The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

The large deviation approach to statistical mechanics

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

Vernon Smith (1927–) received his Ph.D. in economics at Harvard. Although his more than two hundred books and articles address issues in many areas of economics, he is best known for his work on experimental economics, for which he received the Nobel Prize in 2002. After many years at the University of Arizona, Smith is now a professor of economics at George Mason University. Why do economists conduct experiments? To answer that question, it is first necessary briefly to specify the ingredients of an experiment. Every laboratory experiment is defined by an environment , specifying the initial endowments, preferences and costs that motivate exchange. This environment is controlled using monetary rewards to induce the desired specific value/cost configuration (Smith, 1991, 6). An experiment also uses an institution defining the language (messages) of market communication (bids, offers, acceptances), the rules that govern the exchange of information, and the rules under which messages become binding contracts. This institution is defined by the experimental instructions which describe the messages and procedures of the market, which are most often computer controlled. Finally, there is the observed behavior of the participants in the experiments as a function of the environment and institution that constitute the controlled variables. Using this framework of environment, institution, and behavior, I can think of at least seven prominent reasons in the literature as to why economists conduct experiments. Undoubtedly, there are more (Davis and Holt, 1992, Chapter 1 and passim).

https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.8.1.113

Economics in the Laboratory

On the occasion of the hundredth anniversary of Albert Einstein's annus mirabilis, we reflect on the development and current state of research in cavity quantum electrodynamics in the optical domain. Cavity QED is a field which undeniably traces its origins to Einstein's seminal work on the statistical theory of light and the nature of its quantized interaction with matter. In this paper, we emphasize the development of techniques for the confinement of atoms strongly coupled to high-finesse resonators and the experiments which these techniques enable.

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Trapped atoms in cavity QED: coupling quantized light and matter

These scientists prioritized their replies to letters in the same way that people rate their e-mails today.

/pdf/human-dynamics-darwin-and-einstein-correspondence-patterns-4p5ep6m5as.pdf

Human dynamics: Darwin and Einstein correspondence patterns.

List of Texts xiii List of Illustrations xxxiii INTRODUCTORY MATERIAL Introduction to Volume 14 xxxv Editorial Method of the Series lxxxv Acknowledgments xciii Note on the Translation xcv List of Abbreviations xcvi Location Symbols xcvii Descriptive Symbols c TEXTS 1 Alphabetical List of Correspondence 783 Chronology 805 Calendar of Abstracts 821 Appendixes 905 Literature Cited 1007 Index 1031 Index of Citations 1097

The collected papers of Albert Einstein

T HE WILHELMIAN ERA witnessed an enormous transformation in German mathematics, one that manifested itself not only in new research developments in pure mathematics but also in the emergence of a growing concern for areas where mathematics could potentially be applied. Research in pure mathematics moved away from traditional subjects like elliptic and Abelian functions, algebraic geometry, and invariant theory, fields in which a premium was placed on computational techniques and algorithmic virtuosity. The first decades of the twentieth century saw the proliferation of new research fields like set theory, functional analysis, integral equations, point-set topology, and analytic number theory. Interest in the classical fields of nineteenth-century mathematics often gave way to the study of these new abstract subjects or to generalized axiomatic approaches to the older ones. This fascination with abstract structures crystallized during the Weimar era when Emmy Noether and her Gottingen colleagues launched the new subdiscipline of modern algebra. This led to a movement in twentieth-century mathematics that culminated thirty years later in the monumental work of the French mathematicians who wrote under the collective pseudonym of Nicolas Bourbaki. The line between pure and applied mathematics also began to blur at this time. A substantial part of the new mathematics-matrix algebra, groups and their representations, integral equations, and tensor analysis-proved to be of vital importance for relativity theory and quantum mechanics. Moreover, a good deal of activity was taking place on the applied side of the mathematical spectrum, as new mathematical methods were developed for dealing with a variety of problems posed by construction engineering, ship design, aerodynamics, financial investments, and insurance. The demands of a modern industrialized society were particularly acute in Germany, and mathematicians could ill afford to continue embracing the outmoded neohumanist ideology that had guided their profession throughout the nineteenth century. Although this transformation was obviously a complex process involving the interaction of numerous factors, many of which operated on the macro level, the stage on which much of it was played out was the remote little town of Gottingen. It was there, beginning in 1895, that the two principal actors in this drama, Felix Klein (1849-1925) and David Hilbert (1862-1943), forged an intellectual partnership that not only overturned the balance of power within German mathematics but also significantly altered its research orientation and the relationship between mathematics and other scientific and technological disciplines.

Klein, Hilbert, and the Gottingen Mathematical Tradition

1: An overview of American mathematics: 1776--1876. 2: A new departmental prototype: J. J. Sylvester and the Johns Hopkins University. 3: Mathematics at Sylvester's Hopkins. 4: German mathematics and the early mathematical career of Felix Klein. 5: America's wanderlust generation. 6: Changes on the horizon. 7: The World's Columbian exposition of 1893 and the Chicago Mathematical Congress. 8: Surveying mathematical landscapes: The Evanston Colloquium Lectures. 9: Meeting the challenge: The University of Chicago and the American mathematical research community. 10: Epilogue: Beyond the threshold: The American mathematical research community, 1900--1933. Bibliography

The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore

IN 1936 THE GOTTINGER TAGEBLATT, a right-wing newspaper, printed an article with the headline: \"Felix Klein was an Aryan. Which no one, in Gottingen at least, doubted.\" According to the Tageblatt, Klein's ancestry was cleared by his family after the Vcilkischer Beobachter, official mouthpiece for the Nazi party, reported that he was of Jewish descent. The source of the error was said to be the Jewish Encyclopedia, but the Tageblatt added, \"It is well known that the Jews love to stamp famous men as Jewish in order to increase the prestige of their people.\"1 Of course, the Nazi party was very diligent when it came to tracing a suspicious family's ethnic background. Still, one might wonder what led them to chase Felix Klein's ghost some ten years after his death in 1925. What apparently prompted the Nazi investigation was a memorandum sent to the Bavarian Ministry of Culture by Hugo Dingler in November 1933, which quickly found its way to the offices of the Prussian Ministry of the Interior. Its subject was the dominance of Jews in the fields of mathematics and physics; and it was accompanied by a cover letter from Phillip Lenard, the father of \"Deutsche Physik.\"2 The memorandum presented a twenty-page historical synopsis of how Jews invaded the fields of mathematics and physics after being granted legal equality in 1869. According to Dingler, their ringleader was none other than Felix Klein, who \"at least from one side of his family was of Jewish descent.\" Dingler went on to describe how Klein perverted German mathematics by imposing a selfstyled dictatorship over the field. This began when he circulated a proposal among the mathematics professors calling for the organization of research through one central university, namely Gottingen. When this idea failed to win

David E. Rowe

Papers

The collected papers of Albert Einstein

Klein, Hilbert, and the Gottingen Mathematical Tradition

The Emergence of the American Mathematical Research Community, 1876-1900: J. J. Sylvester, Felix Klein, and E. H. Moore

"Jewish Mathematics" at Gottingen in the Era of Felix Klein

Felix Klein's “Erlanger Antrittsrede”: A Transcription with English Translation and Commentary