Dec, 19(T) & 21(R), Sam Bardwell-Evans (IBS-CGP), Topic: Symplectic topology.

Abstract: 

Part One: What and why

Moduli spaces of pseudoholomorphic curves and discs with Lagrangian boundary conditions are fundamental objects in symplectic geometry, and especially in Lagrangian Floer theory. These spaces are in general highly singular and must be regularized in order to be useful. The Kuranishi structure method of Fukaya-Oh-Ohta-Ono provides such a regularization, but these structures are usually themselves highly complicated and difficult to work with. In upcoming work, I will provide a construction of a comparatively simple Kuranishi structure on the moduli space of pseudoholomorphic discs called a global Kuranishi chart. In this first talk, I will define global Kuranishi charts and discuss their use.

Part Two: How

In the second talk, I will provide a sketch of the upcoming construction and discuss some of the technical obstacles.

Nov 27(M), Dongsoo Lee (KAIST), Topic: Knot Floer homology.

Abstract: Heegaard Floer homology is a 3-manifold invariant introduced by Ozsv ́ath-Szab ́o. Knot Floer homology is a version of Heegaard Floer homology applied to knots in 3-manifolds, which was introduced independently by Ozsv ́ath-Szab ́o and Rasmussen. In this talk, I will give a basic introduction to Heegaard Floer homology, knot Floer homology, and some knot concordance invariants derived from the Floer homology.

Oct 23(M)  & 24(T), Hongtaek Jung (SNU), Topic: Teichmuller spaces.

Abstract: This is a two days crash course on Teichmuller spaces. We discuss classical theory of Teichmuller spaces including quasi-conformal mappings and quadratic differentials. We define the Teichmuller metric and Weil-Petersson metric and investigate their basic properties. Then, we discuss compactifications of Teichmuller spaces. We also explore the relationship between Teichmuller spaces and some combinatorial objects, such as curve complexes and Pants complexes. 

Sep 18(M)  & 25(M), Wonbo Jeong (SNU), Topic: Cluster categories.

Abstract: A cluster category appeared first as a categorification of cluster algebra, but it is an interesting theory itself. We recall details about cluster categories associated to acyclic quivers following a foundational work of Buan-Marsh-Reineke-Reiten-Todorov. 

Abstract: The first constructions of cluster categories was generalized in many ways. Especially, we are interested in the work of Iyama-Yang. They introduced a Calabi-Yau triple which induces a generalized cluster category. In this talk, we explain the results in Iyama-Yang and give an example of Calabi-Yau triple from symplectic geometry. This symplectic construction is a joint work with Hanwool Bae and Jongmyeong Kim.  

Aug 31(M), Philsang Yoo(SNU), Topic: Mathematical physics

Abstract: In this talk, which is primarily expository in nature, our main goal is to explain basic ideas of derived geometry. Additionally, we aim to discuss how the ideas can be used to describe certain aspects of classical and quantum field theory. Time permitting, we will discuss how one can obtain a new relationship between geometric representation theory and quantum field theory.

Jul 17(M), Kyongmin Rho(SNU), Topic: Symplectic Topology

Abstract: "Pair-of-pants surface (A-side) <-> Degenerate cusp singularity xyz=0 (B-side)". We explain an explicit correspondence between Lagrangians on the A-side and Cohen-Macaulay modules on the B-side.Then we develop a new concept of degenerate vector bundles to provide a geometric construction of Cohen-Macaulay modules over degenerate cusp singularities. This is based on joint works with Cheol-Hyun Cho, Wonbo Jeong, and Kyoungmo Kim.

Jun 19(M), Joontae Kim(Sogang University), Topic: Symplectic Topology

Abstract: The study of mapping class groups in symplectic manifolds is a central topic of modern symplectic topology. In spite of spectacular advances in symplectic Torelli classes, it is not known yet that there are such classes that are quantitatively exotic. In this talk, we show that there is a symplectic K3 surface which admits Torelli classes with positive topological entropy. This is joint work with Myeonggi Kwon.


May 08(M), Yuki Hirano(TUAT), Topic: Algebraic geometry

Abstract: For a generic quasi-symmetric representation X of a reductive group G,  the GIT quotient stack [X(L)//G] for a generic polarization L  is a (stacky) crepant resolution of the affine quotient X/G, and Halpern-Leistner and Sam proved that  the GIT quotients [X(L)//G] are all derived equivalent, which proved Bondal-Orlov conjecture for [X(L)//G]. One of the key ingredient of Halpern-Leistner--Sam's work is a magic window, which is shown to be equivalent to the derived category of the GIT quotient [X(L)//G]. A magic window is also equivalent to the derived category of a noncommutative crepant resolution (NCCR) of X/G, which is an endomorphism algebra End(M) of some module M over X/G. In this talk, we explain that the modules giving NCCR of X/G are related by certain operations called exchanges, and in the case when G is a torus, the modules are related by Iyama--Wemyss mutations. If time permits, I will explain that certain autoequivalences of a Calabi-Yau hypersurface correspond to the compositions of Iyama--Wemyss mutations via matrix factorizations. 

Apr 24(M), Ruben Louis(Université de Lorraine), Topic: Algebraic geometry

Abstract: We show that there is an equivalence of categories between Lie-Rinehart algebras over a commu tative algebra O and homotopy equivalence classes of negatively graded acyclic Lie infinity -algebroids over O. Therefore, this result makes sense of the universal Lie infinity -algebroid of every singular foliation, without any additional assumption, and for Androulidakis-Zambon singular Lie algebroids. This extends to a purely algebraic setting the construction of the universal Q -manifold of a locally real analytic singular foliation of C. Laurent-Gengoux, S. Lavau & T. Strobl . Also, this allows to associate to any affine variety a universal Lie infinity-algebroid of the Lie-Rinehart algebra of its vector fields. Several explicit examples and are given.

Apr 3(M), Seokbong Seol (KIAS), Topic: Algebra

Abstract: This is a learning seminar on categories/infinity algebra structures associated to the complex singularities. Some basic elements regarding DG manifolds will be covered also. The main reference will be [Lie-Rinehart algebras = acyclic Lie ∞-algebroids] by Camille Laurent-Gengoux and Ruben Louis.

Apr 10(M), Mar 27(M), Dongwook Choa(KIAS), Topic: Algebra


Abstract: This is a learning seminar on categories/infinity algebra structures associated to the complex singularities.  The main reference will be [Lie-Rinehart algebras = acyclic Lie ∞-algebroids] by Camille Laurent-Gengoux and Ruben Louis.

Abstract: This is a learning seminar on categories/infinity algebra structures associated to the complex singularities.  The main reference will be [Lie-Rinehart algebras = acyclic Lie ∞-algebroids] by Camille Laurent-Gengoux and Ruben Louis.