All Posts

Amber

For everyone! Only very basic (elementary/middle school) mathematics background is needed to enjoy these posts.

• On Raising Your Hand
• 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!

Pearl

Some high school level mathematics background would be useful for understanding these posts, but they aim to be generally accessible.

• Hockey sticks on planet ABBABA
• Pythagorean triples on a sphere?
• A faster divisibility rule for 7 (and 13)
• 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

Opal

Some basic undergraduate courses (modern algebra, analysis, or topology) would be useful for understanding this material.

• Sorting sets of binary strings into threes
• 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

Sapphire

Aimed at mathematics graduate students or those with a solid undergraduate foundation in mathematics.

• How many trivalent trees does it take to hold up a moduli space?
• Counting ballots with crystals
• A linear algebra-free proof of the Matrix-Tree Theorem
• On Raising Your Hand
• 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

Diamond

Highly specialized; only mathematicians who specialize in a particular area will have the background to appreciate these posts.

• The Springer Correspondence Part IV: Affine Springer Fibers
• The Garsia-Procesi Modules: Part 2
• Addicted to Crystal Math
• The CW complex structure of the Grassmannian
• Higher specht polynomials
• Shifted partitions and the Orthogonal Grassmannian
• The structure of the Garsia-Procesi modules $R \mu$
• PhinisheD!
• 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

Miscellaneous

Thoughts on education, technology, and other things that are not strictly mathematical.

• Setting up a virtual conferencing room
• Doing mathematics in a pandemic - Part IV: Talks with OBS
• Doing mathematics in a pandemic - Part III: Teaching
• Doing mathematics in a pandemic - Part II: Collaboration
• Doing mathematics in a pandemic - Part I: AlCoVE
• PhinisheD!
• Glencoe/McGraw-Hill doesn't believe this bijection exists
• Expii: Learning, connected
• FindStat and combinatorial statistics
• Olympiad vs. research mathematics
• Rota's Indiscrete Thoughts
• TeXStudio