# 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

Currently I am mostly thinking about special metrics in complex geometry (e.g. extremal Kähler metrics, Kähler metrics with positive scalar curvature). Here are some projects I enjoyed as an undergraduate (click on the paper title to show the abstract):

*Classifying Permutations under Context-Directed Swaps and the CDS Game* (2020), with Angelina Mitchell, Rushil Raghavan, Joseph Rogge, and Marion Scheepers arXiv:2011.00706[math.CO].
**Abstract:**
A special sorting operation called Context Directed Swap, and denoted \(\textbf{cds}\), performs certain types of block interchanges on permutations. When a permutation is sortable by \(\textbf{cds}\), then \(\textbf{cds}\) sorts it using the fewest possible block interchanges of any kind. This work introduces a classification of permutations based on their number of \(\textbf{cds}\)-eligible contexts. In prior work an object called the strategic pile of a permutation was discovered and shown to provide an efficient measure of the non-\(\textbf{cds}\)-sortability of a permutation. Focusing on the classification of permutations with maximal strategic pile, a complete characterization is given when the number of \(\textbf{cds}\)-eligible contexts is close to maximal as well as when the number of eligible contexts is minimal. A group action that preserves the number of \(\textbf{cds}\)-eligible contexts of a permutation provides, via the orbit-stabilizer theorem, enumerative results regarding the number of permutations with maximal strategic pile and a given number of \(\textbf{cds}\)-eligible contexts. Prior work introduced a natural two-person game on permutations that are not \(\textbf{cds}\)-sortable. The decision problem of which player has a winning strategy in a particular instance of the game appears to be of high computational complexity. Extending prior results, this work presents new conditions for player ONE to have a winning strategy in this combinatorial game.

*Spectra of Kohn Laplacians on Spheres* (2018), with John Ahn, Mohit Bansil, Emilee Cardin, and Yunus E. Zeytuncu arXiv:1812.02114[math.CV].
**Abstract:**
In this note, we study the spectrum of the Kohn Laplacian on the unit spheres in \(\mathbb C^n\) and revisit Folland's classical eigenvalue computation. We also look at the growth rate of the eigenvalue counting function in this context. Finally, we consider the growth rate of the eigenvalues of the perturbed Kohn Laplacian on the Rossi sphere in \(\mathbb C^2\).

## Teaching

**Spring 2023:** GSI for Math 54 with Professor John Lott.

**Summer 2022:** Sole instructor for Math 54 lec 004.

**Fall 2021:** GSI for Math 54 with Professor David Nadler.

## Other Links

With Thomas Browning, I have organized the Berkeley math prelim workshop in Fall 2022, Spring 2023, and now Fall 2023. Materials for the current workshop can be found here.