Dec, 19(T) & 21(R), Sam Bardwell-Evans (IBS-CGP), Topic: Symplectic topology.
Dec, 19(T) & 21(R), Sam Bardwell-Evans (IBS-CGP), Topic: Symplectic topology.
Sam will be present two talks explaining his research on a global Kuranish chart for pseudoholomorphic discs. The title and abstract will be updated soon.
Title: Global Kuranishi Charts for Moduli Spaces of Pseudoholomorphic Discs.
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.
Title: A brief introduction to Heegaard 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.
Time and Location: Nov 27 (M), 14:00-16:00, Room 1424.
Notes: Notes.
Title: An overview on 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.
Time and location: Oct 23 (M), 14:00-16:00, Room 1424, and Oct 24 (T), 14:00-16:00, Room 1423.
Notes: Notes
Title: Introduction to Cluster categories (Sep 18)
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.
Time and location: Sep 18(M), 13:30 - 15:00, Room 1423.
Title: Generalized cluster categories (Sep 25)
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.
Time and location: Sep 25(M), 13:30 - 15:00, Room 1423.
Notes: Sep 18, Sep 25.
Title: Derived Geometry and Field Theory
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.
Time and location: Sep 18(M), 14:00 - 16:00, Room 8101.
Title: Homological Mirror Symmetry and Geometry of Degenerate Cusp Singularities
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.
Time and location: Jul 17(M), 14:00 - 16:00, Room 1424.
Title: Symplectic Torelli classes of positive entropy
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.
Time and location: Jun 19(M), 14:00 - 16:00, Room 1424.
Title: Mutations of noncommutative crepant resolutions in geometric invariant theory.
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.
Time and location: May 08(M), 14:00 - 16:00, via Zoom
Title: On Lie-Rinehart algebras and their universal Lie infinity-algebroids
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.
Time and location: Apr 24(M), 17:00 - 19:00, via Zoom
Title: [infinity learning seminar] Lie-Rinehart algebras and matrix factorizations
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.
Time and location: Apr 3(M), 14:00 - 16:00, Room 1423
Title: [infinity learning seminar] Lie-Rinehart algebra and Hochschild cohomology (Apr 10th)
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.
Time and location: Apr 10(M), 14:00 - 16:00, Room 1423
Title: [infinity learning seminar] Lie-Rinehart algebras and matrix factorizations (Mar 27th)
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.
Time and location: Mar 27th(M), 14:00 - 16:00, Room 1423