2026/2027
Контактная топология узлов: от дифференциальных градуированных алгебр Чеканова–Элиашберга к полному инварианту узлов
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Кредиты:
6
Контактные часы:
60
Course Syllabus
Abstract
"A knot is a closed curve in $\mathbb{R}^3$ without self-intersections; two knots are equivalent if one can be smoothly deformed into the other. The central problem of knot theory is distinguishing knots: how can one prove that a given knot cannot be deformed into another? For this one uses invariants -- quantities or algebraic structures assigned to a knot that do not change under deformation. Many invariants are known (the Alexander polynomial, the Jones polynomial, the knot group on its own), but most are incomplete: they fail to distinguish certain non-equivalent knots. A classical complete invariant -- the knot group together with the peripheral subgroup -- is a purely topological construction. In 2018, Ekholm, Ng, and Shende discovered a complete invariant of a very different nature, coming from contact geometry.
Contact geometry has its origins in geometric optics and classical mechanics and studies certain natural geometric structures on odd-dimensional manifolds. Over the last decades its methods have found unexpected applications to topology. In particular, to a knot $K \subset \mathbb{R}^3$ one canonically associates a two-dimensional surface (a torus) inside a certain five-dimensional contact manifold; choosing a point off the knot determines an additional sphere in the same manifold. Counting holomorphic curves with boundaries on these submanifolds produces an algebraic structure -- the KCH-triple equipped with a product map -- from which the knot can be completely recovered.
No prior knowledge of knot theory or contact geometry is assumed; all necessary background is developed from scratch. The course covers: the knot group and the peripheral subgroup; fundamentals of symplectic and contact geometry; the Chekanov–Eliashberg differential graded algebra and its augmentations; augmentation varieties and the $A$-polynomial; knot contact homology; the cord algebra; and enhanced knot contact homology. The proof of the completeness theorem is the final goal of the course."