Advice to Applied Math Graduate Students

I graduated in person this weekend, and in reflecting on my doctoral journey I’d like to share some strategies that helped me as an applied math grad student.

  1. Maintain research logs for yourself every day.
    • By logs, I mean mini progress reports. And write them in LaTeX so you can copy and paste snippets of them into papers and other future logs. This will also help you memorize LaTeX commands more quickly!
    • Here is an example of one of my old logs. Notice the open questions in red text.
    • Structure logs by topic or train of thought. Always motivate each document and link to other documents. Keep them all in the same folder.
    • As always, try to be as concise as possible, although this is not as important as in, say, a paper, since it’s just for you and there is of course no page limit here.
    • Use the natural modularity of math to your advantage, with concrete lemmas, theorems, proofs, conjectures. Make the natural modularity of math work for you. I think of it as building a toolbox (however powerful or minor the tools are!) that you can then deploy on future problems.
    • Use a paper-style format. Cite papers and other logs by file name, as if they were papers in their own right. You can start a log by saying “this builds off of log X.pdf”.
    • It doesn’t have to be a whole new log every day — just try to write something. Write what you tried and whether it failed, and why. Keep adding to it as you have new epiphanies or read relevant papers.
    • Draw your thoughts often, and include those images in the document, with captions.
    • Writing logs has many benefits. When you get stuck on a problem and feel like you’re spinning your wheels (which WILL happen), it can be nice to look back at logs to prove to yourself that you’ve been at least trying, and not wasting time. Also, it’s very easy to send a log to your advisor if want to update them on your progress over a break; your advisor might not read the document in its entirety, but would be able to at least skim it. Furthermore, it’s much easier to pick up where you left off the day/week/month before when you can refresh your entire train of thought by rereading. Many times you will begin a great train of thought that feels like it has hit a dead end, and it’s nice to have that loose end preserved neatly in case you make a breakthrough in the future. This will free up some of your mental headspace as well.
  2. Most people gain intuition in math (form hypotheses) by starting from specific examples (inductive), but tend to write proofs starting from generality (deductive).
    • General teaching principle.
    • For this reason I don’t recommend trying to read a proof cold, i.e. without going through some simple examples of the claim first.
    • Similar idea in computer science: “unit tests” and “base cases”. Even if the example is so simple it seems silly, working through the example can be helpful.
  3. If you can, get a tablet for grad school. My favorite annotation/drawing app is GoodNotes for iPad. It’s indispensable for communicating ideas to yourself and others. It’s useful for:
    • Illustrating your thoughts for logs.
    • Reading papers & commenting inline.
    • Taking notes in class and filing them for future retrieval.
    • Pulling up on the screen for virtual meetings.
  4. A template for establishing a new theory in applied math
    • Introduction: Why is this problem interesting? Consider the audience.
    • Background: What has been previously done in this direction? What are the strengths and weaknesses? Where are there still holes?
    • Definitions and Lemmas: What are the objects that you’ll be talking about? What are their basic properties?
    • Main Result, Theorems, and Corollaries: What new insight do you have about these objects? What does that let you conclude?
    • Application: toy. Prove the main result holds for a super-simplified example.
    • Application: synthetic. Prove the main result holds for a fake example that looks more like real data, e.g. adding noise to the toy data.
    • Application: real data. Prove the main result holds for data one would see “in the wild”.


I am thrilled to share my dissertation, A Lighting-Invariant Approach to Local Shape from Shading.


Heal, Kathryn. 2021. A Lighting-Invariant Approach to Local Shape from Shading. Doctoral dissertation, Harvard University Graduate School of Arts and Sciences.


Shape from shading is a classical problem in computer vision, in which the depth field of an object or a scene is reconstructed from a pattern of intensities in an image. This can be thought of in some sense as the inverse problem to geometry-based graphics rendering. In this context, shading is defined as a function of illumination, surface geometry, and surface characteristics like pattern or texture. Despite its enduring presence in the field, shape from shading is still largely unresolved.

This dissertation shows that under the conventional diffuse shading model with unknown directional lighting, the set of quadratic surface shapes that are consistent with the spatial derivatives of intensity at a single image point is a two-dimensional algebraic variety embedded in the five-dimensional space of quadratic shapes. This work rigorously defines a family of such varieties, describes its geometry, and algebraically proves existence and uniqueness results in the areas of two-shot uncalibrated photometric stereo and coquadratic shape from shading. This work introduces a concise, feedforward model that computes an explicit, differentiable approximation of the variety from the intensity and its derivatives at any single image point. The result is a parallelizable processor that operates at each image point and produces a lighting-invariant descriptor of the continuous set of compatible surface shapes at the point. This processor is demonstrated on the two aforementioned application areas.