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1. Standard monomials and invariant theory of arc spaces II: Symplectic group

   https://doi.org/10.1090/jag/834  

   Linshaw, Andrew; Song, Bailin (October 2024, Journal of Algebraic Geometry)

   This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the Pfaffian variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group. 

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2. Standard monomials and invariant theory for arc spaces III: Special linear group

   https://doi.org/10.1016/j.jalgebra.2024.09.037  

   Linshaw, Andrew R; Song, Bailin (February 2025, Journal of Algebra)

   This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced. 

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3. K-theoretic Gromov–Witten invariants of line degrees on flag varieties

   https://doi.org/10.1142/S0217751X24460138  

   Buch, Anders S; Chen, Linda; Xu, Weihong (November 2024, International Journal of Modern Physics A)

   A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

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4. Residually finite rationally p groups

   https://doi.org/10.1142/S0219199719500160  

   Koberda, Thomas; Suciu, Alexander I. (May 2020, Communications in Contemporary Mathematics)

   In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so. 

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5. Invariant measures in simple and in small theories

   https://doi.org/10.1142/S0219061322500258  

   Chernikov, Artem; Hrushovski, Ehud; Kruckman, Alex; Krupiński, Krzysztof; Moconja, Slavko; Pillay, Anand; Ramsey, Nicholas (August 2023, Journal of Mathematical Logic)

   We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. 

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