Department of Mathematics, UC Davis
One Shields Ave.
Davis, CA 95616-8633
U.S.A.
Office: MSB 2142
About me
I am an Assistant Professor of Mathematics at UC Davis.
In my research, I build and use algebraic tools to study problems and phenomena in low-dimensional topology.
In particular, I am interested in the following:
knots and links in 3-manifolds, surfaces in 4-manifolds
Khovanov, Heegaard Floer, and related homology theories
extensions of the above, e.g. Khovanov and knot Floer stable homotopy types, skein lasagna modules of 4-manifolds
categorification and quantum topology
contact and symplectic topology in dimensions 3 and 4
Of course, I am also interested in other areas of mathematics, especially areas adjacent to my research.
Outside of mathematics, I like creating music (piano, guitar, singing), arts and crafts (painting, knitting), physical activities where you just translate yourself (hiking, swimming, skating, kayaking), and baking (point at anything at a bakery and I can make a worse version). I enjoy these activities in video games as well.
Kirby belts, categorified projectors, and the skein lasagna module of S²xS² Ian A. Sullivan and Melissa Zhang
In Revision. [arXiv/2402.01081]
Khovanov homology and the Involutive Heegaard Floer homology of branched double covers
Akram Alishahi, Linh Truong, and Melissa Zhang
In Revision. [arXiv/2305.07172]
Code :
These are provided as .txt files; change the extension to .py. (Will be moved to github soon.)
OmegaMergeSplit
On Equivariant Khovanov homology
Rostislav Akhmechet and Melissa Zhang
Accepted for publication in the AMS Contemporary Mathematics (CONM) volume associated to the
AMS Special Session on Algebraic Structures in Knot Theory held in Fresno, CA in April 2023.
This article is a survey article based on the following preprint: [arXiv/2210.10731]
On Khovanov homology and related invariants
Carmen Caprau, Nicolle González, Christine Ruey Shan Lee, Adam M. Lowrance, Radmila Sazdanović, and Melissa Zhang
Research directions in symplectic and contact geometry and topology,
273--292, Assoc. Women Math. Ser., 27, Springer, Cham, 2021.
[arXiv/2002.05247]
Annular link invariants from the Sarkar-Seed-Szabó spectral sequence
Linh Truong and Melissa Zhang
Michigan Mathematical Journal,
Advance Publication 1-39 (2021).
DOI: 10.1307/mmj/20205862.
[arXiv/1909.05191]
Localization in Khovanov homology
Matthew Stoffregen and Melissa Zhang
Geometry & Topology
(to appear)
[arXiv/1810.04769]
A rank inequality for the annular Khovanov homology of 2-periodic links
Melissa Zhang
Algebraic & Geometric Topology,
18-2 (2018), 1147--1194.
DOI 10.2140/agt.2018.18.1147.
[arXiv/1707.03279]
Lecture 3: sl(2) action on AKh, annular Khovanov-Lee homology
[slides]
Lecture 4: A further sampling of applications of AKh
[slides]
In Fall 2022, I organized a learning seminar at SLMath called [Homotopy Types in Low-Dimensional Topology]. Click on the link to visit the webpage containing resources for this seminar.
If you are currently in one of my classes, go back to the top of the page and click on the link on the main menu bar corresponding to your course.
In my teaching, my first goal is to pique students' interest in the content, so that learning becomes a means to satisfy curiosity. In the classroom, I am currently focusing on active learning techniques, and like to give my students the time to think about questions — to ask and to answer — during our class meetings. I aim to support my students as a whole persons, as I recognize that everyone has unique struggles and needs which need to be resolved for learning to occur most efficiently.
Past courses at UC Davis:
In Fall 2024, I am currently teaching a graduate topics course in Khovanov homology and related invariants. See the [MAT280] course webpage for my continually updating lecture notes.
Winter Quarter 2024:
Sometimes I don't want to watch the dregs of my "Watch later" playlist and just want to delete them. But YouTube doesn't give you an easy way to do this. You can use AutoIt to bulk-delete unwatched videos on your YouTube "Watch later" playlist.
