• ICM and STAI-X 2026 Notes
• Advanced Calculus by Prof Dafermos

  A collection of simple lemmas that are frequently used in proofs in Fourier analysis, when developed using Riemann integration.

• Fourier Analysis by Prof Dafermos

  A summary of what we covered in Fourier analysis.

• Measure Theory by Dr Glaudo

  A summary of what we covered in measure theory and real analysis.

• Differential Geometry Notes by Prof Dafermos

  An incomplete collection of notes from the differential geometry course at Princeton.

• Differential Geometry Problems by Prof Dafermos

  These were our differential geometry homework problems. (Some, we never got around to solving.)

• Representation Theory Notes by Prof Laga

  An incomplete summary of what we covered in our course on representation theory.

• Probability Review under Prof Shkolnikov

  A summary of what we covered in probability theory.

• Stochastic Calculus Review under Prof Hubert

  A summary of what we covered in stochastic calculus.

• Differential Inequalities for the Random Cluster Model under Prof Aizenman

  My junior paper at Princeton: the derivation of a differential inequality that sheds light on the critical behavior of the Random Cluster Model (RCM) from statistical physics. The RCM is a dependent generalization of Bernoulli percolation, which has direct connections to the Ising and Potts models of ferromagnetism. The key input is a new concentration inequality for "decision trees" from the theoretical computer science community. I plan to elaborate on the broader role of differential inequalities in the study of phase transitions, as at first glance this might not be clear to the reader. I also plan to rewrite Section 4.1, whose exposition at present is a bit rushed.

• Conformal Invariance in Percolation under Prof Aizenman

  One of the most striking results in percolation theory in two dimensions is conformal invariance, which states that when one takes an appropriate scaling limit of Bernoulli percolation on the integer lattice, crossing probabilities are invariant under conformal maps. This junior seminar paper explores implications of conformal invariance. In particular, I derive the crossing probability of an annulus up to constants.

• Random Field Ising Model under Prof van Handel

  The study of disordered models in statistical mechanics is infamously challenging. The ultimate goal is to understand realistic models of magnets with impurities, like the Edwards--Anderson model; but at present, this is hopelessly out of reach. Here, we instead start with one of the simplest disordered models, the Random Field Ising Model (RFIM), and in this note I present an exposition of the proof that the RFIM does not exhibit a phase transition. The key idea is to treat the free energy as a function of the gaussian external field random variables, and then to leverage gaussian integration by parts and gaussian concentration. Once these are applied, bounding the free energy reduces to combinatorics. I plan on rewriting parts of this.

• Cryo-EM Notes under Prof Singer and Prof Moscovich

  Incomplete notes on the math of cryo-electron microscopy.

• Multimode Value Iterations under Prof Gee and Prof Vladimirsky

  Consider a partially-deterministic control problem where an agent navigates two-dimensional space, switching between multiple kinds of dynamics ("modes"). In this incomplete note, we analyze the convergence of value iterations for computing the value function, both in general, as well as under a monotonicity assumption on the mode transitions.

• Elliptic Curve Cryptography under Dr Katharine Woo

  A summary of what I learned about elliptic curves and their applications to cryptography through Princeton's Directed Reading Program. Many of the cryptographic protocols listed in the report can be performed using an arbitrary abelian group. However, the group of rational points on an elliptic curve has a number of nice properties that can be leveraged.

• Elliptic Curve Cryptography Notes

  The incomplete personal notes that were used to prepare the report above. The material on elliptic functions served as motivation for me to learn complex analysis.

• Congruence of Linear Symplectic Forms under Prof Tang
• Degenerate Turán Problem and Its Variants under Prof Jiang
• Math Colloquium: Prof Deligne

  Incomplete notes from a talk on the hidden symmetries of algebraic varieties that went straight over my head.

• Math Colloquium: Prof Fefferman

  Incomplete notes from a talk on interpolation and its relation to metric geometry.

• Math Colloquium: Prof Sarnak

  Incomplete notes from a talk on the relationship between Diophantine equations and optimally navigating the special unitary group, with applications to the design of gates for quantum computing. (I am still trying to find my notes from this talk.)

• PUMaC Problems I Wrote

• An Overview of Statistical Methodology by Prof Katsevich

  A gentle overview of the topics covered in the statistical methodology course at Penn, spanning linear models and their geometry, generalized linear models, statistical inference, resampling methods, model misspecification, robust statistics, the basics of multiple hypothesis testing, and quite a few interesting tidbits from the history of statistics (whose early days seem particularly messy and controversial compared to other parts of the mathematical sciences).

• Mathematical Statistics Review under Prof Cattaneo

  A summary of what we covered in mathematical statistics.

• Mathematical Statistics Notes by Prof Bhattacharya

  Handwritten notes from the mathematical statistics course at Penn.

• High-Dimensional Asymptotics

  Notes on high-dimensional statistics, including the basics of random matrix theory and universality.

• Statistical Physics for ML

  A leisurely overview of topics in equilibrium statistical mechanics, with an emphasis on spin glasses, and with an explanation of how such concepts relate to modern problems in high-dimensional statistics and optimization.

• Estimation in Ising Models under Prof Bhattacharya

  An exposition of results concerning the estimation of the inverse-temperature and external field in Ising models. Maximum likelihood estimation is intractable in this setting due to the usual difficulties (?) posed by the partition function. Luckily, the maximum pseudolikelihood estimator is consistent and achieves low asymptotic variance. Taking a step back, I find this problem a bit unnatural from an applied perspective.

• VAEs and the EM Algorithm on Dwip's Blog

  A blog post Dwip and I wrote about variational autoencoders, an early form of generative AI.

• Diffusion Models

  An incomplete overview of diffusion models, and the related theory of stochastic localization. Since writing these notes, I've greatly clarified my understanding of stochastic localization with the help of Shervin Khalafi, and plan on updating the file accordingly.

• Quantifying single-cell differential abundance with density difference estimation on manifolds under Prof Chitra, Dr Gold, and Prof Raphael
• Power-Law Graph Alignment under Prof Sly

  The subset of my senior thesis that does not appear in existing papers. My broader thesis was on the alignment of correlated random graphs. Graph alignment algorithms can be used to positive ends (aligning protein--protein interaction networks), as well as to less positive ends (deanonymizing social networks).

• AI Alignment Course Notes under Prof Roth
• Quantitative Economics Plan

• Ruminations on AI Safety

  Many mathematicians have begun to think seriously about AI safety, recognizing both the sea change taking place in their native fields, as well as the practical urgency of alignment. In this file, I record some of my preliminary thoughts.

• Perspective on Future of Papers

  A personal perspective on how the academic publishing system in applied mathematics should evolve in this disorienting new era of superhuman AI.

• Perspective on Rating One's Own Papers

  As a beginning graduate student, one aspect of academic culture that I found particularly disorienting was the tendency for all researchers to sell their latest paper as "the next big thing". (The advance of AI has only served to exacerbate this problem, in my experience.) In reality, most of a researcher's scholarly output serves merely to fulfill one of many obligations (either to a grant that funds their lab, or to students who must be trained by working on simple research problems), and such work has no ambition of changing the world. Of course, this doesn't stop researchers from overselling their papers, due to various perverse incentives that exist. In this incomplete note, I brainstorm schemes to address this problem, taking inspiration from Prof Su's "rank your own paper" mechanism for improved peer review in ML conferences. I first imagine an ideal world where tenured researchers are honest and attempt themselves to construct (absolute, not relative) scores for each of their own papers. I then acknowledge that such a system is extremely unlikely to take hold in academia, even among researchers with job security; and as a result, propose a scheme to estimate the unobservable quality of a given lab's papers based on observable factors, like the venue it is submitted to, whether or not it is featured on the lab website, the amount of social media promotion that was attempted, and so on. The goal is not at all to denigrate particular papers, but simply to help new researchers obtain a "calibrated" view of the research literature.

• Letter to Freshman Me

  Recently, a freshman at Princeton wrote to me, asking what I do and how I got here. I'm not sure if he was impressed by or dismayed at my path through life thus far, but I gave him my advice, with the wisdom afforded by two long years outside of the orange bubble. I think some points apply pretty broadly.

My Seminar Notes

• Meena Jagadeesan — 10/29/26
• Tim Roughgarden — 10/27/26
• Lars Peter Hansen — 10/27/26
• Mateo Diaz — 09/17/26
• Zhiheng You — 09/14/26
• Morgane Austern — 09/10/26

• Princeton Lectures in Analysis by Prof Stein and Dr Shakarchi
• The Definition of Numerical Analysis by Prof Trefethen
• A First Course in the Numerical Analysis of Differential Equations by Prof Iserles
• Numerical Linear Algebra by Prof Trefethen and Prof Bau
• Representation Theory Notes by Prof Laga
• Probability in High Dimension by Prof van Handel
• Introduction to Bernoulli Percolation by Prof Duminil-Copin
• Open Problems in Mathematical Physics by Prof Aizenman
• Mathematics of Data Science by Prof Bandeira, Prof Singer, and Prof Strohmer
• The Mean-Field Dynamics of Transformers by Prof Rigollet
• Math of Cryo-EM by Prof Singer
• Foundations and Trends in Optimization, edited by Prof Donoho and Prof Candès
• Foundations and Trends in Communications and Information Theory, edited by Prof Mazumdar
• SolveAll by Edgar

• History of Statistics
• Theory of Statistics by Prof Candès
• Maximum Likelihood: An Introduction by Prof Le Cam
• Semiparametric Estimation Overview by Prof Kennedy
• Hypothesis testing with e-values by Prof Ramdas and Prof Wang
• Theoretical Foundations of Conformal Prediction by Dr Angelopoulos, Prof Foygel Barber, and Prof Bates
• Statistical Optimal Transport by Prof Chewi, Prof Niles-Weed, and Prof Rigollet
• Statistical Physics Methods in Optimization and Machine Learning by Prof Zdeborová and Prof Krzakala
• Graphical Models Concepts in Compressed Sensing by Prof Montanari
• Community Detection and Stochastic Block Models: Recent Developments by Prof Abbe
• A Selective Overview of Deep Learning by Prof Fan and others
• Statistical Perspective on Diffusion Models by Prof Montanari
• Eleven Grand Challenges in Single-Cell Data Science by Prof Raphael and others
• The Visual Display of Quantitative Information by Prof Tufte
• Foundations and Trends in Machine Learning, edited by Prof Roth and Edgar
• Foundations and Trends in Statistics, edited by Edgar
• Edgar's Educational Materials
• Prof Mei's Lecture Notes
• Hamed's Tutorials

• Math Courses by Prof Sargent
• QuantEcon: resources to help people with a math background get started in quantitative economics research.

• Inspirational Interview 3: Prof Elias Stein (Science Lives)
• Inspirational Interview 4: Prof Yves Meyer (Abel Prize)
• Inspirational Interview 5: Prof Lennart Carleson (Abel Prize)
• Inspirational Interview 6: Prof Talagrand at Norwegian Academy
• Inspirational Interview 7: Prof Yakov Sinai (Abel Prize)
• Inspirational Interview 8: Prof Srinivasa Varadhan (Abel Prize)
• Inspirational Interview 10: Dr Jim Simons with Numberphile
• Jim Simons, Businessman and Mathematician — IHES
• Fields Medal to Prof Mirzakhani
• Fields Medal for Prof Duminil-Copin
• Cédric Villani: En route Médaille Fields (2010)
• What's So Sexy About Math? by Prof Villani
• The Joy of Math by Prof Villani
• Théorème vivant by Prof Villani
• Birth of a Theorem by Prof Villani at IAS
• Mathematics: Beauty vs Utility — Numberphile
• Hardy-Littlewood Rules
• Mathematica by Dr Bessis
• Music on a Clear Möbius Strip
• PaganiniBench for physical AI
• Inspirational Documentary on Claude Shannon
• Inspirational Documentary about John von Neumann
• Inspirational Interview 11: Warren McCulloch at his home
• Inspirational Interview 12: J. Robert Oppenheimer at IAS

• Inspirational Interview 1: Prof Donoho at HKUST
• Inspirational Interview 2: Prof Donoho at National University of Singapore

  "At times I practically lived in the computer room almost 24 hours a day. I remember very clearly the room — named after Princeton statistician Sam Wilks. There was an oil painting of Wilks, the minicomputer, Wilks' personal books, the old-fashioned teletype terminals, the fancier pin writer terminals, the tape drives, the clackclack, buzz-buzz sounds everything made. I remember all the strange things that came with staying up late at night working on the computer in this place. Sometimes the computer users held "afternoon tea" at 3 am in the Math lounge!"

• David Donoho: Pioneer of Data Science and Mathematical Foundations
• E-Mail M-words by Prof Donoho
• Carl Friedrich Gauss Prize 2018: Prof Donoho
• Inspirational Interview 9: Charles Stein for Statistical Science
• Memorial: Charles Stein
• Memorial: Lucien Le Cam
• The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century by Prof Salsburg
• The Thinking Eye by Prof Tufte (MIT Sloan Sports Analytics Conference, 2015)

• Inspirational Interview 13: Thomas Sargent at the Federal Reserve Bank of Minneapolis
• Lunch and Conversation with Prof Sargent at UChicago
• What Economic Researchers Do by Prof Sargent
• Lectures on Macroeconometrics by Prof Sargent and Prof Sims in Sweden
• Inspirational Interview 14: Christopher Sims at the Federal Reserve Bank of Minneapolis
• Information Theory in Economics by Prof Sims
• Inspirational Interview 15: Michael Woodford at the Federal Reserve Bank of Minneapolis
• Inspirational Interview 16: Prof Lars Peter Hansen
• Climate Risk by Prof Hansen and Prof Diebold
• Inspirational Interview 17: Michael Lewis on The Big Short, High-Frequency Trading, and the Modern Financial Industry at UC Berkeley Haas
• Inspirational Interview 18: Michael Lewis on The Undoing Project at 92NY
• My Economist Father by Alethea Black

  "He liked philosophy, and he and my mother named me Alethea, the Greek word for truth. For years I believed he was tipping his cap to the idea of absolutes and would cheerfully say as much at parties. Then one day my mother overheard me and said: 'No. We got your name from a TV show'—it was a guest actress in 'Judd, for the Defense.' I laughed, humbled, and as I did, I could have sworn that somewhere, far from the world’s equations and signs, far beyond the manufactured clouds and flaming toasters, way out yonder where light has no speed, my father was laughing, too."