Date  Speaker  Title  Reference 

04/19/19
Virtual fundamental classes are playing essential roles in modern enumerative geometry including GromovWitten and Stable pair invariants. In their paper, Kiem and Li introduced the notion of cosection and used it to construct a localized virtual fundamental class in algebraic geometry. It turns out it enjoys many properties as a usual virtual class including localization and wall crossing. In this talk, we will see their construction of localized virtual cycle. As a byproduct, we will see how to construct a reduced virtual fundamental class for GromovWitten invariants of K3 and abelian surfaces.

Sungwoo Nam  Cosection localization and reduced virtual fundamental class  Reference 
04/12/19
In this talk I’ll start from the basic of perverse sheaves and explain some basic examples. Then I’ll explain the structure theorem of hyperkahler manifold allowing Larangian fibration given by Junliang Shen and Qizhen Ying.

Lutian Zhao  The P=W Conjecture and Topology of Lagrangian fibrations  Reference 
04/05/19
The P=W conjecture stems from the attempt to understand the nonabelian Hodge correspondence and its related topology. In this talk, I’ll give a description of this conjecture and explain the basic idea: the weight filtration associated to the Betti moduli space should coincide with the perverse filtration associated to the Dolbeault moduli space. As a final application, I’ll explain how this conjecture gives rise to a calculation of BPS invariants.

Lutian Zhao  The P=W Conjecture  Reference I, II 
03/29/19
Compact hyperkäher manifolds are one of the building blocks of compact Ricciflat manifolds. It is also a target space for nonlinear sigma model which gives N=4 SCFT. In this talk, I’ll introduce motivations and definitions for compact hyperkähler manifolds. Then I’ll introduce examples and their basic properties. If time permits, I’ll describe a quadratic form on second cohomology group, called BeauvilleBogomolovFujiki form and use it to prove local torelli theorem for hyperkähler manifolds.

Sungwoo Nam  Rudiments of compact hyperkähler manifolds  Reference I, II 
03/08/19
In this talk, I’ll review Bridgeland’s definition of BPS structures. In some sense the KontsevichSoibelman wall crossing formula determines a RiemannHilbert problem. I’ll explain Bridgeland’s solution of this problem and see some examples of this problem.

Lutian Zhao  RiemannHilbert Problem for BPS Structure  Reference 
02/15/19
I’ll review the concept of the heart of a tstructure for K3 surfaces, which is part of the definition of Bridgeland stability condition on K3 surfaces. Then I’ll discuss how much geometric information it contains, especially its relation to derived equivalences.

Sungwoo Nam  Heart of tstructures for K3 surfaces  Reference I, II 
02/08/19
Which Chern class can be realized by slope semistable vector bundles? The Bogomolov’s inequality gives a necessary condition. In this talk, I’ll explain the proof of this inequality. As an application, I’ll do the construction of Bridgeland stability condition for a surface.

Lutian Zhao  The BogomolovGieseker Inequality  Reference I,II 
01/25/19
I’ll present one application of Bridgeland stability condition on a classcial problem in birational geometry. I’ll start by classical BrillNoether theory and Lazarsfeld’s result on a general curve in a K3 surface, and then I’ll describe its proof via stability condition on a K3 surface, using wall crossing argument.

Sungwoo Nam  BrillNoether from Wall Crossing  Reference 
Date  Speaker  Title  Reference 

12/14/18
I'll described period map for family of projective varieties and complete the proof of the theorem that CalabiYau integrable system is analytically completely integrable.

Sungwoo Nam  The Cubic Condition for Integrable Systems, III  Reference I, II 
11/29/18
Building on Matej's talk, I'll introduce the notion of CalabiYau integrable systems and explain how they connect to abelian, Lagrangian fibration(which is also complete integrable system), Matej was talking about last week. Along the way, I'll introduce some notions from Hodge theory such as intermediate Jacobian as a tool connecting two things.

Sungwoo Nam  The Cubic Condition for Integrable Systems, II  Reference 
11/08/18
This is the first in a 2part talk with Sungwoo on the paper https://arxiv.org/abs/alggeom/9408004 . I will focus on section 1 where the cubic condition is introduced to answer an interesting and natural question: Which families of abelian varieties have the structure of a completely integrable systems? It turns out the answer is equivalent to the existence of a field of cubics on the tangent bundle of the base. I will explain this result more precisely, and give an idea of the proof and how it will be used in part 2 of the talk by Sungwoo.

Matej Penciak  The Cubic Condition for Integrable Systems, I  Reference 
11/02/18  Sheldon Katz  Mirror Symmetry for Toric Varieties  Reference I, II 
10/19/18
I’ll give a description of the toric hypersurface by polytopes and produce a calculation of the cohomology. Then we’ll describe Batyrev’s construction of mirror manifold and try to prove the coincidence of Kahler moduli of original CalabiYau family and the complex moduli of mirror family.

Lutian Zhao  Batyrev's Construction, II  Reference I, II 
10/12/18
I’ll try to construct the mirror manifold out of Batyrev’s construction, assuming the knowledge from Joseph’s talk before.

Lutian Zhao  Batyrev's Construction, I  Reference 
10/05/18  Joseph Pruitt  Introduction to Toric Varieties  Reference 
09/28/18  Sheldon Katz  Organizational Meeting and An introduction to mirror symmetry  Reference 
Spring 2018
No seminar due to MSRI program Enumerative Geometry Beyond Numbers
Fall 2017
Date  Speaker  Title  Reference 

11/15/17  Sheldon Katz  GromovWitten Theory  
11/01/17  Hao Sun  GromovWitten theory, Hurwitz numbers, and Matrix models, II  Reference 
10/25/17  Hao Sun  GromovWitten theory, Hurwitz numbers, and Matrix models, I  Reference 
10/18/17  Sungwoo Nam  The Crepant Resolution Conjecture  Reference 
10/11/17  Yun Shi  Introduction to stable pair theory  Reference 
10/04/17  Lutian Zhao  Introduction in Topological String Theory on CalabiYau manifolds, III  Reference 
09/27/17  Sungwoo Nam  The local GromovWitten theory of curves  Reference I, II 
09/20/17  Lutian Zhao  Introduction in Topological String Theory on CalabiYau manifolds, II  Reference 
08/30/17  Lutian Zhao  Introduction in Topological String Theory on CalabiYau manifolds, I  Reference 
Spring 2017
Date  Speaker  Title  Reference 

07/12/17  Michel van Garrel (KIAS)  Rational curves in log K3 surfaces  Reference I, II 
07/05/17  Sungwoo Nam  Relations on moduli spaces of curves  Reference I, II 
06/28/17  Joseph Pruitt  Batyrev's relations in quantum cohomology  Reference 
06/21/17  Mi Young Jang  A Mathematical Theory of Quantum Sheaf Cohomology  Reference 
06/07/17  Lutian Zhao  GopakumarVafa invariants via vanishing cycles  Reference 
05/31/17  Sungwoo Nam  Localization of virtual classes  Reference 
05/24/17  Becca Tramel  Examples of wallcrossing in Bridgeland stability.  
05/10/17  Lutian Zhao  Categorification of DonaldsonThomas invariants via Perverse Sheaves  Reference 
05/03/17  Yun Shi  The intrinsic normal cone  Reference 
04/26/17  Becca Tramel  Bridgeland stability for the quintic threefold  
04/19/17  Sheldon Katz  Mirror Symmetry  
04/12/17  Lutian Zhao  KontsevichSoibelman WallCrossing Formula  Reference I, II 
04/05/17  Yun Shi  Flops and Derived Categories  Reference 
03/29/17  Mi Young Jang  Stable Maps And Quantum Cohomology  Reference 
03/15/17  Becca Tramel  Derived Categories and ZeroBrane Stability  Reference 
03/08/17  Lutian Zhao  Wall Crossing of BPS states by split attractor flows  Reference I, II 
03/01/17  Joseph Pruitt  Enumeration of rational curves via torus actions  Reference 
02/22/17  Mi Young Jang  Localization  
02/15/17  Lutian Zhao  BPS State Counting  Reference I, II 
02/01/17  Becca Tramel  Bridgeland Stability  
01/25/17  Yun Shi  GromovWitten theory and DonaldsonThomas theory  Reference 
01/18/17  Sheldon Katz  Overview of Enumerative Geometry  Reference 