The topics in MAT790 will be presented and developed with the full formality of modern mathematics. The mathematics presented will be characterized by rigor and depth and developed in an abstract manner. The student is expected to be able to read an advanced mathematics text and follow a presentation oriented around theorems and their proofs. Students may be expected to do some creative work in deriving mathematical results and presenting them in a rigorous fashion. 2022-2023 topics: Abstract Algebra (Winter): This course studies the abstract structures that underlie much of algebraic mathematics: Groups, rings and fields. While the origins of the subject involve the quest to solve cubic, quartic and quintic polynomial equations the resulting structures have proved useful in many other quarters. Groups, for instance, allow one to study symmetries of objects and to count them in an organized fashion. Matrices supply examples of non-commutative rings in which the distributive laws still work, finite Fields can be used to do arithmetic for coding problems and more. Real Analysis (Spring): We will study real numbers, functions and sets, and prove the theory behind calculus. Topics will include sizes of infinity, countable and uncountable sets, the fundamental theorem of calculus and many others. The course will be driven by problem sets written by a member of the Mathematics Department.