A long, unsorted list of good topics to learn: (Rota's) umbral calculus model theory, abstract model theory binomial coefficient congruences bezout theorem resultants grobner bases sturm chains real closed fields: Artin's theorem on sums of squares of rational functions, Tarski's decidability results, positivstellansatz Majorization: Karamata, Fuchs, Schur convexity *completeness theorem* universal algebra cross ratios hyperbolic geometry LLL hashlife sturmian words expander/ramanujan graphs modular forms, hecke operators quaternion arithmetic, and more generally arithmetic in division algebras Matiyasevich's theorem, Waring's problem, the S-unit equation nonsingular cubic surfaces Multiplicative functions: Dirichlet convolution, Mobius inversion, average size of a multiplicative function, analogy between L-functions and generating functions, relationship between 1/zeta and the mobius function, ... p-adic analysis: Hensel's lemma, "power lifting", primitive roots, Chevalley-Warning theorem, Newton polygons, Weil bound for elliptic curves over finite fields, Hasse-Minkowski local global principle, p-adic power series (for example, the p-adic exponential and p-adic logarithm)... Reciprocity laws: Quadratic reciprocity, Gauss sums and Jacobi sums, Cubic reciprocity, Quartic reciprocity, ... Point-set topology: image of a compact set under a continuous map is compact, Heine-Borel, ... Convex sets: Helly's theorem, Minkowski sums, Brunn-Minkowski, ... differential galois theory shannon switching game o-minimality of the theory of reals plus exponential function and tarski's exponential function problem and schanuel's conjecture method of interlacing polynomials nonstandard analysis uniformization of Riemann surfaces tropical geometry quantum computation schoof's algorithm for square roots mod p proof theory, resolution proof system, extended resolution, frege systems, natural deduction, applications to SAT-solvers (minisat, glucose, ...) finite projective planes multi-armed bandit problem, upper confidence trees, application to go playing programs Nielsen reduction stable polynomials, positive definite matrices, completely/absolutely monotone functions matroid intersection theorem and its applications linear programming, semidefinite/convex programming newton polytopes and irreducibility spectral theorem forcing (logic) finite topological spaces "frozen boundary" in tiling problems mating post-critically finite rational maps? cayley-menger determinant percolation Glauber dynamics and the Gibbs measure viscosity solutions to pdes graver bases?? context-free grammars markov's inequality, chebyshev's inequality, chernoff bounds maximal inequalities banach-tarski paradox, measure theory nevanlinna theory gauge integral Yang-Baxter equation, quandles, quantum integrability model categories, small object argument, simplicial sets, Dold-Kan correspondence queueing theory