# 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 math prelim workshop in Fall 2022, Spring 2023, Fall 2023, and Spring 2024. Materials can be found here.
REU Projects from undergrad.
Check out Vilas Winstein.