Spectral Flow

"This volume is dedicated to the eightieth birthday of Professor M. Sh. Birman. It contains original articles in spectral and scattering theory of differential operators, in particular, Schrodinger operators, and in homogenization theory. All articles are written by members of M. Sh. Birman's research group who are affiliated with different universities all over the world. A specific feature of the majority of the papers is a combination of traditional methods with new modern ideas."--BOOK JACKET.

This second edition volume expands on the previous edition with discussions on the latest approaches used in the spectral cytometry field. Beginning with a brief history of spectral cytometry development, this book continues with a section addressing new protocols in different areas of imaging cytometry based on the unique technology of Imagestream and also introduces FlowCam – cytometer with capabilities of analyzing phytoplankton and zooplankton. Written for the highly successful Methods in Molecular Biology series, chapters include brief introductions to their respective topics, complete lists of the materials and reagents necessary, and reproducible laboratory protocols with tips on troubleshooting and recommendations. Cutting-edge and thorough, Spectral and Imaging Flow Cytometry: Methods and Protocols, Second Edition is a valuable resources for beginners and professionals interested in learning more about this developing field. Chapters 1, 2, and 5 are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schrödinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis, Spectral Shift Function and Quantum Resonances. The book presents survey articles as well as original research papers on these topics. It will be of interest to researchers and graduate students in Mathematics and Mathematical Physics.

Noncommutative geometry is a new field that is among the great challenges of present-day mathematics. Its methods allow one to treat noncommutative algebras - such as algebras of pseudodifferential operators, group algebras, or algebras arising from quantum field theory - on the same footing as commutative algebras, that is, as spaces. Applications range over many fields of mathematics and mathematical physics. This volume contains the proceedings of the workshop on "Cyclic Cohomology and Noncommutative Geometry" held at The Fields Institute (Waterloo, ON) in June 1995. The workshop was part of the program for the special year on operator algebras and its applications.

Vijay Kumar Patodi was a brilliant Indian mathematicians who made, during his short life, fundamental contributions to the analytic proof of the index theorem and to the study of differential geometric invariants of manifolds. This set of collected papers edited by Prof M Atiyah and Prof Narasimhan includes his path-breaking papers on the McKean-Singer conjecture and the analytic proof of Riemann-Roch-Hirzebruch theorem for Kähler manifolds. It also contains his celebrated joint papers on the index theorem and the Atiyah-Patodi-Singer invariant.

The book gives a systematic exposition of the diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered include: constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, and Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. The text will be a valuable source for both the graduate student and researcher in mathematics and theoretical physics.

This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'. Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd, and a differential operator on an interval. Throughout, attention is paid to the history of the subject and earlier references are provided accordingly. The book is aimed at experts in index theory as well as newcomers to the field.