contractible space is$${\mathbb {Z}}_p$$-acyclic for any prime factor$$p$$of the order of$$G$$. Jones proved the converse of Smith theory for the case$$G$$is a cyclic group acting semi-freely on contractible, finite CW-complexes. We extend the theory to semi-free group actions on finite CW-complexes of given homotopy types, in various settings. In particular, the converse of Smith theory holds if and only if a certain$$K$$-theoretical obstruction vanishes. We also give some examples that show the geometrical effects of different types of$$K$$-theoretical obstructions. 

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2. PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

   https://doi.org/10.1017/S147474802100030X  

   Kamensky, Moshe; Starchenko, Sergei; Ye, Jinhe (May 2023, Journal of the Institute of Mathematics of Jussieu)

   Abstract We considerG, a linear algebraic group defined over$$\Bbbk $$, an algebraically closed field (ACF). By considering$$\Bbbk $$as an embedded residue field of an algebraically closed valued fieldK, we can associate to it a compactG-space$$S^\mu _G(\Bbbk )$$consisting of$$\mu $$-types onG. We show that for each$$p_\mu \in S^\mu _G(\Bbbk )$$,$$\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$$is a solvable infinite algebraic group when$$p_\mu $$is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of$$\mathrm {Stab}\left (p_\mu \right )$$in terms of the dimension ofp. 

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3. Reflective centers of module categories and quantum K-matrices

   https://doi.org/10.1017/fms.2025.10055  

   Laugwitz, Robert; Walton, Chelsea; Yakimov, Milen (January 2025, Forum of Mathematics, Sigma)

   Abstract Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category$${\mathcal {C}}$$and$${\mathcal {C}}$$-module category$${\mathcal {M}}$$, we introduce a version of the Drinfeld center$${\mathcal {Z}}({\mathcal {C}})$$of$${\mathcal {C}}$$adapted for$${\mathcal {M}}$$; we refer to this category as thereflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$of$${\mathcal {M}}$$. Just like$${\mathcal {Z}}({\mathcal {C}})$$is a canonical braided monoidal category attached to$${\mathcal {C}}$$, we show that$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$is a canonical braided module category attached to$${\mathcal {M}}$$; its properties are investigated in detail. Our second goal pertains to when$${\mathcal {C}}$$is the category of modules over a quasitriangular Hopf algebraH, and$${\mathcal {M}}$$is the category of modules over anH-comodule algebraA. We show that the reflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$here is equivalent to a category of modules over an explicit algebra, denoted by$$R_H(A)$$, which we call thereflective algebraofA. This result is akin to$${\mathcal {Z}}({\mathcal {C}})$$being represented by the Drinfeld double$${\operatorname {Drin}}(H)$$ofH. We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangularH-comodule algebras, and we examine their corresponding quantumK-matrices; this yields solutions to the qRE. We also establish that the reflective algebra$$R_H(\mathbb {k})$$is an initial object in the category of quasitriangularH-comodule algebras, where$$\mathbb {k}$$is the ground field. The case whenHis the Drinfeld double of a finite group is illustrated. 

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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. IWASAWA THEORY FOR p -TORSION CLASS GROUP SCHEMES IN CHARACTERISTIC p

   https://doi.org/10.1017/nmj.2022.30  

   BOOHER, JEREMY; CAIS, BRYDEN (June 2023, Nagoya Mathematical Journal)

   Abstract We investigate a novel geometric Iwasawa theory for$${\mathbf Z}_p$$-extensions of function fields over a perfect fieldkof characteristic$$p>0$$by replacing the usual study ofp-torsion in class groups with the study ofp-torsion class groupschemes. That is, if$$\cdots \to X_2 \to X_1 \to X_0$$is the tower of curves overkassociated with a$${\mathbf Z}_p$$-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of thep-torsion group scheme in the Jacobian of$$X_n$$as$$n\rightarrow \infty $$. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of$$X_n$$equipped with natural actions of Frobenius and of the Cartier operatorV. We formulate and test a number of conjectures which predict striking regularity in the$$k[V]$$-module structure of the space$$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$$of global regular differential forms as$$n\rightarrow \infty .$$For example, for each tower in a basic class of$${\mathbf Z}_p$$-towers, we conjecture that the dimension of the kernel of$$V^r$$on$$M_n$$is given by$$a_r p^{2n} + \lambda _r n + c_r(n)$$for allnsufficiently large, where$$a_r, \lambda _r$$are rational constants and$$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$$is a periodic function, depending onrand the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on$${\mathbf Z}_p$$-towers of curves, and we prove our conjectures in the case$$p=2$$and$$r=1$$. 

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