Garrett Brown
About
I am a fifth and final year math PhD student at UC Berkeley, advised by Song Sun (moved from Berkeley to the IASM at Zhejiang University), formally advised by Richard Bamler, and supported by an NSF Graduate Research Fellowship. Postgrad I will begin work as a quantitative researcher at a prop trading firm. My academic 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.
Office: 853 Evans Hall
Email: gm"my last name" at berkeley dot edu
Office Hours: TBA (Spring 2026)
Academic Interests
PhD thesis work: 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.
Most questions I considered in my PhD could be framed generally as searching for equivalent characterizations of purely algebraic conditions on a complex manifold in terms of purely differential geometric (curvature) conditions. My thesis verifies that the existence of a positive scalar curvature Kähler (pscK) metric is equivalent to the algebraic notion of negative Kodaira dimension for algebraic surfaces, yielding a complete classification of pscK surfaces. This was conjectured in this 1995 paper. The question of whether this holds in higher dimensions is still an open question. The most general version of my results suggest an investigation of the stability of scalar curvature under general birational equivalence may be fruitful.
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.