uzu lim

Hi, I'm Uzu Lim. I am a mathematician working on topological and geometric data analysis (TGDA). Currently, I am a postdoctoral researcher at Queen Mary University of London.

My research interests include:

Data manifold
Stochastic topology
Persistent homology
Topology in neuroscience
Singularity in data manifold
Spatial navigation and dementia

I've also co-authored research papers on the following topics:

Number theory (p-adic L-functions)
Algebraic geometry (complete intersections and Hilbert polynomials)

Education

Ph.D. in Mathematics, University of Oxford, 2019 - 2023
- Thesis title: Geometric and Topological Inference from Random Samples
- Thesis Advisors: Vidit Nanda and Harald Oberhauser
- My doctoral thesis explores several topics in the statistical inference of a data manifold from a random sample. Crucially, the data manifold may possess singularities. The thesis combines differential geometry, kernel theory, optimal transport theory, and probability theory to achieve this aim.
B.Sc. in Mathematics and Convergence IT Engineering, Pohang University of Science and Technology (Postech), 2012 - 2019
- During 2015 - 2017, I was in the Korean national military service, which is mandatory for citizens assigned male at birth.
- Convergence IT Engineering was a new bachelor's degree created in Postech. Its coursework basically involves designing each student's own engineering degree by taking various STEM courseworks that fits the student's goals. Much of my coursework was fulfilled by mathematics courses, and additionally I took some courses in electrical engineering and computer science. There were also 4 capstone project courses; I worked on robotics, algorithmic music composition, topological data analysis and machine learning in the capstone projects.
- My mathematics degree was obtained as a double major. To do this, I took all of the standard coursework in a mathematics degree. I also took a large collection of graduate school coursework in mathematics, including commutative algebra, algebraic topology, class field theory, and so on.

Research experience

(Current position) Postdoctoral Research Associate, Queen Mary University of London, 2024 - 2026
- Topic: Universal statistical law of persistent homology, theory and application
- Advisor: Omer Bobrowski and Primoz Skraba
- In this position, I work on the theory and application of the "universality law" of persistent homology statistics. This law, recently discovered by Omer Bobrowski and Primoz Skraba, concerns the empirical distribution of the multiplicative persistence, which follows the Gumbel distribution after some transformations. I am working on both extending their theoretical proof of the weak universality law to new cases, and also applying the statistical methods developed from the universality law to neuroscience.
Postdoctoral Research Associate, University of Oxford, 2023 - 2024
- Topic: Biological applications of topological data analysis
- Advisor: Heather Harrington
- During this position, I worked on the following topics: geometry of scRNA-seq transcriptomics, gradient descent for cover learning, and studying spatial navigation for early dementia screening.

Journal articles and conference proceedings

Luis Scoccola, U.L., and Heather Harrington, Cover Learning for Large-scale Topology Representation, International Conference on Machine Learning (ICML) (2025) - Paper
U.L., Harald Oberhauser, Vidit Nanda, HADES: Fast Singularity Detection with Local Measure Comparison, to appear on SIAM Journal of Mathematics of Data Science (SIMODS) (2025) - Paper, Code
U.L., Harald Oberhauser, Vidit Nanda, Tangent space and dimension estimation with the Wasserstein distance, published on SIAM Journal on Applied Algebra and Geometry (SIAGA) (2023) - Paper, Tweets
U.L., Strange random topology of the circle, Discrete and Computational Geometry (DCG), special issue on SoCG (to appear) - Paper, Tweets, Seminar

Chris Eur, S.H.L., Complete intersections with given Hilbert polynomials, published on Journal of Commutative Algebra (2021) - Paper

Preprints

Jeehoon Park, S.H.L., The Milnor K-theory and the Shintani cocycle, Arxiv preprint - Paper
David Beers, Heather Harrington, Jacob Leygonie, U.L., Louis Theran, Fibers of point cloud persistence, Arxiv preprint (2025) - Paper
{U.L., Rodrigo Leal-Cervantes}, Gillian Coughlan, Renaud Lambiotte, Hugo Spiers, Michael Hornberger, Heather Harrington, Navigation geometry identifies Alzheimer's disease and its genetic risk, MedRXiv preprint (2023) - Paper

In preparation:

U.L., Primoz Skraba, Omer Bobrowski, Universal topological statistics on triangulable spaces
U.L., Yasser Roudi, Omer Bobrowski, Primoz Skraba, Topological hypothesis testing for the toroidal geometry of grid-cell neurons
{U.L., Eng-Jon Ong}, Omer Bobrowski, Primoz Skraba, Universal neighbourhood statistics and intrinsic dimension estimation

Expository articles

I wrote expository articles on classical Hodge theory and local class field theory at the end of my undergraduate studies, although they were uploaded on Arxiv some time later simply because I didn't think about uploading it previously. Note that these articles are written purely for explaining known classical theory, and they are not my original research works.

Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem, Arxiv preprint - Paper
Cohomology of p-adic fields and Local class field theory, Arxiv preprint - Paper

Teaching experience

I enjoy teaching new topics I learn. I had specifically learned mathematics of quantum physics and numerical linear algebra while tutoring the courses below. The courses on algebraic number theory and algebraic topology were taught from my prior knowledge. *Please note that Tutor and TA are both teaching assistant positions.

Tutor, Further quantum theory (2022, University of Oxford)
Tutor, Numerical linear algebra (2022, University of Oxford)
TA, Algebraic number theory (2020, University of Oxford)
TA, Algebraic topology (2019, University of Oxford)

Code repositories

U.L., HADES: Fast singularity detection with local measure comparison: Github repository

Luis Scoccola, U.L., ShapeDiscover: Cover learning with geometric optimisation: Github repository

Other stuff

I like learning about quantum field theory and other topics in physics. There are some interesting applications of TDA to lattice gauge theory. I can follow some scattering amplitude calculations and I can also parse some standard constructions in second quantisation but it's pretty hard to understand this QFT on a level that I can actually apply in practice, so to speak. Best to wait until I can write a physics paper at some point.
My favourite theorems in mathematics are probably the Central limit theorem, Fundamental theorem of calculus, and the Spectral theorem. Some specialty stuff I really like include: the Kodaira embedding theorem, the Prime number theorem, my own singularity detection theorem in the Hades paper (see the long Appendix C featuring 30+ page long proof), application of the Weyl's tube formula for p-value estimation of a Gaussian random field, ...
Musicians I like currently include Japanese Breakfast, Charli XCX, Sungazer, and Balming Tiger. In the past I dug pretty deep into Dream Theater and Porcupine Tree. Then onto jazz fusion like Snarky Puppy and Jacob Collier. (This is sort of predictable) Another persisting musical interest of mine is video game music. Lena Raine of Celeste soundtrack and Danny Baranowsky of Binding of Isaac are two of my favourites. Some other deep cuts include Hakushi Hasegawa, Loro's, Igorrr, ...
Understanding the psychosoma and my complicated identities are a very important self-research topic to me. It's an empirical science with a single scientist with an exclusive access to the observation data known as the qualia. For this I have explored themes in Mahayana Buddhism and worked with the interoception of my own Qi (기).

Contact

email: x@qmul.ac.uk and y@gmail.com, where x = sung.h.lim and y = uzulimsh
I also go by Sung Hyun Lim (임성현), which is my birth name in Korean.

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