DIMACS REU 2026
Name: Herbert Huachaca
Project Mentor: Paul Feehan
Email: hkh29@scarletmail.rutgers.edu
Office Location: CoRE, Room 415
University: Rutgers University
I would like to thank professor Feehan for his guidance and support throughout this project. I'd also like to thank the Rutgers Math Department and the NSF for funding my participation in DIMACS.
I am an undergraduate at Rutgers University studying mathematics. I like topology and differential geometry. If I had to bring one book to a desert island it would be Lee Smooth Manifolds. Aside from math I like reading the other works of mathematicians who are better known for other work. Shoutout Costin Alamariu and Ted K. My favorite tarot card is The Fool (0).
Gauge theory gives useful tools for studying the geometry and topology of four-dimensional manifolds. We develop the differential-geometric background needed to set up the Yang-Mills and anti-self-duality equations, including connections, curvature, differential forms, and the Hodge star operator.
We attempt to find some simple solutions to the Yang-Mills equations on 4-manifolds. In particular I will be looking at a paper by Donaldson and expound on a certain gluing lemma used in the construction. The gluing method constructs new anti-self-dual connections by combining known solutions and correcting the resulting approximate solution.
Got situated with the program. Met my roommates Arnav, John, and Ryan, went over Lee smooth manifolds chapters 1 & 2 as review. Definitions of smooth manifolds, charts, maximal atlases. Attended two REU seminars on fair division and on quantum computing, not my area but were mildly interesting I guess. Used as a guinea pig during the website workshop.
Worked on preparing the initial presentation for the project. Looked at some of the topological properties of projective space. Looked at the Yang-Mills functional, and some background material on some of the special properties of manifolds in dimension 4 specifically (Scorpan, The Wild World of 4-manifolds). Met my officemate Nathan and we got to know each other a bit, his project was pretty interesting. I finally achieved my lifelong dream of becoming a Discord mod!!! Am now owner of the 2026 DIMACS Discord server.
Learned about Vector bundles and Riemannian metrics from the relevant chapters in Lee. Sections, local trivializations, and Hermitian metrics. Looked at the construction of Riemannian metrics using partitions of unity. Met with professor to discuss some sources he wanted me to read through and what sort of background to include in the eventual paper. Played poker with Eli, Nathan, John, Roger, Adelmo, Joshua. Nearly won until everyone went all-in on the last bet.
This week I studied differential forms, the exterior derivative, and the Hodge star operator on manifolds. Looked at the relevant chapters in Lee and also the Introduction to Gauge Theory section in the book Quantum Fields and Manifold Invariants. I was absolutely ecstatic this Friday as it was my favorite holidy, Juneteenth, representing the liberation of unprecedented hordes of maladjusted slaves and second only in our country's history to Independence Day itself.
Learned more about connections and curvature on manifolds, and the local formula for curvature F_A = dA + A /\ A. Looked at Donaldson and Kruth for an invariant definition of the exterior covariant derivative. As an exercise I showed that curvature is C^inf linear. Attended Culture Day at DIMACS where I unfortunately wussed out about making a presentation on The Amazing Digital Circus culture. Went to the Ethics in Research workshop and saw a menagerie of popular scientific experiments that were either done fraudulently or said to be unethical. Was alright I guess.
Started working through the details of the gluing construction using some notes Prof. Feehan posted. The main idea is to start with two anti-self dual connections A_1 and A_2, splice them together using a cutoff function, and get an approximate solution on the connected sum. I proved the curvature expansion F_(A+a) = F_A + d_A + a/\a and started studying the curvature error formula for the spliced connection. We were met by the NYC REU and the students gave us some presentations about their projects. Dr. Z gave a quite interesting talk where he showed us a cool way to approximate Pi with the catalan numbers and then digressed about his philosophy of regressing mathematics to the concrete operational stage through ultrafinitism. As a concrete operational thinker myself, I fully support this. Independence day was pretty mid.
Started organizing the expository writeup on Overleaf. I wrote preliminary sections on the diff geo prerequisites, and the setup of the problem. I added in a section on how the Yang-Mills equation followed from taking the first variation of the Yang-Mills functional and setting it equal to 0. Also included the proof that ASD connections satisfy the Yang-Mills equation using the Bianchi identity. As an application of the contraction mapping theorem I was able to show that by making a slight peturbation a to our approximate connection A_0, we could find an anti-self dual connection on the connected sum of the two manifolds. I soyed out with Eli at the Nokia Bell Labs field trip and the next day we did the photoshoot for DIMACS. Watched Les Miserables with Nathan, Ezme, and Martin. Jean Valjean didn't do nothing doe.
This week I prepared the final presentation about the setup and the gluing construction. Went to Jane Street with Arnav, Doorva, Lucy, Jadyn, John, Joshua, and Roger. They held an Estimathon and our team won, Arnav carried hard. Then we went around NYC afterwards. Jadyn did some makeup on me at a Sephora, I bought an extremely tasteful shirt at the M&M store, and we took a group picture where Arnav bunny eared me. Super fun trip overall.
Typed up the remaining parts of the final paper. Held a watch party for the Odyssey (2026) at the Rutgers Cinema. It's one of those movies that's so bad it's good.