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1. Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

   Casale, Guy; Freitag, James; Nagloo, Joel (November 2020, Annals of mathematics)

   null (Ed.)

   We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups. 

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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. Interpolation Polynomials, Bar Monomials, and Their Positivity

   https://doi.org/10.1093/imrn/rnac049  

   Naqvi, Yusra; Sahi, Siddhartha; Sergel, Emily (March 2022, International Mathematics Research Notices)

   Abstract We prove a conjecture of Knop–Sahi on the positivity of interpolation polynomials, which is an inhomogeneous generalization of Macdonald’s conjecture for Jack polynomials. We also formulate and prove the nonsymmetric version of this conjecture, and in fact, we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade. 

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4. The existence of the Kähler–Ricci soliton degeneration

   https://doi.org/10.1017/fmp.2023.5  

   Blum, Harold; Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan (January 2023, Forum of Mathematics, Pi)

   Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field . 

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5. Some remarks about deformation theory and formality conjecture

   https://doi.org/10.1007/s11565-024-00500-0  

   Chen, Huachen; Pertusi, Laura; Zhao, Xiaolei (February 2024, ANNALI DELL'UNIVERSITA' DI FERRARA)

   Abstract Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel–Mukai threefolds and quartic double solids. 

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