Garrett Brown
About
I am a third year math PhD student at UC Berkeley working with Song Sun. My interests lie in differential geometry/geometric analysis, usually in the setting of complex geometry. From Fall 2017 to Spring 2021, I was an undergraduate at Harvard College. Before that, I resided in southeast Michigan.
I am grateful for the support of an NSF Graduate Research Fellowship.
Office: 935 Evans Hall
Email: gm"my last name" at berkeley dot edu
Office Hours: None this semester
Research Interests and Papers
Special metrics in complex geometry: extremal Kähler metrics, Kähler metrics with positive scalar curvature, non-Kähler analogs.
Gauge theory in holomorphic vector bundles: Hermitian Yang-Mills connections, stable bundles, Kobayashi-Hitchin correspondence.
- The sign of scalar curvature on Kähler blowups arXiv (2024).
Abstract:
We show that if \((M,\omega)\) is a compact Kähler manifold with positive/negative scalar curvature, then the blowup of \(M\) at any point also furnishes a positive/negative scalar curvature Kähler metric in classes which make the exceptional divisor small. In the case of Kähler surfaces with positive scalar curvature, this extends a result of N. Hitchin to surfaces and answers a conjecture of C. LeBrun in the affirmative, as a result completing the classification of such surfaces.
Upcoming/Recent Conference Participation
New Frontiers in Curvature, Special Geometric Structures and Analysis; August 19-December 20; SLMath (MSRI), Berkeley, CA.
4th Symposium on Geometry and Differential Equations; July 1-5; USTC Institute for Geometry and Physics, Hefei, China.
Aarhus Complex Geometry Workshop; June 24-27; Aarhus University, Aarhus, Denmark.
Geometry and Topology Conference; June 2-7; Qingtian, China.
Other Links
With Thomas Browning, I have organized the Berkeley preliminary exam workshop for incoming and continuing math PhD students each semester since Fall 2022. The website for the Fall 2024 workshop can be found here.
REU Projects from undergrad.
Check out Vilas Winstein.