Math and Aesthetics: Algebraic geometry in the mirror with PIMS PDF, Jesse Huang

by Robyn Humphreys, Communications and Event Assistant

Jesse Huang began her first Postdoctoral Fellowship this September at the University of Alberta, after moving from Illinois, USA, in mid-July. She notes her research field can be thought of as the intersection between algebraic geometry and symplectic geometry, “ I am interested in a kind of duality between the two by the name of mirror symmetry”, which she describes as a phenomenon that originally appeared in physics and then developed into a general mathematical philosophy under the Homological Mirror Symmetry conjecture of Kontsevich. Jesse continues, “Building the dictionary between algebraic varieties/Landau-Ginzburg models and their symplectic mirrors has been and still is an active project pursued by many mathematicians. The geometry behind each proven case is interesting and nontrivial on its own, and the fact that connections between two geometries can be expressed in such elegant manner makes mathematics in this field highly aesthetic.”

One of Jesse’s research goals is to understand the meaning of certain structures on one side of the mirror symmetry equivalence, which is the derived category of coherent sheaves, in terms of its mirror geometry. “If this goal can be made sense of and accomplished under certain contexts, I would also like to see some interesting applications to classical questions in algebraic geometry,” Jesse explains.

Tell us how did you get into this field of research?

I first became interested in this field under the influence of several professors at my old alma mater Kansas State University who gave me the impression that there is this whole field of extremely elegant mathematics I did not know about, so I decided to embark on this journey to figure it out. I did my Ph.D. at the University of Illinois Urbana-Champaign and graduated very recently. My Ph.D. advisor is a symplectic topologist also working in mirror symmetry; he really helped me navigate the field in-depth and fortified my background in symplectic geometry to the point I could work on problems. I also put in the effort to learn algebraic geometry, especially the derived categories of GIT quotients, because part of me suspected there is something I could do about the mirrors of these. In the third year of my Ph.D., while I was parsing through various papers about GIT windows, some foundational works on partially wrapped Fukaya categories were put into place, including one on microlocalization which offers the exact tool one would use to think about mirrors of toric varieties the way I would really like to: though the GIT windows I was reading about. After some careful discussions with my Ph.D. advisor about these recent breakthroughs on Fukaya categories, I was convinced that the problem is at least approachable, and I decided to think about it. I would say my mathematics started to build up from there, although very slowly; it has matured a bit during my recent transition from grad student to PDF.

My current PDF supervisor is David Favero. Although I spent a good amount of time in my Ph.D. reading his work with Ballard and Katzarkov, we only met once in Toronto at a conference he co-organized before PIMS brought our brief connection to a different level. The conference was very impressive to me because I could relate my research to most of the talks, which had never happened to me before. Some of the talks made me believe the class of examples (quasi-symmetric cases) I was thinking about at the time is really going to work, and eventually, it did work. It was also a conference I felt very comfortable as a grad student to talk to mathematicians I wanted to talk to.

This is your first PostDoc, do you feel there is a difference between in your graduate research and what you are doing now with your PDF ?

The actual content of my PDF research so far has been a natural extension of my graduate research based around the same circle of ideas. My supervisor and I recently constructed a skeleton out of a cellular resolution of quivers with certain relations, and now we are trying to see if it has any interesting connections to results on BHK mirrors and Bondal-Orlov’s conjecture on rationality. Some of these ideas we came up with also made me revisit what I thought I understood previously and gave me some hope in finding alternative methods to extend the results of my paper with Zhou to the remaining cases. I also hope to think more about the Mori program during my time here as a PIMS PDF.

A topological deformation of skeletons in the cotangent bundle of a 2-torus.

As for the experience doing research, the biggest difference I can tell is a shift in perspectives. In graduate school, especially the earlier years, I worked around the clock trying to “follow up” with the next new thing that emerged in my field. Although a lot of cool mathematics went through my head this way, it did not help me much with what a Ph.D. student is supposed to do ultimately — research. It was only after I started identifying the most basic elements in a problem I realized how much of mathematics is simply organizing and reorganizing basic things from scratch. I am much more practical and problem-oriented now as a PDF. On a more personal level, I feel much happier after an average day of PDF research, because there are always little things I can spend the day working on or questioning myself about, and progress is somewhat more “linear”. In graduate school, I would often end my day in small hours the next morning with a handful of obscure facts from various papers, not knowing what to accomplish with them, and sometimes I got frustrated and stuck for months without progress. Grad school math was more “fun” but less fulfilling compared to PDF math in this regard. On the flip side though, I realized something technical I very painfully read through during grad school but was unable to put into good use has begun to sink in recently, so I guess even though not all hard work in grad school bears fruit immediately in the form of papers, it will not be in vain. Mathematics will unveil itself to the thinking minds after all; one just has to keep going at it and be patient.

Minimal cellular bimodule resolution of the bounded quiver for the Hirzebruch surface (top) and weighted projective stack (bottom) when n=3.

In what ways has the pandemic changed your work processes or your current research?

I adapted to working online while doing my Ph.D. When the pandemic struck, I was really pumped up at first because all of a sudden I had access to all the math talks happening on this planet from my little cozy apartment in the middle of cornfields, but months later quarantine fatigue kicked in hard and I realized I needed some time management tricks. Right now when working from home, I usually schedule one type of activity into a single time block no more than 4 hours, for example, 3 hours of research on one project, with hour-breaks in between activities so that I can eat and exercise properly, and I will almost certainly not be awake at 3:30 am anymore for a talk I am only half-interested in.

I have adapted quite well, and working online has not been an issue at all after I relocated, but now since the province has pretty much opened up, I prefer to go to my office for better productivity. I am not teaching any courses currently, but if I do in upcoming semesters, I can go with either format and/or switch between.

You are in a new country and city—a lot of change has happened. Aside from your research, how have you adapted to your new normal?

I like outdoor activities, which Edmonton obviously has a lot to offer. There are hiking areas along the river, where I go for short trials every other day or so. Hot tent camping is another thing I would be tempted to do when the beloved Alberta winter arrives. As for indoor activities, I hit the gym once a week to get some extra cardio or simply do some full-body training. Very occasionally, I play CS:GO and a few old Blizzard games that still run on my 10-year-old unibody MacBook.

I also like classical music and performance art. After moving to Edmonton, I have been to a concert of the Edmonton Symphony Orchestra featuring Beethoven, Vivaldi, Shostakovich, and Swan Lake performed by Alberta Ballet. They were pretty good.

What is your best discovery since arriving in Canada?

I am sure I saw the northern light earlier this month. It was too beautiful to be anything else.

While In Oglesby, Illinois, Jesse would visit this waterfall to meditate on things; this place knows how much she loves mathematics. Now in Edmonton, hopefully Jesse has found a similarly tranquil place for her mathematical daydreams.

Jesse will be speaking at the PIMS Emergent Research Seminar Series, on December 1, 2021, at 9:30 AM Pacific. Details on her talk, Skeleta for Monomial Quiver Relations, can be found here.

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