Personal Webpage of Lutian Zhao

Spring 2019

Date Speaker Title Reference
04/19/19 Virtual fundamental classes are playing essential roles in modern enumerative geometry including Gromov-Witten 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 Gromov-Witten 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 Ricci-flat 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 Beauville-Bogomolov-Fujiki 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 Kontsevich-Soibelman wall crossing formula determines a Riemann-Hilbert problem. I’ll explain Bridgeland’s solution of this problem and see some examples of this problem.
Lutian Zhao Riemann-Hilbert Problem for BPS Structure Reference
02/15/19 I’ll review the concept of the heart of a t-structure 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 t-structures 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 Bogomolov-Gieseker 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 Brill-Noether 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 Brill-Noether from Wall Crossing Reference

Fall 2018 (You can see abstract when you hover your mouse over the date)

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 Calabi-Yau 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 Calabi-Yau 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 2-part talk with Sungwoo on the paper https://arxiv.org/abs/alg-geom/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 Calabi-Yau 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 Gromov-Witten Theory
11/01/17 Hao Sun Gromov-Witten theory, Hurwitz numbers, and Matrix models, II Reference
10/25/17 Hao Sun Gromov-Witten 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 Calabi-Yau manifolds, III Reference
09/27/17 Sungwoo Nam The local Gromov-Witten theory of curves Reference I, II
09/20/17 Lutian Zhao Introduction in Topological String Theory on Calabi-Yau manifolds, II Reference
08/30/17 Lutian Zhao Introduction in Topological String Theory on Calabi-Yau 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 Gopakumar-Vafa invariants via vanishing cycles Reference
05/31/17 Sungwoo Nam Localization of virtual classes Reference
05/24/17 Becca Tramel Examples of wall-crossing in Bridgeland stability.
05/10/17 Lutian Zhao Categorification of Donaldson-Thomas 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 Kontsevich-Soibelman Wall-Crossing 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 Zero-Brane 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 Gromov-Witten theory and Donaldson-Thomas theory Reference
01/18/17 Sheldon Katz Overview of Enumerative Geometry Reference