Tour of Algebraic Number Theory I: Number Fields and Number Rings (with Timothy Cho)
This is the first part of a hopefully year-long DRP sequence in algebraic number theory. For the first quarter, we will discuss the basic object of study in algebraic number theory, namely number fields: we may regard a number field as a subfield of the complex numbers that has finite degree as a vector space over the field of rational numbers.
The basic principle for this quarter is that Q, the field of rational numbers, is the field of fractions of Z, the integers. Given a number field K (which contains Q), we will find a subring R of K that "behaves like the integers in the rationals" in a certain well-defined sense. Along the way, we will see where the analogies between R and Z break down, which will lead us to consider questions regarding factorization in R, which can be studied using a tool we will develop: the ideal class group.
Prerequisites: 120B, 121A (120C/230B and 121B recommended)
Hands on Quantum Error Correction (with William Ren)
Much of quantum error correction can be understood by constructing circuits whose ideal behavior produces deterministic parity relationships among measurement outcomes. The perspective is similar to discrete mathematics and classical circuit logic: simple operations are combined to create predictable global behavior. Quantum circuits differ because qubits can become entangled, measurements can disturb quantum states, and every physical operation is noisy. Error-correcting circuits address this by measuring carefully chosen correlations among qubits, called syndromes, which reveal information about errors without directly measuring the protected logical state. After developing the necessary background in qubits, Pauli operators, and parity checks, we will build and inspect small examples using Crumble and learn to simulate noisy stabilizer circuits using Stim. We will begin with repetition codes and syndrome-extraction circuits and, depending on the group’s interests, progress toward surface-code circuits, decoding sampled errors, or plotting logical error rates using tools such as PyMatching or Sinter. The goal is practical working knowledge rather than a comprehensive treatment of quantum mechanics or fault tolerance. If this becomes too ambitious we can read Quantum Computing in the Age of Democritus by Scott Aaronson.
Prerequisites: 3A, 13 (121A, familiarity with Python preferred)
PDE Modeling and Simulation (with Jaaziel de la Luz)
This reading will focus on modeling and simulating phenomena using partial differential equations. Simulation is useful in theoretical and industrial applications where no closed-form solution exists (or even when intuition is lacking). Participants will progress from PDE classification and boundary/initial conditions, finite difference methods for the heat, wave, and Laplace/Poisson equations, to nonlinear and conservation-law problems, mesh-based finite element modeling, and coupled real-world systems like reaction-diffusion pattern formation and fluid dynamics. The reading will culminate in an independent capstone project where each participant models and validates a complete simulation of a phenomenon of their choosing.
No formal PDE knowledge is required, but familiarity with the basics of MATLAB (or Python/ C++) are expected.
Prerequisites: 3A, 3D, 9
Deep Hedging, Bellman Methods, and Optimization in Mathematical Finance (with Natanael Alpay)
This reading course will study optimization and decision-making in mathematical finance through the papers Deep Hedging and Deep Bellman Hedging. We will examine how hedging problems are formulated in markets with transaction costs, trading constraints, and risk preferences, and how neural-network and dynamic-programming methods are used to approximate optimal strategies. Particular attention will be given to convex risk measures, stochastic optimization, Bellman equations, and the mathematical arguments supporting the methods. The two papers will provide the main structure of the course, while leaving room to explore related mathematical questions that naturally arise from the readings.
Prerequisites: 130A, 140A
Algebraic Topology: Cohomology of Spaces (with Tyler Perkins and Sireesh Vinnakota)
This quarter will serve as a rigorous introduction to algebraic topology. We’ll begin by studying fundamental and higher homotopy groups of topological spaces. By approximately week four we’ll transition to discussing singular (co)homology and techniques for computing it. By the end of the quarter we’ll discuss de Rham cohomology on smooth manifolds; the ultimate goal is to cover the de Rham theorem. Students should expect to give two short talks throughout the quarter, one a board talk and one a slide talk. Topics will be suggested by the mentors, but will likely look like explaining the solution to a computation or proving a corollary of a foundational theorem discussed during a weekly meeting.
Prerequisites: 13, 120A, 140A (or equivalents)
An Introduction to Algebraic Geometry (with Aurora Vogel)
Algebraic geometry offers a bridge between algebra and geometry for geometric objects which roughly look like the zero sets of polynomials. For such objects, understanding their geometry is studying their algebra. This reading course will illustrate this through theorems such as Hilbert Nullstellensatz (giving a corresponding between these zero sets and ideals of polynomial rings) and Bezout's theorem (see the goals for the course for more detail) which roughly says that the intersection of the zero sets of two polynomials (two plane curves) of degree m and n (when viewed in the right space), intersect exactly at m*n points.
This reading course will use Milne's algebraic geometry notes as well as Fulton's which only assume some familiarity with basic objects of abstract algebra (such as rings and fields) and are both freely available for students.
Prerequisites: 120B, 121A
An Introduction to Smooth Manifolds (with Trent Lucas)
Our plan is to learn the basics of smooth manifold theory: smooth manifolds, smooth maps, tangent spaces, vector fields, and more. These tools give us a way to rigorously study abstract and high-dimensional shapes and spaces that show up in fields like geometry and physics. Time permitting, our project will culminate with the famous Poincar\'e-Hopf theorem, which provides a bridge between analysis, topology, and geometry.
Prerequisites: 140, (160 recommended)