# About Me

I’m a PhD student at University of Waterloo working in algebraic geometry with Prof. Matthew Satriano. Outside school, I enjoy:

• video games: dark souls, mass effect, divinity original sin 2, baldur’s gate 3, dota 2, warframe, etc
• musics: musicals, prog rocks, cantonese pop, classical music, chinese traditional music, etc
• pipa: still in the process of learning it
• photographs
• web novels
• horror films
• biking and swimming
• cooking

# What I Do

# tl;dr:

Things I like to think about:

• toric geometry
• stacks
• tropical geometry
• algebraic enumeration

# Long Version

In short, I’m working with polynomials, and ask questions like when do a set of polynomials vanish simultaneously. Given a finite set of polynomials $I=\{f_1,...,f_n\}\subseteq k[x_1,...,x_m]$ over field $k$, the set $V_k(I):=\{x\in k^m:\forall f\in I, f(x)=0\}$ is called an algebraic variety, and this is one of the things I work with most of the day. Perhaps one of the most famous example of this would be Fermat’s last theorem:

No three positive integers $a,b,c$ satisfy the equation $a^n+b^n=c^n$ for any integer value of $n$ greater than $2$

Equivalently, we are saying that $V_{\mathbb{Q}}(x^n+y^n-z^n)=\emptyset$ for $n\geq 3$. This theorem is in fact proved by Weil, using heavy machinary from modern algebraic geometry.

More specifically, I’m currently focusing on toric geometry and stacks, and on the side I also pondering on algebraic enumeration and tropical geometry.

Toric geometry concerns with algebraic varities that can be encoded in combinatorial objects called fans. This provides us with a very convenient testing ground for theorems in algebraic geometry, as almost everything about a toric variety can be computed explicitly.

On the other hand, stacks are more abstract objects than varieties. In ordinary people’s life, we will probably only see stacks when we think about moduli spaces, e.g. the moduli spaces of elliptic curves. Since the definition of this requires a few pages to explain, you should just think of this as a more flexible framework than varieties.

Algebraic enumeration deals with finding exact formulas for the number of combinatorial objects of a given type. One classical question in this area would be finding the number of trees, up to isomorphism, labelled by the set $[n]=\{1,2,...,n\}$.

Exercise: show the number of trees on $n$ vertices is $n^{n-2}$

At last, we have tropical geometry. This area also lies in the field of combinatorial algebraic geometry, just like toric geometry. The simplest way to think about them is the “algebraic geometry over the tropical ring”. Here the tropical ring is a semiring defined by $(\mathbb{R}\cup\{+\infty\},\oplus,\otimes)$ where $x\oplus y=\min(x,y)$ and $x\otimes y=x+y$. Once you do this, and ask when the polynomials vanishes simultaneously over this ring, you will see nice little pictures pop up.