Victor G. Kac Books

This revised edition delves into Kac-Moody algebras, a specialized area of infinite-dimensional Lie algebras, and their representations. It draws on extensive teaching experience from courses at MIT and Paris, making it an ideal resource for graduate-level studies. The book offers a comprehensive exploration of the subject, providing valuable insights for students and researchers in the field.

Simply put, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics. This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. It is based on lectures and seminars given by MIT Professor Kac over the last few years at MIT.

Based on the third International Conference on Symmetries, Differential Equations and Applications (SDEA-III), this proceedings volume highlights recent important advances and trends in the applications of Lie groups, including a broad area of topics in interdisciplinary studies, ranging from mathematical physics to financial mathematics. The selected and peer-reviewed contributions gathered here cover Lie theory and symmetry methods in differential equations, Lie algebras and Lie pseudogroups, super-symmetry and super-integrability, representation theory of Lie algebras, classification problems, conservation laws, and geometrical methods. The SDEA III, held in honour of the Centenary of Noether’s Theorem, proven by the prominent German mathematician Emmy Noether, at Istanbul Technical University in August 2017 provided a productive forum for academic researchers, both junior and senior, and students to discuss and share the latest developments in the theory andapplications of Lie symmetry groups. This work has an interdisciplinary appeal and will be a valuable read for researchers in mathematics, mechanics, physics, engineering, medicine and finance .

This volume honors the memory of the esteemed American mathematician Bertram Kostant and contains 19 invited papers from leading experts in Lie theory, representation theory, algebra, geometry, and mathematical physics. Kostant's groundbreaking contributions have significantly advanced these fields, fostering new insights and research areas. The collection includes the only published articles detailing recent important results from the contributors, complete with proofs. Key topics addressed in the volume encompass Poisson structures and potentials, vertex algebras, modular irreducible representations of semisimple Lie algebras, and asymptotic Hecke algebras. Additional discussions cover tensor categories and quantum groups, nil-Hecke algebras and Whittaker D-modules, Toeplitz operators, Kashiwara crystals, and characters of highest weight modules. The volume also explores alcove polytopes, representation theory of quantized Gieseker varieties, generalized Bruhat cells and integrable systems, almost characters, Verlinde formulas, the Dirac operator and equivariant index, modality of representations, and the geometry of θ-groups, as well as distributions on homogeneous spaces and the reduction of orthogonal representations. This rich compilation reflects the depth and breadth of contemporary mathematical research inspired by Kostant's legacy.