Syllabus | 21-317 Functional Inequalities

Course scope, prerequisites, format, and policies for 21-317 in Fall 2026.

Course information

At a glance

Course: 21-317 Topics in Analysis
Title: Functional Inequalities: From Analysis to Probability
Instructor: Matthew Rosenzweig
Meetings: Monday, Wednesday, Friday · 12:00–12:50 p.m.
Term: Fall 2026 · August 24–December 4
Level: Undergraduate · 9 units
Classroom: To be announced
Office hours: To be announced
Contact: mrosenz2@andrew.cmu.edu

Course description

This course introduces several central functional inequalities in modern analysis and probability, including Poincaré, Sobolev, logarithmic Sobolev, and isoperimetric inequalities, together with selected applications. Topics include spectral gap and variance decay, concentration of measure, entropy methods, and convergence to equilibrium for Markov processes. Examples will be drawn from Euclidean, Gaussian, and discrete settings, with selected applications to partial differential equations, mathematical physics, statistics, and machine learning.

Course goals

By the end of the course, students should be better able to:

state and interpret central functional inequalities, including their hypotheses, scaling, and equality or near-equality cases;
explain how spectral gaps, concentration, entropy decay, and convergence to equilibrium are connected;
read a research-level mathematical source and reconstruct its proof architecture;
present advanced mathematics clearly and design useful exercises for an audience; and
write a precise, self-contained expository paper.

Prerequisites

The official required prerequisites are

analysis: 21-355 with minimum grade A, or 21-235; and
probability: one of 21-325, 15-259, 36-218, or 36-225.

The official listing recommends 21-356. Additional coursework in analysis or probability, such as 21-720 or 21-326, may be helpful but is not necessary. See the official CMU special-topics listing for the controlling prerequisite language.

Course format

The course is designed both to develop mathematical understanding and to strengthen the ability to read, explain, and write research-level mathematics. Much of the semester will be organized around student presentations of foundational and current research papers. Students will prepare presentations in consultation with the instructor, design short exercises based on their material, and receive structured feedback from peers and the instructor.

Students will also complete an expository paper related to the course theme. Short modules on mathematical writing and related professional skills, together with regular individual meetings, will support the preparation of talks and papers. In place of a final examination, students will work in teams on a final project applying concepts and methods from the course.

Assessment and course policies

The grading weights, detailed specifications for presentations and the expository paper, attendance and late-work policies, collaboration rules, submission procedures, and other course policies have not yet been posted. This page will be updated when those details are finalized; any substantive update will also receive a dated announcement.

Source note. The course description, format, and prerequisite language above follow the current CMU Fall 2026 special-topics listing. Meeting time and units follow the Fall 2026 Schedule of Classes.