Resources

The following are several resources which may prove valuable to undergraduage and graduate students alike. I have uploaded various notes from directed reading courses/self study as well as many research/personal statements for applications. I hope you find these useful!

Subject Notes:

During Spring 2022 I took a course on Optimal Transportation at the Fields Institute by Robert McCann. Below you can find condensed notes focusing on entropic curvature dimension spaces and stability under Gromov-Hausdorff convergence. For a final project we had to read a paper and write exposition on it. I chose the 2014 paper by Figalli-Maggi-Pratelli proving stability for the anisotropic isoperimetric inequality via optimal transport. You can find my final report below too.

Entropic Curvature Dimension Spaces and Stability under Gromov-Hausdorff Convergence.
Optimal Transportation and Stability of the Anisotropic Perimeter.

These are expository notes written as a result of UChicago's REU program.

Optimal Mass Transport and the Isoperimetric Inequality.
Stable Minimal Surfaces.

Textbook & Prelim Solutions:

I've compiled solutions for textbook problems (and other various homework problems) assigned in some courses while at UT Austin.

Fall 2020:
- M382C: Algebraic Toplogy (Hatcher).
Spring 2021:
- M382D: Differential Topology (Guillemin and Pollack).
Fall 2022:
- M385C: Probability I (Zitkovic Notes).

While studying for prelims at UT Austin, I worked many textbook problems and typed my solutions. I also typed up solutions for many past prelims. These are provided below, along with scans of my actual exams solutions.

Real Analysis:
- Past prelim solutions.
- Folland solutions.
- Prelim Attempt 1 (Jan. 2021), failed.
- Prelim Attempt 2 (Aug. 2021), passed.
Complex Analysis:
- Past prelim solutions.
- Ahlfors solutions.
- Prelim Attempt 1 (Aug. 2021), passed.
Methods of Applied Mathematics I:
- Arbogast solutions.
- Prelim Attempt 1 (Aug. 2020), passed.

Graduate School:

When I was applying to grad school, I didn't get the chance to see anyone's personal statements. I think having some sort of guidance is essential, so I share the one I sent to UT here. Upon reflection, I think it helped to share my experiences and specifically what I learned from them. It also probably helped to list people I wanted to work with (and being genuine about it!) and researching available opportunities there that I could make use of.

UT grad personal statement.

The following website is good to keep track of which schools have sent out offers and for making connections. Do not read into it too much, it can be very toxic. Don't compare yourself to other students. Don't check it religiously.

Resource for math grad info.

NSF GRFP:

The National Science Foundation Graduate Research Fellowship Program (NSF GRFP) provides graduate students with $34,000 over three years. Eligible students include undergraduates heading to grad school and first and second year graduate students. You can apply twice, once during undergrad and once during grad, so you might as well apply during undergrad! This gives you some experience with the process and what to look out for, and also allows you to practice writing proposals.

The NSF GRFP judges candidates based on two criteria, loosely defined as Intellectual Merit and Broader Impacts. These essentially amount to how promising of a researcher you are (past experience, internships, coursework, etc.) and how your research will impact society (diversity, health/knowledge benefits, etc.)

The NSF GRFP application consists 2-3 letters of rec of two statements: the personal and research statement. Both should address intellectual merit and broader impacts. More information on the application process and the judgement criteria can be found on the NSF GRP website below.

NSF GRFP Overview.

Here are some key points:

  1. Each reviewer has maybe 5 minutes to read over your app.
  2. "NSF funds the researcher, not the research" -- an NSF reader. Meaning, they're looking for potential.
  3. Letters of rec are super important. You should at least have one who knows you as a researcher and one who knows you as a person. This helps with fulfill the intellectual merit/broader impacts criteria.
  4. Broader impacts is half the application. You need to make a statement about how your research is meaningful. You could even write something like being funded will alleviate duties and allow you more time to help underrepresented students. Then, show how you've used your time already to do so.
  5. Know your audience. Not everyone will know the technical jargon from your field of interest. Try and state things loosely or intuitively.
  6. Pictures say 1000 words! In an application where you need to express so much information in little space, use pictures to help clarify what may take longer with words.

The following are my personal + research statements, as well as my ratings and comments. Feel free to read them to get a feel for how to write your own.

Sample NSF GRFP research statement.
Sample NSF GRFP personal statement.
Sample NSF GRFP ratings and comments.

Alex Lang has an amazing writeup of the NSF GRFP on his website. You can find it below.

Alex Lang's Website.

REUs:

Research Experiences for Undergraduates (REUs) provide exciting, informative opportunities to undergraduate students. For math majors, these take the form of collaborative research projects or directed self study programs.

Below is a presentation on REUs I gave to UF's University Math Society in 2019. It describes many of the possible REUs and the application process. I also share the programs I applied to and to which ones I was accepted.I encourage all undergraduates to apply to REUs -- epecially those of you coming from underrepresented groups! These are incredible opportunities and you are more than qualified to participate, I promise.

Info on REUs and applying.

I have some personal opinions on specific REUs that I have heard from attendees. I will not post them here, but if you would like to hear them, email me.

The application process starts at this website where you'll find program listings. Most REUs have an application on this website, but some are on NSF's website. Generally speaking, you'll need a cover sheet (basic background info), transcript, letters of rec, and a personal statement. CVs are optional, but I'd reccommend them. You can find two personal statements below.

UChicago REU personal statement.

Lena Ji also has a good website on REUs, see below.

Lena Ji's REU Info.

Goldwater Scholarship:

The Goldwater Scholarship is an opportunity for budding sophomore and junior year researchers to obtain funding. The scholarship provides $7500/yr until graduation. For anyone interested in pursuing research, this is certainly a good scholarship to apply to.

I will not post my application statements here. They are far too personal, but I will share them upon email request.