Lectures

Combinatorial properties of subspace arrangements
- Abstract: In this talk, we describe combinatorial properties of subspace arrangements in terms of corresponding simplicial complexes.

(Factorial) Schur functions and weighted Grassmannians
Abstract: It is a well-known fact that the so-call Schur functions that form a basis of the algebra of symmetric functions represent the Schubert classes of the cohomology of Grassmannians. After a brief introduction around this fact, I will explain how to generalize this picture to the case of weighted Grassmannians.

Small covers over polytopes with a few facets
- Abstract: We classify small covers over an n-polytope with n+3 facets. This is a joint work with Suyoung Choi.

Some properties of the permutation tableaux of type B
- Abstract: Corteel, Josuat-Verges and Kim introduced some q-Eulerian polynomials for symmetry of crossings of singed permutations. They proved the symmetry using the pignose diagram. We give an algorithm to prove the polynomial using the permutation tableaux of type B. Morover, we comment some interesting results for the permutation tableaux of type B.

Strong cohomological rigidity of quasitoric manifolds
Abstract: Any cohomology ring isomorphism between two quasitoric manifolds with second Betti number 2 is realizable by a homeomorphism. In particular, for non-singular complete toric varieties with second Betti number 2, that is, $2$-stage generalized Bott manifolds, any cohomology ring isomorphism between them is realizable by a diffeomorphism. This is a joint work with Suyoung Choi.

T^2-cobordism of quasitoric 4-manifolds.
- Abstract: We show the T^2-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the T^2-cobordism classes of CP^2 . We construct nice oriented T^2 manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.

Computing signed a-polynomials of complete multipartite graphs
Abstract: Recently, Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. In this talk, we introduce a signed a-polynomial which is a generalization of the signed a-number. The signed a-polynomial of a finite simple graph G is related to the Poincaré polynomial of real toric manifolds associated to the graph associahedron P_B(G) which is the nestohedron as the Minkowski sum of simplices obtained from connected induced subgraphs of G. We give the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. This is a joint work with Seunghyun Seo.

M(J)-manifolds and its rigidity problem.
Recently, Bahri, Bendersky, Cohen and Gitler introduced a new construction of quasitoric manifolds, say M(J)-manifolds. From this construction, we can produce infinite family of quasitoric manifolds from the given one. Suppose we have (equivariant) homeomorphic or equivalent two quasitoric manifolds M and N, then are M(J) and N(J) (equivariant) homeomorphic or equivalent? Here I answer to that question.