William
Chang
PhD student in Applied Mathematics at UCLA.
About
I'm a PhD student in Applied Mathematics at the University of California, Los Angeles, where I've been since Fall 2023. Before that I earned a B.A. in Mathematics from the University of Southern California in 2022.
My research interests are broad and span mathematics and machine learning. See my papers below for more.
I'm also the founder of BruinML, a research group at UCLA where I mentor students on projects in machine learning and mathematics.
I enjoy tutoring and take students in math and physics at every level, from elementary school through university. Email me to book a session.
Outside of research I practice Taekwondo and sing. My covers are on YouTube and Spotify.
Research
Mentoring
I run BruinML, a UCLA machine learning and mathematics research group.
Since Spring 2024 I've mentored undergraduates through research projects from first idea to submission. Many of the papers above were led by students in the group, with work appearing at NeurIPS, UAI, EWRL, Asilomar and TMLR.
If you're a student interested in a project, send me an email with a bit about your background.
ProjectsSmall things I've built
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Proof Paths
How the Lean proof assistant checks a proof, one tactic at a time; what a theorem rests on; why two proofs of the same fact can be genuinely different; and a game where the shortest proof is usually a lemma someone already proved.
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Crease Lab
Fold paper by hand in a bar-and-hinge simulator, teach a folded tube to crawl with no wheels, and download a printable template, servo plan and Arduino code to build the real robot.
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PartForge
Build parts out of logic gates, assemble them into bigger parts, and wire one up as a robot's brain. The example robot follows a line using two inverters made from NAND gates.
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Model Router
Which requests should the $0.001 model get, and which the frontier model? Drop in an eval log, set a quality target, and get the cheapest cascade or router that meets it, certified on held-out prompts. Runs in your browser.
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Room Map
Everyone in the room scans a QR code and types one phrase; the phrases land on a big-screen map where similar ones sit together. Find yourself, find your neighbours, and see what “similar” means to a model.
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Feedback Loop
A recommender only learns from what it shows, so it keeps showing what it already knows. Rate a dozen things, watch 400 readers' feeds narrow over 30 rounds, then switch on exploration and see the catalogue open back up.
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Shortcut Hunt
A puzzle game about data leakage: eight models that score too well, and you guess what each is really using before it retrains without its shortcut. Then run the same checks on your own CSV.
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Robot Sumo
Two-wheeled robots shoving each other out of a ring. Small neural nets trained by evolution strategies and self-play. Drive one yourself and try to beat each level.
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Selfplay Chess
A chess engine that learned its own piece values by playing itself, starting from all pieces equal. Play against any of its four levels.
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Pianoify
Drop in a song and get a simplified piano arrangement: the chords, the melody and sheet music you can play. Beat, chord and melody detection run in your browser.
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Scorepad
Sheet music you write together, like a shared doc. Type or tap in notes, hear them played back, and invite people to edit the same score live, with cursors and version history.
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Cover Me
Sing six notes and a few words, pick any song, and hear its melody sung back in your own voice. Pitch shifting by PSOLA, all in the browser.
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Unmix
Pull the singer out of a song, right in your browser: drop in a track and get the voice and the backing as separate files. No training data, nothing uploaded.
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Liner Lyric Studio
Load a song, paste timestamped lyrics (or tap along to time them), and turn them into an animated lyric video whose scene follows the song's mood, read by a small model that runs in your browser. Exports the finished video.
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Dataset Doctor
Drop in a CSV and find the problems that quietly break machine-learning models: leaking columns, test rows copied from training, contradictory labels, hidden placeholders. Then download the fix.
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Agent Blame
When a multi-step LLM pipeline gets an answer wrong, find which step caused it. Swap steps for better versions with the randomness held fixed, and watch Shapley values split the blame.
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Bandits Lab
Race bandit algorithms against the Lai–Robbins lower bound in your browser, and plan an A/B test: see whether a fixed split or Thompson sampling costs fewer conversions.
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Classifier Toolkit
See how far to trust a classifier: honest confidence, prediction sets with a coverage guarantee, abstention with a certified error rate and drift alarms. Try it on three models or your own CSV.
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Theory Demos
Five ideas from ML theory you can play with in the browser: double descent, high-dimensional geometry, diffusion models that copy their training data, private polls and certified robustness.
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Run List
A shared bucket list sorted like ski runs, from easy nights in to passport trips. Make an account, start a list and invite people to it.
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Figure Lab
Describe a math figure in words and get an interactive one you can rotate, slice and share. Starts with a drilled sphere, solids of revolution, Riemann sums and more.
TeachingTeaching assistant, UCLA Mathematics
- Fall 2024Math 131A Real Analysis
- Fall 2024Math 134 Linear and Nonlinear Systems of Differential Equations
- Spring 2024Math 156 Machine Learning
- Winter 2024Math 32B Calculus of Several Variables
- Fall 2023Math 31A Differential and Integral Calculus
I've also taught at the UCLA Math Circle since Fall 2023.
Study toolsInteractive figures, worked examples and concept checks, one subject at a time
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Smooth Manifolds
Charts, tangent vectors, immersions, level sets, flows and the Lie bracket, forms and Stokes. Drag the figures, watch the animation, then test yourself. Follows Lee's Introduction to Smooth Manifolds.
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Connections and Curvature
Curves, the shape operator, connections, parallel transport, geodesics, Gauss–Bonnet and characteristic classes. Bend the surfaces, carry vectors around loops, then test yourself. Follows Tu's Differential Geometry: Connections, Curvature, and Characteristic Classes.
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Modular Forms
The modular group and its fundamental domain, lattices and j, congruence subgroups, dimension formulas, Eisenstein series, Hecke operators, the group law and point counts mod p. Reduce points, test identities on q-expansions, then test yourself. Follows Diamond and Shurman's A First Course in Modular Forms.
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Algebraic Geometry
Zariski topology, projective closure, blowing up, Bézout, Spec ℤ, cohomology of projective space, Riemann–Roch, Hurwitz, the group law on a cubic and the 27 lines. Drag the curves, watch the blow-up, then test yourself. Follows Hartshorne's Algebraic Geometry.
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Real Algebraic Geometry
Descartes, Sturm and Hermite root counting, Puiseux series, cylindrical decomposition, Morse bounds, subresultants, stability, root isolation and quantifier elimination. Edit the polynomials, step through the algorithms, then test yourself. Follows Basu, Pollack and Roy's Algorithms in Real Algebraic Geometry.
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Quantum Computation
Qubits and gates on the Bloch sphere, teleportation, Bell tests, circuits, period finding, Grover search, noise, error correction, entropy and key distribution. Click gates, run the simulators, then test yourself. Follows Nielsen and Chuang's Quantum Computation and Quantum Information.
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Computer Systems
Bits and floats, the stack, pipelines, caches, linking, fork, virtual memory, malloc and thread races. Toggle bits, step through calls, run a cache, then test yourself. Follows Bryant and O'Hallaron's Computer Systems: A Programmer's Perspective.
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Probability and Information
Counting, Bayes, random variables, entropy and mutual information, Huffman codes, channel capacity, the central limit theorem and Markov chains. Run the simulations, watch a code being built, then test yourself. Follows Applebaum's Probability and Information.
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High-Dimensional Probability
Hoeffding and Chernoff, random vectors and matrices, Johnson–Lindenstrauss, Hanson–Wright, Gaussian width, VC dimension and sparse recovery. Run the simulations, place the points, then test yourself. Follows Vershynin's High-Dimensional Probability.
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Continuous-Time Markov Chains
Generators and sample paths, the forward equation, fast chains and quasi-stationary limits, aggregation, occupation measures, and control on two time scales. Run the chains, slide ε, then test yourself. Follows Yin and Zhang's Continuous-Time Markov Chains and Applications.
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Information Geometry
Divergences, dual coordinates, the Fisher metric, e- and m-geodesics, EM, natural gradient and ICA. Drag distributions around the simplex, run small learning algorithms, then test yourself. Follows Amari's Information Geometry and Its Applications.
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Gradient Flows
Minimizing movements, λ-convexity, optimal transport and the Wasserstein distance, displacement convexity, and heat and Fokker–Planck as gradient flows of entropy. Run the schemes, edit the histograms, then test yourself. Follows Ambrosio, Gigli and Savaré's Gradient Flows.
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Graph Ramsey Theory
Colouring games, Paley graphs, random colourings, the local lemma, Turán graphs, Shannon capacity, quasi-randomness, van der Waerden and bipartite Ramsey numbers. Colour the edges, run the experiments, then test yourself. Follows Li and Lin's Elementary Methods of Graph Ramsey Theory.
Background
Education
- University of California, Los Angeles 2023–2027PhD, Applied Mathematics. Passed qualifying exams in Basic (Fall 2023), Applied Differential Equations (Fall 2023) and Geometry/Topology (Spring 2024).
- University of Southern California 2018–2022B.A., Mathematics
Experience
- Northrop Grumman Summer 2022Intern, Communications Systems and Mission Payload, Space Systems. Analyzed slant ranges and channel losses.
- Baruch College Discrete Math REU Summer 2021Asymptotic bounds on cliques in Kt-minor-free graphs, with Fan Wei.
- UCLA Applied Math REU 2020–2021Thin-film equations for pleural fluid in the lungs, with Hangjie Ji.
- USC Math Center 2019–2020Tutor for linear algebra, calculus, probability, number theory and analysis.
Fellowships and awards
- NSF EAGER REU Fellowship 2021–2022Multi-agent bandits with information asymmetry, with Rahul Jain.
- USC CURVE Fellowship 2021, 2021–2022Decentralized multiplayer bandits (Rahul Jain); stability of physics-informed neural networks (Assad Oberai).
- USC Provost Undergraduate Research Fellowship 2020–2022Matrix varieties (Robert Guralnick); e-positivity of chromatic polynomials (Greta Panova).
- Putnam Competition 2018–2021Top 500 in 2020; USC representative four years.
Talks and service
- ReviewerICLR and NeurIPS (2023, 2024); IFAC and CDC (2023).
- IEEE CDC 2022Online learning for cooperative multi-player multi-armed bandits.
- Asilomar 2022Approximation capabilities of neural networks using morphological perceptrons.
- Joint Mathematics Meetings 2022Undergraduate poster: maximal cliques in Kt-minor-free graphs.