**Venue changed: Lectures on May 14 and May 21 will be changed to Rm 440, Astro-Math. Bldg.**

1. Background & Purposes

This lecture series explores nonlinear PDE on Riemannian manifolds: non-linear elliptic equations on Kähler manifolds, an L^2-estimate for the Dirac-Dolbeault operator for line bundles with mixed curvature and the elliptic theory of G_2-structures.

The first part focuses on the work of G. Székelyhidi concerning a priori estimates for fully non-linear elliptic equations on compact Kähler manifolds. These estimates play a crucial role in understanding the regularity of solutions to such equations.

The second part explores the ellipticity of the G_2 holonomy equation on manifolds with boundaries, developed by S.K. Donaldson. This approach leads to a deformation theory and the existence of certain geometric objects called G_2 cobordisms.

2. Outline & Descriptions

The series comprises four lectures: Professor Chao-Ming Lin (05/14, 05/21 11:00-12:30) will give two talks and Ming-Yuan Chang Chang (05/14, 05/21 13:30-14:30) will deliver two talks. There is lunch break from 12:30-13:30. The titles and abstract are below.

This lecture series is targeted towards students or mathematicians with a background in differential geometry and analysis on manifolds. Familiarity with basic concepts of Kähler manifolds and elliptic equations would be beneficial.

3. Registration
https://forms.gle/uXo4yuJbT7hi6tLP7

**Deadline: May 10, lunch box is provided

4. Lectures Abstract

2024/5/14
11:00-12:30
Online Meeting Link: https://ntucc.webex.com/ntucc-en/j.php?MTID=md24203053e2012420d8fb83076ab1943
Speaker: Chao-Ming Lin (Ohio State University)

Title: A priori estimates for fully non-linear elliptic equations on compact Kähler manifolds (I)

Abstract:In this series of talks, we will go over Székelyhidi's work [S18]. We will show some technical details of a priori estimates for fully non-linear elliptic equations on compact Kähler manifolds therein. In the first talk, we will mention the settings and some classic results in Székelyhidi [S18] and Guan [G14] and a $C^0$-estimate by Błocki [B05].

References:

[B05] Z.~Błocki, On uniform estimate in Calabi--Yau theorem, Science in China Series A: Mathematics, vol.~48, no.~1, pp.~244--247, 2005.

[S18] G.~Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, Journal of Differential Geometry, vol.~109, no.~2, pp.~337--378, 2018.

[G14] B.~Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Mathematical Journal, vol.~163, no.~8, pp.~1491--1524, 2014.

13:30-14:30
Speaker: Ming-Yuan Chang (Institute of Mathematics, Academia Sinica)

Title: An L^2-estimate for the Dirac-Dolbeault operator for line bundles with mixed curvature

Abstract: Classical L^2-estimates for semi-positive curvatures are obtained by the well-known Bochner-Kodaira-Nakano identity and twisting the canonical bundle.

For mixed curvature case, we generalize this method by twisting the (p,0)-form bundle and choosing a canonicalsection to help.

After computing the extra error term, we can obtain an L^2 estimate theorem.

As an application, we show that this can be used to obtain an asymptotic expansion of the generalized Bergman kernel on the nondegenerate set.

2024/5/21
11:00-12:30
Online Meeting Link: https://ntucc.webex.com/ntucc-en/j.php?MTID=m58c5317ff6f9f4d49a45fece51cbae0c
Speaker: Chao-Ming Lin (Ohio State University)

Title: A priori estimates for fully non-linear elliptic equations on compact Kähler manifolds (II)

Abstract: In this series of talks, we will go over Székelyhidi's work [S18]. We will show some technical details of a priori estimates for fully non-linear elliptic equations on compact Kähler manifolds therein. In the second talk, we will mainly focus on the technical details of a priori estimates. We will talk about a $C^1$-estimate by Hou--Ma--Wu [HMW10] and Collins--Jacob--Yau [CJY20], a $C^2$-estimate by Székelyhidi [S18], and a $C^{2, alpha}$-estimate by Tosatti--Wang--Weinkove--Yang [TWWY15] and Siu [S12].

References:

 [CJY20] T.~C.~Collins, A.~Jacob, and S.-T.~Yau, (1, 1) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions, Cambridge Journal of Mathematics, vol.~8, no.~2, pp.~407--452, 2020.

[HMW10] Z.~Hou, X.-N. Ma, and D.~Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Mathematical Research Letters, vol.~17, no.~3, pp.~547--561, 2010.

[S12] Y.-T.~Siu, Lectures on Hermitian--Einstein metrics for stable bundles and Kähler--Einstein metrics: delivered at the German Mathematical Society Seminar in Düsseldorf in June, 1986, vol.~8, 1987.

[S18] G.~Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, Journal of Differential Geometry, vol.~109, no.~2, pp.~337--378, 2018.

[TWWY15] V.~Tosatti, Y.~Wang, B.~Weinkove, and X.~Yang, $C^{2, alpha}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calculus of Variations and Partial Differential Equations, vol.~54, no.~1, pp.~431--453, 2015.

13:30-14:30
Speaker: Ming-Yuan Chang (Institute of Mathematics, Academia Sinica)

Title: An elliptic boundary value problem for G2-structures

Abstract:This talk explores the work of Donaldson concerning the ellipticity of the G_2

holonomy equation on a manifold with a boundary. The equation is considered alongside a prescribed 3-form defined on the boundary.

A key aspect of this work involves establishing a suitable linear elliptic boundary value problem. This approach leads to the development of a deformation theory.

As a specific application, the existence of certain G_2 cobordisms is established. These cobordisms connect two slightly deformed versions of a Calabi-Yau 3-fold.