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1. Geography of symplectic Lefschetz fibrations and rational blowdowns

   https://doi.org/10.1090/tran/8968  

   Baykur, R; Korkmaz, Mustafa; Simone, Jonathan (July 2024, Transactions of the American Mathematical Society)

   We produce simply-connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides asymplecticextension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$–manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{C}\mathbb{P}{}^2\mathrm{\#<\#comment/>}5\stackrel{¯<\#comment/>}{\mathbb{C}\mathbb{P}}{}^2$through this procedure applied to a genus– $3$fibration. 

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2. Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

   https://doi.org/10.15673/pigc.v16i2.2648  

   Abebaw, Tilahun; Gatsinzi, Jean-Baptiste; Yeruk, Smegnsh Demelash (June 2024, Proceedings of the International Geometry Center)

   The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. 

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3. Koszul modules with vanishing resonance in algebraic geometry

   https://doi.org/10.1007/s00029-023-00912-4  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Weyman, Jerzy (April 2024, Selecta Mathematica)

   Abstract We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace$$K\subseteq \bigwedge ^2 V$$ $K\subseteq {\bigwedge }^{2}V$, whereVis a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves onK3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. 

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4. Bounds on cohomological support varieties

   https://doi.org/10.1090/btran/182  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (March 2024, Transactions of the American Mathematical Society, Series B)

   Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$of finitely generated $R$-modules an algebraic variety ${\mathrm{V}}_R\left(M\right)$that encodes homological properties of $M$. We give lower bounds for the dimension of ${\mathrm{V}}_R\left(M\right)$in terms of classical invariants of $R$. In particular, when $R$is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$has finite projective dimension, we also give an upper bound for $\dim <\#comment/>{\mathrm{V}}_R\left(M\right)$in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of $R$. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. 

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5. Surfacing the complex conceptions of equity across making and tinkering spaces

   https://doi.org/10.1108/ILS-10-2022-0115  

   Roque, Ricarose; Hladik, Stephanie; Moreno, Celeste; Hayden, Ronni (July 2023, Information and Learning Sciences)

   PurposeRelatively few studies have examined the perspectives of informal learning facilitators who play key roles in cultivating an equitable learning environment for nondominant youth and families in making and tinkering spaces. This study aims to foreground the perspectives of facilitators and highlight the complexities and tensions that influence their equity work. Design/methodology/approachInterviews were conducted with facilitators of making and tinkering spaces across three informal learning organizations: a museum, a public library system and a network of community technology centers. This study then used a framework that examined equity along dimensions of access to what, for whom, based on whose values and toward what ends to analyze both the explicit and implicit conceptions of equity that surfaced in these interviews. FindingsAcross organizations, this study identified similarities and differences in facilitators’ conceptualizations of equity that were influenced by their different contexts and had implications for practice at each organization. Highlighting the complexity of enacting equity in practice, this study found moments when dimensions of equity came together in resonant ways, while other moments showed how dimensions can be in tension with each other. Practical implicationsThe complexity that facilitators must navigate to enact equity in their practice emphasizes the need for professional development and support for facilitators to deepen their conceptions and practices around equity beyond access – not just skill building in making and tinkering. Originality/valueThis study recognizes the important role that facilitators play in enabling equity-oriented participation in making and tinkering spaces and contributes the “on the ground” perspectives of facilitators to highlight the complexity and tensions of enacting equity in practice. 

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