The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.Comment: 46 pages. Final version, to appear in Advances in Mathematic

Optimal devaluations

According to the conventional wisdom, when an economy enters a recession and nominal prices adjust slowly, the monetary authority should devalue the domestic currency to make the recession less severe. The reason is that a devaluation of the currency lowers the relative price of non-tradable goods, and this reduces the necessary adjustment in output relative to the case in which the exchange rate remains constant. This paper uses a simple small open economy model with sticky prices to characterize optimal fiscal and monetary policy in response to productivity and terms of trade shocks. Contrary to the conventional wisdom, in this framework optimal exchange rate policy cannot be characterized just by the cyclical properties of output. The source of the shock matters: while recessions induced by a drop in the price of exportable goods call for a devaluation of the currency, those induced by a drop in productivity in the non-tradable sector require a revaluation.Economic Theory&Research,Debt Markets,Emerging Markets,Currencies and Exchange Rates,Economic Stabilization

Tur\'an and Ramsey problems for alternating multilinear maps

Guided by the connections between hypergraphs and exterior algebras, we study Tur\'an and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lov\'asz (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992). Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.Comment: 20 pages. v3: rewrite introductio

On the Genealogy of Universals: The Metaphysical Origins of Analytic Philosophy

The concepts of particular and universal have grown so familiar that their significance has become difficult to discern, like coins that have been passed back and forth too many times, worn smooth so their values can no longer be read. On the Genealogy of Universals seeks to overcome our sense of over-familiarity with these concepts by providing a case study of their evolution during the late nineteenth century and early twentieth century, a study that shows how the history of these concepts is bound up with the origins and development of analytic philosophy itself. Understanding how these concepts were taken up, transfigured, and given up by the early analytic philosophers, enables us to recover and reanimate the debate amongst them that otherwise remains Delphic. This book begins from the early, originating texts of analytic philosophy that have hitherto baffled commentators, including Moore's early papers, and engages afresh with the neglected contributions of philosophical figures that historians of analytic philosophy have mostly since forgotten, including Stout and Whitehead. This sheds new light upon the relationships of Moore to Russell, Russell to Wittgenstein, and Wittgenstein to Ramsey

Optimal Monetary Policy with Endogenous Entry and Product Variety

We show that deviations from long-run stability of product prices are optimal in the presence of endogenous producer entry and product variety in a sticky-price model with monopolistic competition in which price stability would be optimal in the absence of entry. Specifically, a long-run positive (negative) rate of inflation is optimal when the benefit of variety to consumers falls short of (exceeds) the market incentives for creating that variety under flexible prices, governed by the desired markup. Plausible preference specifications and parameter values justify a long-run inflation rate of two percent or higher. Price indexation implies even larger deviations from long-run price stability. However, price stability (around this non-zero trend) is close to optimal in the short run, even in the presence of time-varying flexible-price markups that distort the allocation of resources across time and states. The central bank uses its leverage over real activity in the long run, but not in the short run. Our results point to the need for continued empirical research on the determinants of markups and investigation of the benefit of product variety to consumers.

On the isomorphism question for complete Pick multiplier algebras

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra \mv = \{f\big|_V : f \in \cM_d\}, where $d$ is some integer or $\infty$, \cM_d is the multiplier algebra of the Drury-Arveson space $H^2_d$, and $V$ is a subvariety of the unit ball. For finite $d$ it is known that, under mild assumptions, every isomorphism between two such algebras \mv and \mw is induced by a biholomorphism between $W$ and $V$. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where $V$ is the proper image of a finite Riemann surface. The second deals with the case where $V$ is a disjoint union of varieties.Comment: 17 pages. Final version, to appear in Integral Equations and Operator Theor