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Abstract The problem of analyzing interconnectedness is one of today’s premier challenges in understanding systemic risk. Connections can both stabilize networks and provide pathways for contagion. The central problem in such networks is establishing global behavior from local interactions. Jiang-Lim-Yao-Ye (Jianget al2011Mathematical Programming 1271203–244) recently introduced the use of theHodge decomposition(see Lim 2020SIAM Review62685–715 for a review), a fundamental tool from algebraic geometry, to construct global rankings from local interactions (see Barbarossaet al2018(2018 IEEE Data Science Workshop (DSW), IEEE)pp 51–5; Haruna and Fujiki 2016Frontiers in Neural Circuits1077; Jiaet al2019(Proc. of the XXV ACM SIGKDD International Conf. on Knowledge Discovery & Data Mining, pp 761–71 for other applications). We apply this to a study of financial networks, starting from the Eisenberg-Noe (Eisenberg and Noe 2001Management Science47236–249) setup of liabilities and endowments, and construct a network of defaults. We then use Jiang-Lim-Yao-Ye to construct a global ranking from the defaults, which yields one way of quantifying ‘systemic importance’. 

Abstract A hyperplane arrangement in $$\mathbb P^n$$ is free if $R/J$ is Cohen–Macaulay (CM), where $$R = k[x_0,\dots ,x_n]$$ and $$J$$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $$ J^{un}$$, the intersection of height two primary components, and $$\sqrt{J}$$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $$\mathbb P^3$$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $$r$$, there is an arrangement for which $$R/J^{un}$$ (resp. $$R/\sqrt{J}$$) fails to be CM in only one degree, and this failure is by $$r$$. We draw consequences for the even liaison class of $$J^{un}$$ or $$\sqrt{J}$$. 

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]. 

CBMS notes from a series of 10 Lectures by David Cox, with supplemental lectures by C. D'Andrea, A. Dickenstein, J. Hauenstein, H. Schenck, J. Sidman. Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert.