Logic & Foundations
- VideoLogic & Foundations What Makes a Function Well-Defined? Students are told, “Make sure your functions are well-defined!” but what does that really mean? Watch ↗
- VideoLogic & Foundations 3 Ways to Think About Injections and Surjections In this video I cover three of the most common ways to think about injective and surjective functions. Watch ↗
- InteractiveLogic & Foundations Functions on a 3-Element Set There are $27$ functions from a 3-element set to itself. Explore all of them at once by viewing them in a cube formation or by grouping them by type. Open ↗
- VideoLogic & Foundations Equivalence Relations Practice Quiz and Solutions A practice quiz on equivalence relations with solutions. Watch ↗
- VideoLogic & Foundations Canonical Decomposition for Functions Every function decomposes as a surjection followed by a bijection followed by an injection. This is a quick video lecture explaining this decomposition. Watch ↗
Real Analysis
- VideoReal Analysis Undergraduate Real Analysis Topic List A tour of the major topics in a first course in real analysis. Watch ↗
- VideoReal Analysis The Completeness Axioms The completeness axioms of the real numbers explained in pictures. Watch ↗
- VideoReal Analysis Two Consequences of Completeness Completeness implies that the natural numbers are unbounded and that the rational numbers are dense in the real line. Watch ↗
- VideoReal Analysis The Nested Interval Property An introduction to the nested interval property, with a proof that no sequence lists every real number. Watch ↗
- VideoReal Analysis Sequences and Subsequences Practice Quiz and Solutions A True / False practice quiz on sequences and subsequences with solutions. Watch ↗
- WorksheetReal Analysis Attracting and Repelling Fixed Points The derivative test for fixed points ($|f'(p)| < 1$ attracts, $|f'(p)| > 1$ repels), applied to recursively defined sequences and their limits. Open ↗ Key
- VideoReal Analysis Riemann Integration Examples The definition of the Riemann integral with worked examples and a counterexample. Watch ↗
Linear Algebra
- InteractiveLinear Algebra Definition of a Vector Space An interactive map of Section 1B of Sheldon Axler’s Linear Algebra Done Right, covering the definition of a vector space. Axler’s numbered statements are arranged in an explorable diagram. Open ↗
Abstract Algebra
- VideoAbstract Algebra Abstract Algebra Study Guide A tour of the main topics in a first course in abstract algebra. Watch ↗
- VideoAbstract Algebra Hi, I’m the Klein Group An introduction to the Klein group, the smallest noncyclic group. Watch ↗
- VideoAbstract Algebra Abelian Groups Practice Quiz and Solutions A practice quiz on abelian groups with solutions. Watch ↗
- VideoAbstract Algebra Cayley’s Theorem A short lecture on Cayley’s theorem from group theory. Watch ↗
- VideoAbstract Algebra Classify Groups of Order 65 A classification of all groups of order 65 using the Sylow theorems. Watch ↗
- VideoAbstract Algebra Ring Theory Practice Exercises and Solutions Two practice exercises on the characteristic of a ring with solutions. Watch ↗
- WorksheetAbstract Algebra Sets and Categories Problems on products, coproducts, and universal properties, following Aluffi’s Chapter 0. Open ↗ Key
Number Theory
- WorksheetNumber Theory Farey Sums We study the Farey sum $a/b \oplus c/d = (a+c)/(b+d)$ and the fractions it generates between $0/1$ and $1/1$. Open ↗ Key
Research
- ThesisResearch Deformations and Products of Polish Groups My PhD thesis (University of Illinois at Chicago, 2019) on the large-scale geometry of Polish groups, with special attention to deformation retracts and knit products. Open ↗
