The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

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Cluster algebras I: Foundations

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

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Trees, parking functions, syzygies, and deformations of monomial ideals

In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E. Laguerre on uniform limits of sequences of real-rooted polynomials and by an interesting result of G. H. Hardy. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities, one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular; J. I. Hutchinson has shown that an entire function p(x) = a(0) + a(1)x + ... + a(n)x(n) + ... with strictly positive coefficients has the property that all of its finite segments a(i) x(i) + a(i+1)x(i+1) + ... + a(j)x(j) have only real roots if and only if a(i)(2)/a(i-1)a(i+1) >= 4 for i = 1, 2,.... In the present paper, we give sharp lower bounds on the ratios a(i)(2)/a(i-1)a(i+1) (i = 1, 2,...) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when i -> infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class L - PI. We also explain the relation between Newton's and Hutchinson's inequalities and the logarithmic image of the set of all real-rooted polynomials with positive coefficients.

Hardy-petrovitch-hutchinson's problem and partial theta function

The goal of the paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk i=1 Qi ...

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Algebro-geometric aspects of Heine–Stieltjes theory

For more than one hundred years, Schubert decomposition of complete and partial flag manifolds and Schubert calculus have been an intertwining point of algebra, geometry, representation theory, and combinatorics. In this paper, we are concerned with the further refinement of Schubert decomposition theory. Our far and ambitious goal is to describe the topology of intersection of two arbitrary Schubert cells. This problem seems to be very important in connection with representation theory, in particular, with calculation of Kazhdan–Lusztig polynomials, which is known to be a very hard problem. As a minor step in this direction, we calculate in this paper the number of connected components in the intersection of two open opposite Schubert cells in the space of real n-dimensional flags. This question was raised in [A], and later, in connection with criteria of total positivity, in [SS]. Other topological characteristics of pairwise intersections of Schubert cells (not necessarily opposite) are considered in our previous papers [SV], [SSV1], [SSV2]. Despite the fact that in this paper we consider only real algebraic varieties, we hope that our approach can give some information about complex geometry as well. The reasons for this hope are as follows. (a) The calculation method is based on the beautiful and very deep (“very algebraic”) results by Berenstein, Fomin, and Zelevinsky [BFZ] on the Lusztig parametrization of the unipotent lower-triangular matrices,which form an open cell in the flag manifold. It looks like the same type of parametrization and cell decompositions might be used

Connected Components in the Intersection of Two Open Opposite Schubert Cells in SLn(ℝ)/B

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.

Boris Shapiro

Papers

Trees, parking functions, syzygies, and deformations of monomial ideals

Hardy-petrovitch-hutchinson's problem and partial theta function

Algebro-geometric aspects of Heine–Stieltjes theory

Connected Components in the Intersection of Two Open Opposite Schubert Cells in SLn(ℝ)/B

Hardy-Petrovitch-Hutchinson's problem and partial theta function