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1. Quadratic Gorenstein Rings and the Koszul Property II

   https://doi.org/10.1093/imrn/rnab297  

   Mastroeni, Matthew ; Schenck, Hal ; Stillman, Mike ( October 2021 , International Mathematics Research Notices)

   Abstract Conca–Rossi–Valla [6] ask if every quadratic Gorenstein ring $R$ of regularity three is Koszul. In [15], we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three, which are not Koszul. In this paper, we study the analog of the Conca–Rossi–Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having ${\operatorname {codim}} \ R = c$ and ${\operatorname {reg}} \ R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda [16] and Migliore–Nagel [19]. 

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2. A PIE Representation of Scalar Quadratic PDEs and Global Stability Analysis Using SDP

   Jagt, Declan ; Seiler, Peter ; Peet, Matthew ( December 2023 , Proceedings of the IEEE Conference on Decision Control)

   It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed the `fundamental state') lies in $L_2$ - requiring no auxiliary boundary conditions or continuity constraints. Such a representation (termed a Partial Integral Equation or PIE) is then defined in terms of an algebra of bounded integral operators (termed Partial Integral (PI) operators) and is constructed by identifying a unitary map from the fundamental state to the state of the original PDE. Unfortunately, when the PDE is nonlinear, the dynamics of the associated fundamental state are no longer parameterized in terms of PI operators. However, in this paper we show that such dynamics can be compactly represented using a new tensor algebra of partial integral operators acting on the tensor product of the fundamental state. We further show that this tensor product of the fundamental state forms a natural distributed equivalent of the monomial basis used in representation of polynomials on a finite-dimensional space. This new representation is then used to provide a simple SDP-based Lyapunov test of stability of quadratic PDEs. The test is applied to three illustrative examples of quadratic PDEs. 

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3. Higher resonance schemes and Koszul modules of simplicial complexes

   https://doi.org/10.1007/s10801-024-01313-2  

   Aprodu, Marian ; Farkas, Gavril ; Raicu, Claudiu ; Sammartano, Alessio ; Suciu, Alexander I. ( March 2024 , Journal of Algebraic Combinatorics)

   Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$$k$of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$$A=k⟨\Delta ⟩$is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$$\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

    

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4. Cyclic pairings and derived Poisson structures.

   Ramadoss, Ajay C ; Zhang, Yining ( April 2019 , New York journal of mathematics)

   null (Ed.)

   By a fundamental theorem of D. Quillen, there is a natural duality - an instance of general Koszul duality - between differential graded (DG) Lie algebras and DG cocommutative coalgebras defined over a field k of characteristic 0. A cyclic pairing (i.e., an inner product satisfying a natural cyclicity condition) on the cocommutative coalgebra gives rise to an interesting structure on the universal enveloping algebra Ua of the Koszul dual Lie algebra a called the derived Poisson bracket. Interesting special cases of the derived Poisson bracket include the Chas-Sullivan bracket on string topology. We study the derived Poisson brackets on universal enveloping algebras Ua, and their relation to the classical Poisson brackets on the derived moduli spaces DRep_g(a) of representations of a in a finite dimensional reductive Lie algebra g. More specifically, we show that certain derived character maps of a intertwine the derived Poisson bracket with the classical Poisson structure on the representation homology HR(a, g) related to DRep_g(a). 

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5. Explicit isomorphisms of quaternion algebras over quadratic global fields

   https://doi.org/10.1007/s40993-022-00380-3  

   Csahók, Tímea ; Kutas, Péter ; Montessinos, Mickaël ; Zábrádi, Gergely ( December 2022 , Research in Number Theory)

   Abstract Let L be a separable quadratic extension of either $${\mathbb {Q}}$$ Q or $${\mathbb {F}}_q(t)$$ F q ( t ) . We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over L . Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L -algebra. 

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