Courses 2025-26

This course will introduce students to research areas in CS through weekly overview talks by Caltech faculty and aimed at first-year undergraduates. More senior students may wish to take the course to gain an understanding of the scope of research in computing and mathematical sciences. Graded pass/fail.

Part a: Covers the basic algebraic, geometric, and topological properties of normed linear spaces, inner-product spaces and linear maps. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations. Topics: Completeness, Banach spaces (l_p, L_p), Hilbert spaces (weighted l_2, L_2 spaces), introduction to Fourier transform, Fourier series and Sobolev spaces, Banach spaces of linear operators, duality and weak convergence, density, separability, completion, Schauder bases, continuous and compact embedding, compact operators, orthogonality, Lax-Milgram, Spectral Theorem and SVD for compact operators, integral operators, Jordan normal form. Part b: Continuation of ACM 107a, developing new material and providing further details on some topics already covered. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.Topics: Review of Banach spaces, Hilbert spaces, Linear Operators, and Duality, Hahn-Banach Theorem, Open Mapping and Closed Graph Theorem, Uniform Boundedness Principle, The Fourier transform (L1, L2, Schwartz space theory), Sobolev spaces (W^s,p, H^s), Sobolev embedding theorem, Trace theorem Spectral Theorem, Compact operators, Ascoli Arzela theorem, Contraction Mapping Principle, with applications to the Implicit Function Theorem and ODEs, Calculus of Variations (differential calculus, existence of extrema, Gamma-convergence, gradient flows) Applications to Inverse Problems (Tikhonov regularization, imaging applications).

This course offers a rigorous introduction to probability theory with applications to computational mathematics. Emphasis is placed on nonasymptotic properties of probability models, rather than classical limit theorems. Topics include measure theory, integration, product measures, probability spaces, random variables and expectation, moments, Lp spaces, orthogonality, independence, concentration inequalities, distances between probability measures, the Berry-Esseen theorem, conditional expectation, and conditioning for Gaussian families.

This class develops mathematical optimization from the perspective of certificates, duality, and convexity. Topics covered include Fenchel and Lagrange duality; representation of convex sets; linear and semidefinite programming; integer programming; convex relaxations for intractable problems; and numerical methods. Throughout the course, applications of optimization to problems arising in various areas of science and engineering are presented.

This course develops core principles for the analysis and design of algorithms. Basic material includes mathematical techniques for analyzing performance in terms of resources, such as time, space, and randomness. The course introduces the major paradigms for algorithm design, including greedy methods, divide-and-conquer, dynamic programming, linear and semidefinite programming, randomized algorithms, and online learning.

Social networks, the web, and the internet are essential parts of our lives, and we depend on them every day. CS/EE/IDS 143 and CMS/CS/EE/IDS 144 study how they work and the "big" ideas behind our networked lives. In this course, the questions explored include: What do networks actually look like (and why do they all look the same)?; How do search engines work?; Why do epidemics and memes spread the way they do?; How does web advertising work? For all these questions and more, the course will provide a mixture of both mathematical analysis and hands-on labs. The course expects students to be comfortable with graph theory, probability, and basic programming.

This course will cover popular methods in machine learning and data mining, with an emphasis on developing a working understanding of how to apply these methods in practice. The course will focus on basic foundational concepts underpinning and motivating modern machine learning and data mining approaches. We will also discuss recent research developments.

This course is a research seminar course covering topics at the intersection of mathematics, computation, and their applications. Students are asked to attend one seminar per week (from any seminar series on campus) on topics related to computing and mathematical sciences. This course is a requirement for first-year PhD students in the CMS department.

Research in the field of computing and mathematical science. By arrangement with members of the staff, properly qualified graduate students are directed in research.