Dr. Holmfridur Sigridar Hannesdottir is a member of the School of Natural Sciences at the Institute for Advanced Study in Princeton, NJ. She received a Ph.D. in physics from Harvard University, after completing a B.Sc. in physics from the University of Iceland. Her work focuses on theoretical aspects of quantum field theories, including the study of infrared divergences in gauge theories and constraining the analytic structure of Feynman integrals.Dr. Sebastian Mizera is a long-term member at the Institute for Advanced Study in Princeton, NJ. He graduated from the University of Cambridge with a bachelor's degree in natural sciences and a master's in mathematics. He obtained a Ph.D. in theoretical physics from the Perimeter Institute and the University of Waterloo in Canada. Dr. Mizera's work focuses on the interconnections between scattering amplitudes in quantum field theories and the mathematics of algebraic geometry and topology.

• 1. Introduction.-  2. Unitarity implies anomalous thresholds.-  3. Primer on the analytic S-matrix 4.-  Singularities as classical saddle points.-  5. Branch cut deformations.-  6. Glimpse at generalized dispersion relations.-  7. Fluctuations around classical saddle points.-  8. Conclusion Appendix. Review of Schwinger parametrization.

Dr. Holmfridur Sigridar Hannesdottir is a member of the School of Natural Sciences at the Institute for Advanced Study in Princeton, NJ. She received a Ph.D. in physics from Harvard University, after completing a B.Sc. in physics from the University of Iceland. Her work focuses on theoretical aspects of quantum field theories, including the study of infrared divergences in gauge theories and constraining the analytic structure of Feynman integrals.Dr. Sebastian Mizera is a long-term member at the Institute for Advanced Study in Princeton, NJ. He graduated from the University of Cambridge with a bachelor's degree in natural sciences and a master's in mathematics. He obtained a Ph.D. in theoretical physics from the Perimeter Institute and the University of Waterloo in Canada. Dr. Mizera's work focuses on the interconnections between scattering amplitudes in quantum field theories and the mathematics of algebraic geometry and topology.

• 1. Introduction.-  2. Unitarity implies anomalous thresholds.-  3. Primer on the analytic S-matrix 4.-  Singularities as classical saddle points.-  5. Branch cut deformations.-  6. Glimpse at generalized dispersion relations.-  7. Fluctuations around classical saddle points.-  8. Conclusion Appendix. Review of Schwinger parametrization.

Dr. Holmfridur Sigridar Hannesdottir is a member of the School of Natural Sciences at the Institute for Advanced Study in Princeton, NJ. She received a Ph.D. in physics from Harvard University, after completing a B.Sc. in physics from the University of Iceland. Her work focuses on theoretical aspects of quantum field theories, including the study of infrared divergences in gauge theories and constraining the analytic structure of Feynman integrals.Dr. Sebastian Mizera is a long-term member at the Institute for Advanced Study in Princeton, NJ. He graduated from the University of Cambridge with a bachelor's degree in natural sciences and a master's in mathematics. He obtained a Ph.D. in theoretical physics from the Perimeter Institute and the University of Waterloo in Canada. Dr. Mizera's work focuses on the interconnections between scattering amplitudes in quantum field theories and the mathematics of algebraic geometry and topology.

• 1. Introduction.-  2. Unitarity implies anomalous thresholds.-  3. Primer on the analytic S-matrix 4.-  Singularities as classical saddle points.-  5. Branch cut deformations.-  6. Glimpse at generalized dispersion relations.-  7. Fluctuations around classical saddle points.-  8. Conclusion Appendix. Review of Schwinger parametrization.