- Demonstrate mastery of the foundations of one or more advanced area of mathematics.
- Write extended passages of mathematical prose following modern conventions of precision and clarity.
- Explain advanced mathematics orally following modern conventions of precision and clarity
- Produce mathematical proofs in advanced areas of mathematics
- Understand, and critique for accuracy, complex mathematical proofs.
- Understand, produce, and critique mathematical models and algorithms appropriate to their (field(s) of speciality, utilizing appropriate software where necessary.
- Understand, appreciate, and explain the motivation and culture of their field(s) of specialty. This includes the major historical developments of the field, and the connections between the field and other areas of mathematics and science.
- Master the techniques, proofs, and applications of differential and integral calculus, and apply the methods of calculus in a variety of situations, such as analyzing numerical methods, ordinary differential equations, partial differential equations, measure theory, complex analysis, applicable analysis, and differential geometry.