강연영상 8 페이지 > 서울대학교 수리과학부

Concordance is a relation which classifies knots in 3-space via surfaces in 4-space, and it is closely related with low dimensional topology. Satellite operators are one of the main tools in the study of knot concordance, and it has been widely used to reveal new structures of knot concordance. In this talk, I will explain interplay between concordance and low dimensional topology, and discuss recent developments on satellite operators. The talk is based on joint work with Jae Choon Cha.

.

.

작용수대수에서 순서구조가 중요한 역할을 한다. C*-대수의 시작이라 할 수 있는 Gelfand-Naimark-Segal 표현정리는 양선형범함수로부터 *-준동형을 만들어내는데,
그 표현정리 이후 여러 가지 종류의 양사상에 대한 연구가 이루어졌다. 최근 활발하게 연구되고 있는 양자정보이론의 핵심 개념은 꼬임(enatnglement)인데, 선형사상과 텐서곱의 쌍대를 이용하면 이를 적절한 행렬대수 사이의 양사상들의 쌍대 개념으로 이해할 수 있다.이번 강의에서는 여러가지 양사상들 사이의 포함관계를 비롯하여 최근 연구자들의 관심을 끌고 있는 몇몇 문제와 관련된 경험을 소개한다.

※ 강연 뒷부분이 녹화되지 않았습니다. 

 A symplectic manifold is a space with a global structure on which Hamiltonian equations are defined. A classical result by Darboux says that every symplectic manifold locally looks standard,

so it has been interesting to study global properties of symplectic manifolds.

Since Gromov invented his famous theory of J-holomorphic curves in 1985, symplectic rigidity phenomena have been found in many different ways. In this talk, we explore it in terms of the symplectic mapping class groups and entropies.

 

It is a fundamental problem in algebraic geometry to study equations defining algebraic curves. In 1984, Mark Green formulated a famous conjecture on equations defining canonical curves and their syzygies. In early 2000's, Claire Voisin made a major breakthrough by proving that Green's conjecture holds for general curves. A few years ago, Aprodu-Farkas-Papadima-Raicu-Weyman gave a new proof of Voisin's theorem by studying equations defining tangent developable surfaces, and recently, I obtained a simple geometric proof of their result using equations defining secant varieties. In this talk, I first review the geometry of algebraic curves, and then, I explain the main ideas underlying the recent work on syzygies of algebraic curves and their tangent and secant varieties.

.

.

.

.