Shashikant B. Mulay is a Professor at the Department of Mathematics, University of Tennessee. Mulay earned his Ph.D. in Mathematics in 1982 from Purdue University under the supervision of Professor Shreeram Abhyankar. His research interests are in Algebra. He has held visiting positions at Purdue University, University of Kentucky, POSTECH (South Korea), MSRI Berkeley and the Max Plank Institute (Germany).

Mark A. Shattuck is a Researcher at the Institute for Computational Science, Ton Duc Thang University in Vietnam (2017-present) and an Adjunct Assistant Professor in the Department of Mathematics, University of Tennessee (2013-present). Shattuck earned his Ph.D. in Mathematics in 2005 from the University of Tennessee under the supervision of Professor Carl Wagner. His research interests are primarily in Enumerative and Algebraic Combinatorics. He has been a Visiting Researcher at the University of Haifa (Israel) in 2011-2012 and in 2015.

Provides a coherent and systematic account of the authors' important contributions to understanding strongly interacting fermions in fractional quantum Hall states

Is closely related to the research for which Haldane was awarded the Nobel Prize recently and Laughlin many years ago

Sheds light on the amazing phenomenon of FQHE (fractional quantum Hall effect) and correlated electrons in general

1. (Chapter 1 Title: Introduction to Quantum Hall States.)Historical background of many-body interactions in the context of integral and fractional quantumHall effect. Review of experimentally observed families of IQL states and their interpretations2. (Chapter 2 Title: The composite Fermion hierarchy and justification of the CF approach.)Jain's mean field composite Fermion picture, Laughlin-Jastrow type correlations. The workof the University of Tennessee group in condensed matter physics justifying MFCF undercertain conditions on the interaction energy. The concept of ¿Effective CF angular momentum¿associated to the lowest CF Landau level. Partially filled shells and Quasi-electons. Haldane'sheirarchy of Laughlin correlated daughter-states and Jain's sequence of IQL states3. (Chapter 3 Title: Correlation diagrams and the Algebra of correlation functions.)Justification of the CF approach. Establishing (with rigorous proofs) conditions under whichJain's elegant CF approach correctly predicts the angular momentum multiplets. Correlationdiagram approach to Moore-Read states provides a new insight and a more general [...] symmetric correlation functions for N Fermions in an IQL state for any filing factor lessthan ¿ , obtained from balanced correlation diagrams. Detailed treatment of the algebraiccorrelation functions and diagrams for small values of N; overlap with the numerical workon diagonalization.4. (Chapter 4 Title: Invariant-theoretic essentials.)A self-contained brief introduction to the mathematical theory of invariants of binary formsin context of correlation diagrams and their associated correlation functions. Semi-invariantsof binary forms and dimension counting theorems. The relationship between the semi-invariantsand angular momentum multiplets in presence of quasi-electrons.5. (Chapter 5 Title: Constructions of correlation diagrams and their correlation functions.)Theorems proving existence of nontrivial fully symmetric correlation functions fora class of correlation diagrams. Symmetry groups of the balanced correlation diagramsand their utility from the computational point of view.6. (Chapter 6 Title: Trial wave functions for systems of Fermions in IQL state with generalfilling factor $n / (2 p n + - 1) < 1/2$.) Concrete constructions of fully symmetric correlationfunctions and detailed examination of their special algebraic features.7. (Chapter 7 Title: Correlation functions for some Configurations with quasi-electrons andrelated open questions. ) Concrete constructions of fully symmetric correlation functions forconfigurations of N interacting Fermions some of which are quasi-electrons, for a familytotal angular momenta. Various open mathematical questions directly related to suchconstructions.