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1. Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

   https://doi.org/10.1017/fms.2021.16  

   Howe, Sean (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We construct a $$(\mathfrak {gl}_2, B(\mathbb {Q}_p))$$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $$0$$ of a sheaf on $$\mathbb {P}^1$$ , landing in the compactly supported completed $$\mathbb {C}_p$$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $$1$$ . For classical weight $$k\geq 2$$ , the Verma has an algebraic quotient $$H^1(\mathbb {P}^1, \mathcal {O}(-k))$$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $$\mathbb {P}^1$$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology. 

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2. Erratic birational behavior of mappings in positive characteristic

   https://doi.org/10.1002/mana.202200003  

   Cutkosky, Steven Dale (November 2023, Mathematische Nachrichten)

   Abstract Birational properties of generically finite morphisms of algebraic varieties can be understood locally by a valuation of the function field ofX. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. 

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3. Locally homogeneous non-gradient quasi-Einstein 3-manifolds

   https://doi.org/10.1515/advgeom-2021-0036  

   Lim, Alice (January 2022, Advances in Geometry)

   Abstract In this paper, we classify the compact locally homogeneous non-gradient m -quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m -quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m -quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S 1 is the only compact manifold of any dimension which admits a metric which is nontrivially m -quasi Einstein and Einstein. 

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4. The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron

   https://doi.org/10.1093/imrn/rnad010  

   Łukowski, Tomasz; Parisi, Matteo; Williams, Lauren K (March 2023, International Mathematics Research Notices)

   Abstract The positive Grassmannian $$Gr^{\geq 0}_{k,n}$$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$$\mu $$ onto the hypersimplex [ 31] and the amplituhedron map$$\tilde{Z}$$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $$\mathcal{N}=4$$ super Yang-Mills. We define a map we call T-duality from cells of $$Gr^{\geq 0}_{k+1,n}$$ to cells of $$Gr^{\geq 0}_{k,n}$$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $$\Delta _{k+1,n}$$ to positroid dissections of the amplituhedron $$\mathcal{A}_{n,k,2}$$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $$\mathcal{A}_{n,k,2}$$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $$Gr_{k,k+2}$$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $$m$$, we define the momentum amplituhedron for any even $$m$$. 

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5. Abstract Extensionality: On the properties of incomplete abstract interpretations

   https://doi.org/10.1145/3371096  

   Bruni, R; Giacobazzi, R; Gori, R; Garcia-Contreras, I; Pavlovic, D (January 2020, Proceedings of the ACM on programming languages)

   In this paper we generalise the notion of extensional (functional) equivalence of programs to abstract equivalences induced by abstract interpretations. The standard notion of extensional equivalence is recovered as the special case, induced by the concrete interpretation. Some properties of the extensional equivalence, such as the one spelled out in Rice’s theorem, lift to the abstract equivalences in suitably generalised forms. On the other hand, the generalised framework gives rise to interesting and important new properties, and allows refined, non-extensional analyses. In particular, since programs turn out to be extensionally equivalent if and only if they are equivalent just for the concrete interpretation, it follows that any non-trivial abstract interpretation uncovers some intensional aspect of programs. This striking result is also effective, in the sense that it allows constructing, for any non-trivial abstraction, a pair of programs that are extensionally equivalent, but have different abstract semantics. The construction is based on the fact that abstract interpretations are always sound, but that they can be made incomplete through suitable code ransformations. To construct these transformations, we introduce a novel technique for building incompleteness cliques of extensionally equivalent yet abstractly distinguishable programs: They are built together with abstract interpretations that produce false alarms. While programs are forced into incompleteness cliques using both control-flow and data-flow transformations, the main result follows from limitations of data-flow ransformations with respect to control-flow ones. A further consequence is that the class of incomplete programs for a non-trivial abstraction is Turing complete. The obtained results also shed a new light on the relation between the techniques of code obfuscation and the precision in program analysis. 

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