Tentative topics being considered for 2027+

Here is a preview of some topics we're considering for future years of AMM. Stay tuned for further updates!

Topics in calculus and real analysis

Broadly, calculus and real analysis at the level of a first-semester undergraduate course is permissible. We expect to be fairly similar to the Putnam in content coverage.

• This includes limits, convergence, continuity, differentiation, integration, infinite series, Rolle's theorem / mean value theorem, and Taylor expansions.

• The concept of compactness may be particularly convenient (e.g., sequential compactness, extreme value theorem, Heine-Borel, Bolzano-Weierstrass).

• For more advanced problems, knowledge of the basic definitions of point-set topology may be helpful.

• Occasionally, multivariable calculus (at a first-year college level) may be helpful. This includes partial derivatives, gradients, multiple integrals change of variables and Jacobians, Green's theorem, Divergence theorem, Stokes' theorem.

Topics in linear algebra

We expect a typical first-semester course in linear algebra to be sufficient.

• Vector spaces and subspaces, spans, linear independence and basis, dimensions, linear maps, kernels and images, rank-nullity theorem, inner products are included.

• Matrices and determinants (and how to calculate them) are included.

• For more advanced problems, understanding of eigenvalues and eigenvectors and the spectral theorem can be helpful.

Topics in abstract algebra

We expect a typical first-year undergraduate course in abstract algebra covering basics of groups, rings and fields would be sufficient. For some advanced problems, we may use relevant terminology even in problem statements.

• Groups: Groups and subgroups, cyclic groups, cosets, Lagrange theorem, normal subgroups, quotient groups, homomorphisms, group actions, orbit-stabilizer theorem.

• Commutative rings: Rings and ideals, polynomial rings and commutative rings, integral domains, ring homomorphisms, quotient rings, PID's, UFD's.

• Fields: Field extensions and their degrees, finite fields.