Numerical Linear Algebra (with Elisha Dayag)
Most continuous problems in the sciences (e.g. problems involving derivatives or integration) can be reduced to matrix equations. From a computational perspective, numerical linear algebra is the study of how to solve these matrix equations on a computer both as efficiently and with as little rounding errors as possible. From a theoretical perspective, numerical linear algebra is functional analysis on finite dimensional spaces. In our study we will learn about matrix norms, QR and SVD decompositions, and stability and accuracy of numerical algorithms. We will learn to use numpy (the package for scientific computing with python) to write matrix algorithms. If time permits we can explore other topics such as eigenvalue problems and iterative methods.
Prerequisites: 3A, 9A, 13, (105A, 121A, 130A strongly preferred)
Mean-Variance Analysis and Portfolio Optimization (with Michael Womack)
This course will introduce basic financial instruments, how to price them and basic ideas in portfolio optimization. The goal is to understand and implement the CPAM for a portfolio in Python. If time permits we will also discuss the Black-Scholes pricing model.
Prerequisites: 130A, (130B, 134B strongly preferred)
Introduction to Representation Theory: Lie Groups and Lie Algebras (with Timothy Cho)
This is a continuation of the representation theory DRP from last quarter, though last quarter is not a strict prerequisite. In this course, we lay the groundwork for studying Lie groups and Lie algebras, and we will look at the representations of certain examples of Lie groups towards the second half of the quarter.
Prerequisites: 120A, 121B, (120B strongly preferred)
Dynamical Systems and Mathematical Logic (with Jaaziel de la Luz)
This is a continuation of last quarter's DRP on mathematical control theory. Topics to be covered include the nonlinear controls, calculus of variations, the Pontryagin maximization principle, dynamic programming, and game theory. The reading will culminate in a group project.
Prerequisites: 3AD, 13, 140AB
High-Dimensional Probability and Machine Learning (with Natanael Alpay)
The DRP will study random vectors and random matrices in high dimensions, with an emphasis on tools used in modern machine learning and theoretical computer science. We will focus on concentration of measure (e.g., subgaussian/subexponential tails), random projections and dimension reduction (e.g., Johnson–Lindenstrauss–type results), and basic random matrix bounds (e.g., singular values of random matrices). The goal is to understand the main proof techniques (nets, symmetrization, Bernstein/Hoeffding-type inequalities, and Gaussian width ideas) and then connect them to applications such as compressed sensing, covariance estimation, and randomized algorithms.
Prerequisites: 13, 121A, 130A, 140A
Topics in Algebraic and Complex Geometry (with Sireesh Vinnakota)
It was known as far back as the Babylonians that it was possible to write down a function that took in coefficients of a quadratic polynomial and returned the roots. The 1500s-era Italian school brought similar formulas for the cubic and quartic, and the Abel-Ruffini theorem and Galois theory presented us with a concrete obstruction to such a formula using radicals for any degree five or higher. The goal of this directed reading is to acquire the topology, algebra, category theory, and geometry skills necessary to understand this story, as well as the developments since its reintroduction to the literature in the past thirty years.
Prerequisites: 13, 120A, (147 is strongly preferred)
Algebraic Geometry and Moduli Spaces (with Aurora Vogel and Brooke Wei)
We plan on continuing reading Shafarevich and build up the necessary background knowledge to understand some basic notions of moduli theory, including geometric intuitions, functorial descriptions, moduli spaces of curves, and if possible weighted moduli spaces
Prerequisites: 120BC, (some algebraic geometry from previous quarters is highly recommended)