Speaker: Dr. Lin Chen (YMSC)
Time: 10:30-11:30 May 16, 2024 (Thursday)
Place: MCM110
Title: The global unramified geometric Langlands equivalence
Abstract: Recently, Gaitsgory's school (to which I am honoured to belong) announced their proof of the global unramified geometric Langlands conjecture. I will explain the history, motivation and statement of this conjecture and the main ingredients used in this proof. If time permits, I will also introduce some questions in this field that remain open after this proof.
Speaker: Prof. Zhengyu Zong (Tsinghua Univ)
Time: 10:00-11:00 May 15, 2024 (Wednesday)
Place: MCM410
Title: Open WDVV equations for toric Calabi-Yau 3-folds
Abstract: The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations is an important system of equations in the study of genus zero Gromov-Witten invariants. It implies the associativity of the quantum product. The associativity of the quantum product has many important applications including the recursive formula given by Kontsevich and Manin that calculates the Gromov-Witten invariants of the projective plane. The system of open WDVV equations plays an important role in the study of open Gromov-Witten invariants. It can be viewed as an extension of the WDVV equation to the open sector. The natural structure that captures the WDVV equation is that of a Frobenius manifold. Similarly, the system of open WDVV equations determines the structure of an F-manifold, a generalization of a Frobenius manifold.
In this talk, we prove two versions of open WDVV equations for toric Calabi-Yau 3-folds. The first version leads to the construction of a semi-simple (formal) Frobenius manifold and the second version leads to the construction of a (formal) F-manifold. This is a joint work with Song Yu.
Speaker: Prof. Zhongwei Shen (University of Kentucky)
Time: 10:00-11:00 May 17th, 2024 (Friday)
Place: MCM110
Title: Resolvent Estimates for the Stokes Operator
Abstract: This talk is concerned with the study of resolvent estimates in $L^p$ for the Stokes operator. Such estimates play an essential role in the functional analytic approach of Fujita and Kato to the nonlinear Navier-Stokes equations in bounded domains. In the case of smooth domains ($C^2$), the resolvent estimate is well known and holds for all $1<p<\infty$. If the domain is Lipschitz, the estimate was established for a limited range of $p$, depending on the dimension, using the method of layer potentials and a real-variable argument. In this talk, I will discuss some recent work, joint with Jun Geng, for the case of $C^1$ domains. Starting with the upper half-space, using a perturbation argument, we are able to show that the resolvent estimate holds for all $1<p<\infty$. The case of exterior $C^1$ domains is also studied.
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