Mini-course: Convex Geometric Methods in Commutative Algebra
Thời gian: 09:30 đến 11:30 ngày 03/07/2026, 09:30 đến 11:30 ngày 10/07/2026, 09:30 đến 11:30 ngày 17/07/2026, 09:30 đến 11:30 ngày 24/07/2026,
Địa điểm: Vietnam Institute for Advanced Study in Mathematics (VIASM), 161 Huynh Thuc Khang Street, Lang Ward, Hanoi.
- Lecture 1: July 3, 2026
- Lecture 2: July 10, 2026
- Lecture 3: July 17, 2026
- Lecture 4: July 24, 2026
Lecturer: Prof. Ha Huy Tai, Tulane University, USA.
Abstract
This lecture series introduces convex-geometric methods in commutative algebra, with an emphasis on how asymptotic algebraic questions can be translated into problems about semigroups, cones, and convex bodies. Starting from concrete examples of monomial ideals and Newton polyhedra, the lectures will develop the role of valuations, value semigroups, and Newton–Okounkov bodies in measuring algebraic growth. We will see how classical invariants such as Hilbert–Samuel multiplicity, integral closure, analytic spread, and asymptotic behavior of graded families of ideals can often be understood through volumes, surface areas, and vertices of naturally associated convex regions. The series will also discuss applications to containment problems, including symbolic powers and resurgence, where algebraic containment questions become questions about inclusion and scaling of convex bodies. The lectures are designed to be accessible to graduate students and young researchers, while also highlighting current research directions at the interface of commutative algebra, algebraic geometry, and convex geometry. Each session will consist of a one-hour formal lecture followed by an informal discussion, providing participants with opportunities to ask questions, work through examples, and explore possible research problems.
The lectures are based on the theory of Newton–Okounkov bodies and value semigroups developed by Okounkov, Lazarsfeld–Mustaţă, and Kaveh–Khovanskii, together with related ideas from the study of Newton polyhedra, integral closure, Rees valuations, multiplicities, and graded families of ideals. Rather than giving a comprehensive survey, the lectures will focus on the parts of this theory that are especially useful in commutative algebra: how valuations turn algebraic growth into semigroup-counting problems, and how convex bodies and their volumes encode asymptotic invariants.
Mode of Participation: Hybrid (online participation is available only for participants outside the Hanoi area)
Target participants: This series is primarily aimed at final-year undergraduate students preparing for graduate studies, early-career Ph.D. students, and young researchers (within a few years of completing their Ph.D.). Participants should have completed coursework in General Algebra and possess basic knowledge of Commutative Algebra.
Language: English and Vietnamese
Sponsors
- National Program for the Development of Mathematics in the period 2021-2030 (NPDM);
- Vietnam Innovation of Education Foundation (VIEF).
Contact
Ms. Truong Thuy Linh
Email: truongthuylinh@viasm.edu.vn