The 50-year history of weak interaction since Fermi's pro­ posal of this coupling has been marked with striking direct inter­ plays between experimental results and theoretical understanding, e.g. the discoveries of neutrinos, parity violation, and CP vio­ lation. The recent discoveries of the quark hierarchy, the charm and the beauty, and the intermediate vector bosons W± and ZO have truly made a splendid page in the history of particle physics. It is the purpose of this conference to discuss the questions of quark and lepton generations and mixing, their relations to CP violation, and to ask the questions about what are inside the quarks and the leptons in view of the present and future exper­ imental situation. Dr. Ling-Lie Chau Brookhaven National Laboratory Upton, New York vii ACKNOWLEDGMENTS I would like to thank all the Advisory and Organizing Committee members for their advice and suggestions during the organization of the conference. The running of the conference could not have gone so smoothly without the help of many participants, I sincerely thank: L. Becker, F.J. Botella, S. Gentile, P. Le Comte, M.E. Machacek, L. Lanceri, W.M. Morse, F.J. Olness, Y.-X. Pham, G. Poulard, K.J. Sliwa, and J.N. Webb.

The 50-year history of weak interaction since Fermi''s pro­ posal of this coupling has been marked with striking direct inter­ plays between experimental results and theoretical understanding, e.g. the discoveries of neutrinos, parity violation, and CP vio­ lation. The recent discoveries of the quark hierarchy, the charm and the beauty, and the intermediate vector bosons W± and ZO have truly made a splendid page in the history of particle physics. It is the purpose of this conference to discuss the questions of quark and lepton generations and mixing, their relations to CP violation, and to ask the questions about what are inside the quarks and the leptons in view of the present and future exper­ imental situation. Dr. Ling-Lie Chau Brookhaven National Laboratory Upton, New York vii ACKNOWLEDGMENTS I would like to thank all the Advisory and Organizing Committee members for their advice and suggestions during the organization of the conference. The running of the conference could not have gone so smoothly without the help of many participants, I sincerely thank: L. Becker, F.J. Botella, S. Gentile, P. Le Comte, M.E. Machacek, L. Lanceri, W.M. Morse, F.J. Olness, Y.-X. Pham, G. Poulard, K.J. Sliwa, and J.N. Webb.

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit­ ful again. Many of the leading specialists of both fields became involved in this devel­ opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im­ portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me­ chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth­ while.

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit­ ful again. Many of the leading specialists of both fields became involved in this devel­ opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im­ portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me­ chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth­ while.

Stanley Mandelstam (1928–2016) was one of the most influential and respected particle theorists. Coming as a young chemical engineer from South Africa to study theoretical physics in England, he quickly became a leading physicist in his field. With his deep understanding of quantum field theory, he pioneered the development of the analytic S-matrix theory as well as the path-dependent formulations for quantum gauge theories and for quantum general relativity. They are being actively used for the electroweak theory and having their imprints in lattice gauge theory and loop quantum gravity. Also he elucidated the mechanisms for quark confinement in quantum chromodynamics, constructed non-perturbative bosonization methods in 1+1 dimensions, and proved the perturbative finiteness and β=0 of N=4 supersymmetric Yang–Mills theory. His work also led to the discovery of dual resonance models, which in turn became superstring theory. He was a leader in these developments, devoting much of his later years to the proof that the theory is perturbatively finite so it can be considered as a contender for the theory of quantum gravity.He was also a very modest and friendly man, impressing everyone with his sharp intellect as well as his humanity. This volume contains essays written by many of his friends and students, including both detailed reports on his scientific achievements as well as personal reminiscences. Also collected in the volume are some selected reprints of Mandelstam's early seminal papers and abstracts of selected papers representing the full spectrum of his contributions.

The 50-year history of weak interaction since Fermi's pro­ posal of this coupling has been marked with striking direct inter­ plays between experimental results and theoretical understanding, e.g. the discoveries of neutrinos, parity violation, and CP vio­ lation. The recent discoveries of the quark hierarchy, the charm and the beauty, and the intermediate vector bosons W± and ZO have truly made a splendid page in the history of particle physics. It is the purpose of this conference to discuss the questions of quark and lepton generations and mixing, their relations to CP violation, and to ask the questions about what are inside the quarks and the leptons in view of the present and future exper­ imental situation. Dr. Ling-Lie Chau Brookhaven National Laboratory Upton, New York vii ACKNOWLEDGMENTS I would like to thank all the Advisory and Organizing Committee members for their advice and suggestions during the organization of the conference. The running of the conference could not have gone so smoothly without the help of many participants, I sincerely thank: L. Becker, F.J. Botella, S. Gentile, P. Le Comte, M.E. Machacek, L. Lanceri, W.M. Morse, F.J. Olness, Y.-X. Pham, G. Poulard, K.J. Sliwa, and J.N. Webb.

The 50-year history of weak interaction since Fermi''s pro­ posal of this coupling has been marked with striking direct inter­ plays between experimental results and theoretical understanding, e.g. the discoveries of neutrinos, parity violation, and CP vio­ lation. The recent discoveries of the quark hierarchy, the charm and the beauty, and the intermediate vector bosons W± and ZO have truly made a splendid page in the history of particle physics. It is the purpose of this conference to discuss the questions of quark and lepton generations and mixing, their relations to CP violation, and to ask the questions about what are inside the quarks and the leptons in view of the present and future exper­ imental situation. Dr. Ling-Lie Chau Brookhaven National Laboratory Upton, New York vii ACKNOWLEDGMENTS I would like to thank all the Advisory and Organizing Committee members for their advice and suggestions during the organization of the conference. The running of the conference could not have gone so smoothly without the help of many participants, I sincerely thank: L. Becker, F.J. Botella, S. Gentile, P. Le Comte, M.E. Machacek, L. Lanceri, W.M. Morse, F.J. Olness, Y.-X. Pham, G. Poulard, K.J. Sliwa, and J.N. Webb.

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit­ ful again. Many of the leading specialists of both fields became involved in this devel­ opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im­ portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me­ chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth­ while.

After several decades of reduced contact, the interaction between physicists and mathematicians in the front-line research of both fields recently became deep and fruit­ ful again. Many of the leading specialists of both fields became involved in this devel­ opment. This process even led to the discovery of previously unsuspected connections between various subfields of physics and mathematics. In mathematics this concerns in particular knots von Neumann algebras, Kac-Moody algebras, integrable non-linear partial differential equations, and differential geometry in low dimensions, most im­ portantly in three and four dimensional spaces. In physics it concerns gravity, string theory, integrable classical and quantum field theories, solitons and the statistical me­ chanics of surfaces. New discoveries in these fields are made at a rapid pace. This conference brought together active researchers in these areas, reporting their results and discussing with other participants to further develop thoughts in future new directions. The conference was attended by SO participants from 15 nations. These proceedings document the program and the talks at the conference. This conference was preceded by a two-week summer school. Ten lecturers gave extended lectures on related topics. The proceedings of the school will also be published in the NATO-AS[ volume by Plenum. The Editors vii ACKNOWLEDGMENTS We would like to thank the many people who have made the conference a success. Furthermore, ·we appreciate the excellent talks. The active participation of everyone present made the conference lively and stimulating. All of this made our efforts worth­ while.

Stanley Mandelstam (1928–2016) was one of the most influential and respected particle theorists. Coming as a young chemical engineer from South Africa to study theoretical physics in England, he quickly became a leading physicist in his field. With his deep understanding of quantum field theory, he pioneered the development of the analytic S-matrix theory as well as the path-dependent formulations for quantum gauge theories and for quantum general relativity. They are being actively used for the electroweak theory and having their imprints in lattice gauge theory and loop quantum gravity. Also he elucidated the mechanisms for quark confinement in quantum chromodynamics, constructed non-perturbative bosonization methods in 1+1 dimensions, and proved the perturbative finiteness and β=0 of N=4 supersymmetric Yang–Mills theory. His work also led to the discovery of dual resonance models, which in turn became superstring theory. He was a leader in these developments, devoting much of his later years to the proof that the theory is perturbatively finite so it can be considered as a contender for the theory of quantum gravity.He was also a very modest and friendly man, impressing everyone with his sharp intellect as well as his humanity. This volume contains essays written by many of his friends and students, including both detailed reports on his scientific achievements as well as personal reminiscences. Also collected in the volume are some selected reprints of Mandelstam's early seminal papers and abstracts of selected papers representing the full spectrum of his contributions.