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1. Combinatorics and Real Lifts of Bitangents to Tropical Quartic Curves

   https://doi.org/10.1007/s00454-022-00445-1  

   Cueto, Maria Angelica; Markwig, Hannah (April 2023, Discrete & Computational Geometry)

   Abstract Smooth algebraic plane quartics over algebraically closed fields of characteristic different than two have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real. 

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2. Quasi-projectivity of images of mixed period maps

   https://doi.org/10.1515/crelle-2023-0070  

   Bakker, Benjamin; Brunebarbe, Yohan; Tsimerman, Jacob (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasi-projective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions.Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating an ${ℝ}_{\mathrm{an},\exp }${\mathbb{R}_{\mathrm{an},\exp}}-definable structure to mixed period domains and admissible mixed period maps. 

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3. Persistent cup product structures and related invariants

   https://doi.org/10.1007/s41468-023-00138-5  

   Mémoli, Facundo; Stefanou, Anastasios; Zhou, Ling (October 2023, Journal of Applied and Computational Topology)

   Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length$$\ell \ge 0$$ $\ell \ge 0$and the other is the filtration parameter. This new persistence structure, called thepersistent cup module, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter$$\ell $$ $\ell$, we obtain a 1-dimensional persistence module, called the persistent$$\ell $$ $\ell$-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of apersistent invariant, which extends both therank invariant(also referred to aspersistent Betti number), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-definedpersistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called thepersistent LS-category invariant. 

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4. Free Stein irregularity and dimension

   https://doi.org/10.7900/jot.2019aug29.2271  

   Charlesworth, Ian; Nelson, Brent (January 2021, Journal of operator theory)

   null (Ed.)

   We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray–von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu’s free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples. 

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5. Associahedra as moment polytopes

   https://doi.org/10.1112/blms.13212  

   Gekhtman, Michael; Thomas, Hugh (December 2024, Bulletin of the London Mathematical Society)

   Abstract Generalized associahedra are a well‐studied family of polytopes associated with a finite‐type cluster algebra and a choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani‐Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster ‐variety with universal coefficients by its maximal natural torus action. We prove our result by showing that the construction of Padrol, Palu, Pilaud, and Plamondon can be understood on the basis of the way that moment polytopes behave under symplectic reduction. 

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