For everyone! Only very basic (elementary/middle school) mathematics background is needed to enjoy these posts.
- Halloween Candy and Counting
- Glencoe/McGraw-Hill doesn’t believe this bijection exists
- PEMDAS is broken (and how we can fix it)
- Days of the week and modular arithmetic
- Tropical polynomials and your federal tax return
- Growth patterns of desert bushes?
- The Mad Veterinarian
- Early Easter Egg
- The 3x+1 problem!
Some high school level mathematics background would be useful for understanding these posts, but they aim to be generally accessible.
- Ellipses, parabolas, and infinity
- What do Schubert curves, Young tableaux, and K-theory have in common? (Part I)
- Deriving the discriminant of a cubic polynomial through analytic geometric means
- Can you prove it?
- Enumerating positive walks
- Positive and recurrent walks
- Teeter-totter transpositions
- Knuth equivalence on a necklace
- Circles of Apollonius… and magnetism!
Some basic undergraduate courses (modern algebra, analysis, or topology) would be useful for understanding this material.
- Can you Prove it… combinatorially?
- What is a $q$-analog? (Part 2)
- What is a $q$-analog? (Part 1)
- The r-major index
- Symmedians and circumcircles
- Geometry of the real projective plane
- The q-factorial in terms of the major index
- What happens in characteristic p?
- Equilateral triangles in the complex plane
- Theme and variations: the Newton-Girard identities
- Addendum: An alternate proof of the FTSFT
- The Fundamental Theorem of Symmetric Function Theory
Aimed at mathematics graduate students or those with a solid undergraduate foundation in mathematics.
- Writing polynomials in terms of invariants and coinvariants
- A proof of the existence of finite fields of every possible order
- Schubert Calculus
- A better way: Carlitz’s bijection
- Yet another definition of the Schur functions
- Combinatorial species
- A bridge between two worlds: the Frobenius map
- The hidden basis
- Characters of the symmetric group
Highly specialized; only mathematicians who specialize in a particular area will have the background to appreciate these posts.
- Shifted partitions and the Orthogonal Grassmannian
- The structure of the Garsia-Procesi modules $R_\mu$
- What do Schubert curves, Young tableaux, and K-theory have in common? (Part III)
- What do Schubert curves, Young tableaux, and K-theory have in common? (Part II)
- The Springer Correspondence, Part III: Hall-Littlewood Polynomials
- The Springer Correspondence, Part II: The Resolution
- The Springer Correspondence, Part I: The Flag Variety
- Digging deeper: The isotypic components
- Molien’s Theorem and symmetric functions
- A q-analog of the decomposition of the regular representation of the symmetric group
- Summary: Symmetric functions transition table
Thoughts on education, technology, and other things that are not strictly mathematical.