In the table below you can find the planned course offerings for the current and future semesters. The courses beyond Spring 2020 might be subject to slight variations.

Electives (or advanced topics) will usually be taken by students past their second year, who have more flexible schedules. The actual elective courses offerings are decided on a year-by-year basis, and they reflect the individual faculty member's research interests.

The highlighted portions depict the progression from core courses (first three semesters) to the more advanced ones.

Course Offerings

Fall 2021	Spring 2022	Fall 2022	Spring 2023
Measure & Integration I	Measure & Integration II	Measure & Integration I	Measure & Integration II
GRV I	GRV II	GRV I	GRV II
Topology I	Topology II	Topology I	Topology II
Complex Analysis I	Partial Differential Equations	Complex Analysis I	Functional Analysis
Abstract Algebra I (GRV III)	Number Theory	Abstract Algebra I (GRV III)	Algebraic Geometry
Algebraic Topology	Geometry of fiber bundles	Differential Topology	Symplectic Geometry
Algebraic Geometry (Schemes)		Computer Algebra
Ergodic Theory		Algebraic foundations for Topological Data Analysis

Beyond second year (F21)  Second year (F21)  First year (F21)  First year (F22)

Algebra core sequence

The Algebra core sequence consists of threee courses: Groups, Rings and Vector Spaces I & II, and Abstract Algebra I. (The latter is often nicknamed GRV III.)

This sequence of courses covers basic material on classical algebraic structures such as categories, groups, rings, modules, including standard results as as the Sylow theorems, the classification of modules over a PID, factorization in integral domains, linear algebra over fields and more general rings, Galois theory, and basic notions in homological algebra, as well as multilinear algebra (symmetric and alternating algebras), Kähler differentials, the de Rham complex. The three semesters fit into a seamless continuum.

GRV I & II make up the material for the qualifier exam in Algebra. The specific topics are listed in detail at the Algebra qualifier page.

Analysis core sequence

The Analysis core sequence consists of two semesters of Real Analysis (under the name of Measure & Integration I and II) followed by one semester of Complex Analysis.

Real Analysis covers various aspects of modern measure and integration theory; in particular, passing to the limit under the integral, double vs iterated integration, Riesz representation theorem, foundations of operator theory, and locally convex topological spaces. Complex Analysis covers the fundamentals of Analysis in the complex domain in one variable, including complex differentiability, power series, complex integration, and classification of singularities and residues; it also covers more advanced topics, such as Möbius transformations, the Riemann and the Open mapping theorems.

Measure & Integration I and Measure & Integration II make up the material for the qualifier exam in Analysis. Exam topics are listed at the Real Analysis qualifier page.

Topology core sequence

The Topology core sequence consists of three courses: Topology I & II, and one of Topology IIIa (Algebraic Topology—MTG5346) or Topology IIIb (Differential Topology—MTG5932) (they alternate, see table above).

Topology I covers point set topology, in particular basic notions such as bases, limits, continuous functions, fundamental separability and countability properties, connectedness and compactness, and important results including the Tietze extension theorem, Urysohn’s lemma, and various metrization theorems. Topology II focuses on homotopy, homotopy equivalence, CW structures, and a detailed discussion of the fundamental group with an emphasis on its functorial properties. Fundamental results such as 2-dimensional Brouwer fixed point theorem, Borsuk-Ulam theorem, and Van Kampen’s theorem are discussed as well. Covering spaces are discussed in detail too, with an emphasis on relationships between fundamental group and covering spaces. Topology IIIa covers Algebraic Topology, in particular simplicial homology, singular homology, and cohomology. Topology IIIb covers Differential Topology with an emphasis on differentiable structures, the inverse function theorem, differentiable maps, tangent vectors and tangent bundle, differential forms and cotangent bundle, exterior differentiation, and flows and Lie derivative.

Topology I & II make up the material for the qualifier exam in Topology. The specific topics are listed in detail at the Topology qualifier page.

Intermediate topics courses are designed to complement the core courses with more advanced, but not super-specialistic material. Each of these courses is (roughly) offered every other year.

Algebra intermediate topics

• Number Thery (Spring 2022)
• Commutative Algebra and Algebraic Geometry (Spring 2023)

Analysis intermediate topics

• Partial Differential Equations (Spring 2022)
• Functional Analysis (Spring 2023)

Topology intermediate topics

• Geometry of fiber bundles (Spring 2022)
• Symplectic Geometry (Spring 2023)

Advanced topics courses are meant to provide an in-depth presentation of more specialistic research areas or to provide an introduction to new mathematical research areas.

• Advanced Algebraic Geometry (Fall 2021)
• Ergodic Theory (Fall 2021)
• Computer Algebra (Fall 2022)
• Algebraic Foundations for Topological Data Analysis (Fall 2022)