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1. Decategorified Heegaard Floer theory and actions of both E and F

   https://doi.org/10.4171/QT/204  

   Manion, Andrew (January 2024, Quantum Topology)

   We define larger variants of the vector spaces one obtains by decategorifying bordered (sutured) Heegaard Floer invariants of surfaces. We also define bimodule structures on these larger spaces that are similar to, but more elaborate than, the bimodule structures that arise from decategorifying the higher actions in bordered Heegaard Floer theory introduced by Rouquier and the author. In particular, these new bimodule structures involve actions of both odd generatorsEandFof\mathfrak{gl}(1|1), whereas the previous ones only involved actions ofE. Over\mathbb{F}_2, we show that the new bimodules satisfy the necessary gluing properties to give a 1+1 open-closed TQFT valued in graded algebras and bimodules up to isomorphism; in particular, unlike in previous related work, we have a gluing theorem when gluing surfaces along circles as well as intervals. Over the integers, we show that a similar construction gives two partially defined open-closed TQFTs with two different domains of definition depending on how parities are chosen for the bimodules. We formulate conjectures relating these open-closed TQFTs with the\mathfrak{psl}(1|1)Chern–Simons TQFT recently studied by Mikhaylov and Geer–Young. 

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2. 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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3. Rank bounds in link Floer homology and detection results

   https://doi.org/10.4171/QT/197  

   Binns, Fraser; Dey, Subhankar (November 2023, Quantum Topology)

   Viewing the BRAID invariant as a generator of link Floer homology, we generalize work of Baldwin–Vela-Vick to obtain rank bounds on the next-to-top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight and prove a variant of a classification of links with Khovanov homology of low rank due to Xie–Zhang. In another direction, we use a variant of Ozsváth–Szabó's classification ofE_2collapsed\mathbb{Z}\oplus\mathbb{Z}filtered chain complexes to show that knot Floer homology detectsT(2,8)andT(2,10). Combining these techniques with the spectral sequences of Batson–Seed, Dowlin, and Lee, we can show that Khovanov homology likewise detectsT(2,8)andT(2,10). 

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4. Distribution of Kloosterman paths to high prime power moduli

   https://doi.org/10.1090/btran/98  

   Milićević, Djordje; Zhang, Sichen (June 2023, Transactions of the American Mathematical Society, Series B)

   We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power $p^n$of a fixed odd prime $p$, a pure depth-aspect analogue of theorems of Kowalski–Sawin and Ricotta–Royer–Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as $n\to <\#comment/>\mathrm{\infty <\#comment/>}$to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every $p$converges in law as $p\to <\#comment/>\mathrm{\infty <\#comment/>}$, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the $p$-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments. 

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5. Extremal weights and a tameness criterion for mod $$p$$ Galois representations

   https://doi.org/10.4171/JEMS/1513  

   Le, Daniel; Le_Hung, Bao Viet; Levin, Brandon; Morra, Stefano (September 2024, Journal of the European Mathematical Society)

   We study the weight part of Serre’s conjecture for genericn-dimensional modpGalois representations. We first generalize Herzig’s conjecture to the case where the field is ramified atpand prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated ton-dimensional local modprepresentations which we callextremal weights. Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations. 

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