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1. NORMAL REFLECTION SUBGROUPS OF COMPLEX REFLECTION GROUPS

   https://doi.org/10.1017/S1474748021000323  

   Arreche, Carlos E.; Williams, Nathan F. (July 2021, Journal of the Institute of Mathematics of Jussieu)

   Abstract We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author. 

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2. A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

   https://doi.org/10.1112/mtk.12231  

   Cook, Brian; Hughes, Kevin; Li, Zane Kun; Mudgal, Akshat; Robert, Olivier; Yung, Po‐Lam (November 2023, Mathematika)

   We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory. 

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3. On the parameterized Tate construction

   https://doi.org/10.1112/topo.70018  

   Quigley, J_D; Shah, Jay (March 2025, Journal of Topology)

   Abstract We introduce and study a genuine equivariant refinement of the Tate construction associated to an extension of a finite group by a compact Lie group , which we call the parameterized Tate construction . Our main theorem establishes the coincidence of three conceptually distinct approaches to its construction when is also finite: one via recollement theory for the ‐free ‐family, another via parameterized ambidexterity for ‐local systems, and the last via parameterized assembly maps. We also show that uniquely admits the structure of a lax ‐symmetric monoidal functor, thereby refining a theorem of Nikolaus and Scholze. Along the way, we apply a theorem of the second author to reprove a result of Ayala–Mazel‐Gee–Rozenblyum on reconstructing a genuine ‐spectrum from its geometric fixed points; our method of proof further yields a formula for the geometric fixed points of an ‐complete ‐spectrum for any ‐family . 

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4. Stability in the high-dimensional cohomology of congruence subgroups

   https://doi.org/10.1112/s0010437x20007046  

   Miller, Jeremy; Nagpal, Rohit; Patzt, Peter (April 2020, Compositio Mathematica)

   We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $$\mathbf{SL}_{n}(\mathbb{Z})$$ . This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $$\mathbf{SL}_{n}(K)$$ for $$K$$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $$\mathbf{SL}_{n}(\mathbb{Z})$$ . 

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5. Stratified Noncommutative Geometry

   https://doi.org/10.1090/memo/1485  

   Ayala, David; Mazel-Gee, Aaron; Rozenblyum, Nick (May 2024, Memoirs of the American Mathematical Society)

   We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable $\mathrm{\infty <\#comment/>}$-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as ${\mathbb{E}}_n$-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group $G$, we give a symmetric monoidal stratification of genuine $G$-spectra. In the case that $G$is finite, this expresses genuine $G$-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory. 

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