I'm a first-year graduate student at Princeton studying mathematics. I am particularly interested in low-dimensional topology and symplectic geometry. Check out my resume here.
Previously, I was an undergraduate at Harvard. What this means is that I have been stuck on the East Coast for several years, and will continue to be stuck here for the foreseeable future. Someday, I will be happy. That day is not today.
In my free time, I like doing the crossword and drinking smoothies. I also like imagining a world without Skittles.
You can contact me at jz3485@princeton.edu. Why the "3485"? I don't know. I did not pick this email. I struggle daily to remember this email. Fortunately, I rarely, if ever, email myself. This is because I am normal.
In summer 2024, I worked with Ciprian Manolescu to find an efficient algorithm to compute the Khovanov homology of class-1 knots in RP3. The algorithm mimics Bar-Natan's tangle-theoretic algorithm, but uses the TQFT for RP3, as defined by Manolescu and Willis, rather than the standard Khovanov TQFT.
In summer 2023, I worked with Peter Kronheimer on understanding higher differentials in the Lee spectral sequence. I showed that the map taking a knot K to the n-th page of its Lee spectral sequence is functorial. One consequence is that the Lee spectral sequence of a band sum decomposes quite nicely. I presented this research at the Joint Mathematics Meetings in 2024.
A persistent Googler may also discover that I have a preprint on tight contact structures. Unfortunately, Lemma 3.4 from that preprint is false. The closed form formula is probably still true though, based on computations using Honda's algorithm for tight contact structures. I sadly cannot offer much more information than that. I still have dreams of fixing this proof.
There is also a paper on regular matroids which has my name on it. I am sorry to say that I no longer know what a matroid is, nor what makes one regular. Please do not talk to me about this paper. It only reminds me of all the knowledge I've lost.
I wrote my senior thesis with Denis Auroux on bordered Heegaard Floer homology, with a particular emphasis on proving compactness results typically relegated to symplectic field theory texts. It is an at-times structurally unsound thesis. Think: the Leaning Tower of Pisa. Do not think: the (fortunately off-season) 2002 collapse of the 300-foot tall amusement park ride VertiGo. Absolutely do not think: the collapse of a PepsiCo warehouse in western Russia which resulted in a 7.4-million gallon flood of fruit and vegetable juices.
I have published an expository paper on elliptic bootstrapping and the nonlinear Cauchy-Riemann operator. It is called, creatively enough, "Elliptic bootstrapping and the nonlinear Cauchy-Riemann operator." Subtlety has always been a strong suit of mine. It assumes relatively little background, mostly because I had relatively little background when writing this.
I also have a short paper on Khovanov homology and Lee homology, as well as an even shorter paper on the relationship between knot Floer homology and grid homology. These were both written for classes. I have written other papers for math classes, but they are all so awful that I refuse to share them. I have also written other papers for non-math classes, but I won't even comment on their quality.
I have solutions to (most of) the first two parts of Bott and Tu's Differential Forms in Algebraic Topology. I'm not sure if I'll ever get around to typing the rest, but I'm sure solutions are all available if you Google intently enough.
I also have written up solutions to Rotman's Introduction to Algebraic Topology. It's missing Chapter 12 (on cohomology) and will definitely never be updated, because I am lazy.
