I received my Ph.D. in math from Stanford in August 2020, advised by Ravi Vakil and Jarod Alper.

My academic research interests are largely in algebraic geometry, with a focus on moduli theory and the formal geometry of algebraic stacks. See my papers below.

Before coming to Stanford, I was an undergraduate in math at the Australian National University, advised by Jarod Alper and David Ishii Smyth.

In my free time, I enjoy writing expository articles on topics close to my area of research, i.e. algebraic geometry. Outside of math, I enjoy mastering variations of the clean and snatch.

# Projects

## COVID-19

Since the start of the total lockdown in Malaysia on 1/6/2021, I have become increasingly interested in trying to understand the spread of COVID-19 in Malaysia.

## The failure of the 2021 FMCO

I argue that the failure to stop workplace clusters from happening was why the 2021 FMCO failed in the Klang Valley, Johor, Pulau Pinang and Negeri Sembilan. Please see this pdf here. In addition, please see this press statement (with IDEAS and Bait Al-Amanah), urging the federal government to address this issue ASAP.

## A reopening criterion for Penang

A reopening criterion for Penang (as well as other states in Malaysia) at the district level, based on the basic reproduction number and daily new cases per 100,000 population.

The gif above shows the evolution over time of ($R_0$, daily new cases/100k) for each of the 5 districts in Penang, i.e. Timur Laut, Barat Daya, Seberang Perai Utara, Seberang Perai Tengah and Seberang Perai Selatan. These have been abbreviated respectively as TL, BD, SPU, SPT, SPS.

The criterion for reopening that we suggest depends on determining thresholds shaded green, yellow and red in the figure above. In theory, these thresholds are a function of the state of the healthcare system, e.g. ICU occupancy and vaccination rates. However, since I do not have access to the data of ICU occupancy rates in Penang, I shaded in thresholds determined empirically from past data: Go back in time and record the date that is two weeks or so prior to the healthcare system being overwhelmed. Record ($R_0$, daily new cases/100k) around then, and set that to be your threshold.

The code to calculate $R_0$ for each district of (of any state!) in Malaysia is available on my GitHub here. The basic reproduction number was calculated using the method of ordinary least squares applied to daily new case numbers over a short time frame.

## 2021 Bayan Baru Donation Drive

The goal is to raise RM 50,000 (approx. USD 12,000) to be distributed as cash to B40 families around the area of Bayan Baru, Penang, Malaysia during the 2021 FMCO (Full Movement Control Order). The GIVE.asia page to donate is here. This project is joint with the member of parliament for Bayan Baru, YB Sim Tze Tzin.

# Papers

## Coherently complete algebraic stacks in positive characteristic

This paper generalizes the main algebraization theorem of Alper-Hall-Rydh to the reductive case. Currently, the result is only proven in the case that $A^G$ is a field, but not in the general case that $A^G$ is a complete Noetherian local ring.

With Jarod Alper and Jack Hall.
In preparation.   Jarod's talk on our result (video)

## Grothendieck's existence theorem for relatively perfect complexes on algebraic stacks

I extend Lieblich's algebraization result for complexes on algebraic spaces to the setting of algebraic stacks.

arXiv

## Algebraization theorems for coherent sheaves on stacks

My Ph.D. thesis. It is essentially an integrated version of the two papers above.

PDF

## Moduli functors, GIT and semistability of the gradient morphism for ternary forms

My undergraduate thesis. I generalize a result of Alper and Isaev on the semistability of the gradient morphism from the case of binary forms to cubics forms of degree $d$, for $3 \leq d \leq 9$.

A year or so after I graduated, Fedorchuk generalized my result to arbitrary forms of any degree. See Theorem 1.4.1 here and the paragraph below it acknowledging my work.
My undergraduate thesis is available on request.

# Notes and Expository Articles

## The rank of $y^2 = x^3 - 2$ via Mazur-Tate methods

Inspired by the paper Points of order $13$ on elliptic curves, I show that the rank of the elliptic curve $y^2 = x^3 - 2$ is $1$ using the method of $2$-descent.

PDF

## The Picard number of a Kummer surface

I give a self-contained proof that the Picard number of the Kummer surface $X$ associated to an abelian surface $A$, satisfies $\rho(X)= 16 + \rho(A)$.

PDF

## A proper scheme with infinite-dimensional fppf cohomology

It is a fundamental fact in étale cohomology that the cohomology of any constructible sheaf on a proper scheme is finite. In this article, I give an example showing this is false in the fppf topology.

PDF   Blog article on Thuses

## The étale cohomology of curves over a finite field

I compute the étale cohomology (with coefficients in $\mathbf{G}_m$) of a curve over a finite field. The calculation is more subtle than the case over an algebraically closed field, because theorems such as Artin vanishing and Poincaré duality cannot be applied.

Blog article on Thuses

## Semicontinuity of weights

Notes for a lecture I gave during the 2016-2017 edition of Stanford's Number Theory Learning Seminar. Given a variety over a finite field, I prove that the weight of a lisse sheaf (in the sense of Deligne) is invariant under restriction to a dense open.

PDF   Version on learning seminar website (My notes start from page 9)

## The Peter-Weyl theorem

Essay I wrote for the 2013 edition of ANU's Analysis 3 (Functional Analysis and Spectral Theory). I prove the "analytic" version of the Peter-Weyl Theorem, namely that the matrix coefficients of a compact Lie group $G$ are dense in $L^2(G)$.

PDF

## Schur-Weyl duality and irreducible representations of $\mathfrak{sl}_n$

Essay I wrote for the 2012 edition of ANU's Algebra 3 (Lie Groups and Lie Algebras). I prove Schur-Weyl duality for $\mathrm{SL}_n(\mathbf{C})$, and using this I deduce the characters and dimensions of the highest weight representations of its Lie algebra.

PDF

# Education

Ph.D. in Mathematics, Stanford University, 2015-2020.

Ph.B. in Mathematics, Australian National University, 2011-2014.

# Honors

University Medal, Australian National University, 2014.