Garrett Brown
About
I am a fourth year math PhD student at UC Berkeley working with Song Sun, who recently moved to the IASM at Zhejiang University. 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: 853 Evans Hall
Email: gm"my last name" at berkeley dot edu
Office Hours: Visiting another university Spring 2025
Research Interests and Papers
Some key phrases occurring in problems I am thinking about: extremal Kähler metrics, Kähler metrics with positive scalar curvature and relation with birational geometry, non-Kähler analogs, Hermitian Yang-Mills connections, and applications of complex geometry such as to minimal surfaces.
The sign of scalar curvature on Kähler blowups arXiv (2024). Submitted.
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.
Talks/Activities
Giving a talk at: Yau Mathematical Sciences Center; April 9; Tsinghua University, Beijing, China.
Visiting/Giving a talk at: Academy of Mathematics and Systems Science; April 7-11; Chinese Academy of Sciences, Beijing, China.
Visiting: Institute for Advanced Study in Mathematics; February 16th-May 12th; Zhejiang University; Hangzhou, China.
Giving a talk at: Vanderbilt Geometry/Topology Seminar; September 27th; Vanderbilt University, Nashville, TN.
Participating in: Workshops on Geometric Analysis; August 19-December 20; SLMath (MSRI), Berkeley, CA.
Participating in: 4th Symposium on Geometry and Differential Equations; July 1-5; USTC Institute for Geometry and Physics, Hefei, China.
Participating in: Aarhus Complex Geometry Workshop; June 24-27; Aarhus University, Aarhus, Denmark.
Participating in: 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.
In Fall 2024 I organized a learning seminar on the basics of the minimal model program in birational geometry. One of my main research interests is concerned with the relationship between certain curvature conditions and birational geometry.
REU Projects from undergrad.
Check out Vilas Winstein.