Schedule & Talk Information

김민훈 교수님 (이화여자대학교)  13:30-14:20

Title: Calegari homotopy 4-spheres are standard

Abstract: The smooth 4-dimensional Poincare conjecture which states that every smooth homotopy 4-sphere is diffeomorphic to the standard 4-sphere is a fundamental open problem in geometric topology. In the first part of the talk, I will give an overview of the classification results/problems of manifolds including the smooth 4-dimensional Poincare conjecture. In the second part of the talk, I will describe a construction of Calegari homotopy 4-spheres which are potential counterexamples to the smooth 4-dimensional Poincare conjecture constructed by Danny Calegari in 2009. I will also discuss a proof that all Calegari homotopy 4-spheres satisfy the smooth 4-dimensional Poincare conjecture, which is based on joint work with Jae Choon Cha.

송종백 교수님 (부산대학교) 14:40-15:30

Title: Integral cohomology ring of toric surfaces

Abstract: It is well-known that the rational cohomology ring of a toric variety with orbifold singularities, say toric orbifolds, behaves similarly to the integral cohomology ring of smooth toric varieties. However, the information about the integral cohomology ring of a (underlying topological spaces of) toric orbifold or a singular toric variety in general is somewhat restrictive compared to the smooth case. In this talk, we consider toric surfaces, namely the toric varieties of complex dimensions 2, which have at worst orbifold singularities. The main result determines the integral cohomology ring structure of toric surfaces in terms of  “bases” and “relations”, which can be easily read off from the underlying combinatorial data. This is a joint work with Xin Fu (BIMSA) and Tseleung So (Western University).

김영락 교수님 (부산대학교) 15:50-16:40

Title : Determinant vs. Permanent

Abstract : There are two elementary concepts associated to a square matrix. The first one is the determinant, defined as the signed sum of the product of entries over the permutations, and the other one is the permanent which defined as the unsigned sum. They look quite similar in their structure, however, they behave apart in so many places. In this talk, we first review some important problems comparing these two notions including a famous conjecture of Valiant. After that, we compare their behavior with a viewpoint of the tensor rank, by considering them as an alternating and a symmetric n-linear maps. In particular, we introduce some recent observations on the tensor rank of these tensors, as an application of Koszul flattening methods developed by Landsberg-Ottaviani and Hauenstein-Oedding-Ottaviani-Sommese. If time permits, we will discuss some open problems on the geometry of determinantal and permanental hypersurfaces. A part of this talk is based on a joint work with Jong In Han and Jeong-Hoon Ju.