At Stanford OHS, advanced mathematics is not a set of formulas to memorize—it is a living conversation. Our courses invite students to explore open-ended questions, construct examples and counterexamples, and engage in proof writing that develops deep, first-principles understanding. Students learn to think like mathematicians: to conjecture, test, revise, and communicate ideas clearly to peers.
In Geometry of Numbers, students investigate the geometric and number theoretic structures of the lattice plane—the array of integer points (x, y) ∈ Z2—and discover the surprising ways that algebra and geometry intertwine. Beginning with questions that are simple to pose but rich to pursue, the class develops tools from number theory, linear algebra, graph theory, and abstract algebra to prove deep results. Along the way, students meet landmark theorems (Pick, Blichfeldt, Minkowski), study the algebraic structure of lattice transformations, and discover and prove a necessary and sufficient condition for existence of a regular lattice n-gon. Through carefully scaffolded questions and ideas, students arrive at open research directions, including connections to the Riemann Hypothesis, advances in Ehrhart theory, and the study of lattice polytopes in higher dimensions.
During this one-semester course, each student builds a research portfolio of original proofs, constructions, and reflections. Students complete the course with advanced experience reading and writing mathematical proofs, cross-disciplinary insight, and the confidence to approach open problems. They are prepared not only to succeed in advanced university mathematics, but to participate meaningfully in it.