Handbook of Combinatorial Algebraic Geometry

Erik Insko received his Ph.D. in Mathematics in 2012 under the guidance of Julianna Tymoczko. He served on the faculty at Florida Gulf Coast University from 2012 to 2023 before joining Central College as a Professor of Mathematics and Computer Science. He enjoys mentoring student researchers and was awarded the Mid-Career Faculty Mentor Award by the Council on Undergraduate Research in the Mathematics and Computer Science Division in 2021. His research interests lie in Algebraic Combinatorics and Graph Theory. He has 26 publications indexed on MathSciNet.Martha Precup received her Ph.D. in Mathematics from the University of Notre Dame in 2013 under the supervision of Samuel Evens. She subsequently held postdoctoral fellowships at Baylor University and Northwestern University. Since 2018, she has been a faculty member at Washington University in St. Louis, where she is now an Associate Professor. Precup has an active and growing research program, with 19 publications listed on MathSciNet. She previously held a single–PI NSF research grant and currently holds an NSF CAREER award. She was recently awarded the 2026–2027 Ruth I. Michler Prize by the Association for Women in Mathematics.Edward Richmond received his Ph.D. in Mathematics in 2008 from the University of North Carolina at Chapel Hill under the direction of Prakash Belkale. He has held postdoctoral appointments at the University of Oregon and the University of British Columbia. Since 2014, he has been a faculty member at Oklahoma State University, where he is now a Professor and the Director of Graduate Studies. His research lies in algebraic combinatorics and Lie theory, with a focus on Schubert varieties, flag varieties, and related combinatorial and geometric objects. He has 23 publications listed on MathSciNet and his work has been supported by the NSA and the Simons Foundation.

• Part 1: Flag varieties and Schubert varieties 1. Introduction to the Cohomology of the Flag Variety Part 2: Subvarieties of the flag variety 2. Schubert Geometry and Combinatorics 3. Richardson varieties, projected Richardson varieties and positroid varieties 4. Torus orbit closures in the flag variety 5. Pattern avoidance and K-orbit closures Part 3: Hessenberg Varieties 6. An Introduction to Hessenberg Varieties 7. The cohomology rings of regular nilpotent Hessenberg varieties 8. Hessenberg varieties and algebraic combinatorics of hyperplane arrangements 9. Combinatorics and Hessenberg Varieties Part 4: Additional topics 10. Generalizations of the flag variety tied to the Macdonald-theoretic delta operators 11. Nil-Hecke rings and Schubert calculus 12. Coxeter groups and Billey–Postnikov decompositions