https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

The Geometry of the Zeros of a Polynomial in a complex variable. By M. Marden. Pp. ix, 183. $5. 1949. Mathematical Surveys, 3. (American Mathematical Society)

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

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Triangulated categories of mixed motives

http://www.math.huji.ac.il/~temkin/lectures/Hironaka80.pdf

Desingularization of quasi-excellent schemes in characteristic zero

Integral, measure, and derivative; a unified approach

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A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms

By using the approach via the aspect of form and general topology, as well as basic notions of abstract algebra, a theoretical framework has been developed which elucidates the mechanism of the additivity relationships between structure and properties in molecules having many identical moieties. The main theorem, the Asymptotic Linearity Theorem (ALT), together with an auxiliary theorem, the a Independence Theorem, implies that the zero-point vibrational energy (or total pi-electron energy for the case of alternant hydrocarbons)En of a linearly extended system B-An-B′ having n repeating identical moieties has the asymptotic expansionEn = αn +β +o(1) as n → ∞, where α ∈ ℝ is independent of the choice of the end moieties B and B′. The theorem being formulated in a general context, the actual implication of the ALT is much broader than the above two applications would indicate.

The asymptotic linearity theorem for the study of additivity problems of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternant hydrocarbons

The present article is the first part of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. Four key problems are formulated whose affirmative solutions imply the formation of the initial investigative bridge between the research field of nanotubes and that of the additivity and other network problems studied and solved by using the RST. All of these four problems are solved affirmatively by using tools from the RST. The Piecewise Monotone Lemmas (PMLs) are cornerstones of the proof of the Fukui conjecture concerning the additivity problems of hydrocarbons. The solution of the fourth problem gives a generalized analytical formula of the pi-electron energy band curves of nanotube (a, b), with two new complex parameters c and d. These two parameters bring forth a broad class of analytic curves to which the PMLs and associated theoretical devices apply. Based on the above affirmative solutions of the problems, a central theorem in the RST, called the asymptotic linearity theorem (ALT) has been applied to nanotubes and monocyclic polyenes. Analytical formulae derived in this application of the ALT illuminate in a new global context (i) the conductivity of nanotubes and (ii) the aromaticity of monocyclic polyenes; moreover an analytical formula obtained by using the ALT provides a fresh insight into H¨ uckel’s (4n +2) rule. The present article forms a foundation of the forthcoming articles in this series.

Repeat space theory applied to carbon nanotubes and related molecular networks. III

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1,..., xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points ${{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}$ , the existence of connected sets in the real spectrum of R[x1,..., xn], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points ${{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}$ , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.

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Mark Spivakovsky

Papers

A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms

The asymptotic linearity theorem for the study of additivity problems of the zero-point vibrational energy of hydrocarbons and the total pi-electron energy of alternant hydrocarbons

Repeat space theory applied to carbon nanotubes and related molecular networks. III

On connectedness of sets in the real spectra of polynomial rings

Non-existence of the Artin function for Henselian pairs