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1. 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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2. Integral models for spaces via the higher Frobenius

   https://doi.org/10.1090/jams/998  

   Yuan, Allen (January 2023, Journal of the American Mathematical Society)

   We give a fully faithful integral model for simply connected finite complexes in terms of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$is the data of its Spanier-Whitehead dual, as an ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\mathrm{\infty <\#comment/>}$-category of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we callpartial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of ${\mathbb{F}}_p$up to $p$-completion. 

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3. The two-phase problem for harmonic measure in VMO and the chord-arc condition

   https://doi.org/10.1090/btran/197  

   Tolsa, Xavier; Toro, Tatiana (November 2024, Transactions of the American Mathematical Society, Series B)

   Let ${\mathrm{\Omega <\#comment/>}}^{+}\subset <\#comment/>{\mathbb{R}}^{n+1}$be a bounded $\mathrm{\delta <\#comment/>}$-Reifenberg flat domain, with $\mathrm{\delta <\#comment/>}>0$small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}={\mathbb{R}}^{n+1}\setminus <\#comment/>\stackrel{¯<\#comment/>}{{\mathrm{\Omega <\#comment/>}}^{+}}$is an NTA (non-tangentially accessible) domain as well and denote by ${\mathrm{\omega <\#comment/>}}^{+}$and ${\mathrm{\omega <\#comment/>}}^{-<\#comment/>}$the respective harmonic measures of ${\mathrm{\Omega <\#comment/>}}^{+}$and ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}$with poles $p^{\pm <\#comment/>}\in <\#comment/>{\mathrm{\Omega <\#comment/>}}^{\pm <\#comment/>}$. In this paper we show that the condition that $\log <\#comment/>\frac{d{\mathrm{\omega <\#comment/>}}^{-<\#comment/>}}{d{\mathrm{\omega <\#comment/>}}^{+}}\in <\#comment/>\mathrm{VMO}<\#comment/>\left({\mathrm{\omega <\#comment/>}}^{+}\right)$is equivalent to ${\mathrm{\Omega <\#comment/>}}^{+}$being a chord-arc domain with inner unit normal belonging to $\mathrm{VMO}<\#comment/>\left({\mathcal{H}}^n{|}_{\mathrm{\partial <\#comment/>}{\mathrm{\Omega <\#comment/>}}^{+}}\right)$. 

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4. On the Boundary Behavior of Mass-Minimizing Integral Currents

   https://doi.org/10.1090/memo/1446  

   De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

   Let $\mathrm{\Sigma <\#comment/>}$be a smooth Riemannian manifold, $\mathrm{\Gamma <\#comment/>}\subset <\#comment/>\mathrm{\Sigma <\#comment/>}$a smooth closed oriented submanifold of codimension higher than $2$and $T$an integral area-minimizing current in $\mathrm{\Sigma <\#comment/>}$which bounds $\mathrm{\Gamma <\#comment/>}$. We prove that the set of regular points of $T$at the boundary is dense in $\mathrm{\Gamma <\#comment/>}$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\mathrm{\Sigma <\#comment/>}$and $\mathrm{\Gamma <\#comment/>}$. As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $\mathrm{\Gamma <\#comment/>}$is connected, then $T$has at least one point $p$of multiplicity $\frac{1}{2}$, namely there is a neighborhood of the point $p$where $T$is a classical submanifold with boundary $\mathrm{\Gamma <\#comment/>}$; we generalize Almgren’s connectivity theorem showing that the support of $T$is always connected if $\mathrm{\Gamma <\#comment/>}$is connected; we conclude a structural result on $T$when $\mathrm{\Gamma <\#comment/>}$consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\mathrm{\Sigma <\#comment/>}={\mathbb{R}}^{m+1}$and $T$is $m$-dimensional. 

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5. The structure of arbitrary Conze–Lesigne systems

   https://doi.org/10.1090/cams/31  

   Jamneshan, Asgar; Shalom, Or; Tao, Terence (February 2024, Communications of the American Mathematical Society)

   Let $\mathrm{\Gamma <\#comment/>}$be a countable abelian group. An (abstract) $\mathrm{\Gamma <\#comment/>}$-system $\mathrm{X}$- that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\mathrm{\Gamma <\#comment/>}$- is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor ${\mathrm{Z}}^2\left(\mathrm{X}\right)$. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian $\mathrm{\Gamma <\#comment/>}$, namely that they are the inverse limit of translational systems $G_n/{\mathrm{\Lambda <\#comment/>}}_n$arising from locally compact nilpotent groups $G_n$of nilpotency class $2$, quotiented by a lattice ${\mathrm{\Lambda <\#comment/>}}_n$. Results of this type were previously known when $\mathrm{\Gamma <\#comment/>}$was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3\left(G\right)$norm for arbitrary finite abelian groups $G$. 

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