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Topics courses – syllabi

WQ26 Math Math 515-1, Topics in Geometry and Topology: Prof. Ben Antieau

Topic: derived commutative rings.

WQ26 Math 520-2, Topics on Mathematical Physics: Prof. Ethan Sussman

A (rigorous!) introduction to quantum field theory accessible to mathematics graduate students, as well as physicists. The first half of the course will be devoted to the full construction of free fields, including non-scalar fields and massless fields. For this part of the course, we will follow Weinberg's Quantum Theory of Fields, Volume I, Chapters 2--5 and Chp. 9--10 of Talagrand's What is a Quantum Field Theory?, supplemented by my own notes.

The second half of the course will be an introduction to (perturbative) interacting theories, focusing on quantum electrodynamics (QED) as the paradigmatic example. QED is often touted as the most precisely verified theory in science. The QED prediction for the anomalous magnetic moment of the electron matches experiment to more than ten (!) significant figures. The goal of this course is to get to the algorithm via which these predictions are derived, without ever writing down an ill-defined integral. 

By this point, the standard textbook treatment has abandoned rigor. There is one exception among textbooks following this line --- Scharf's Finite QED (Chp. 3 & 4), on which our treatment will be based. The formalism of Bogolyubov and Stuckelberg, often known as ``causal perturbation theory,'' is used to axiomatize the scattering matrix and make everything rigorous (but only at the level of perturbation theory).

FQ25 MATH 477 Commutative Algebra:  Prof. Jakub Witaszek

This course follows the Northwestern University Syllabus Standards. Students are responsible for familiarising themselves with this information. Course description: review of foundational theory (integrality, affine algebraic geometry, primary decomposition, Dedekind domains), dimension theory, theory of depth (regular sequences, Koszul complexes, local cohomology, Cohen-Macaulay modules), homological methods (including Serre’s criterion for regularity).

All course materials will be posted on Canvas. Check for updates and announcements on a regular basis. Information about office hours will be announced and updated on Canvas.

Textbook: Matsumura, Commutative Ring Theory.

Grading: Students must submit solutions to at least 12 posted problems of their choice by the last week of classes. At least 10 of these solutions must be reasonably correct to earn an A. Students are strongly encouraged to submit problems regularly each week in order to receive feedback on their work. The requirement to submit homework solutions is waived for graduate students who haeave passed the math qualifying exam or an equivalent.

Supports for Wellness and Mental Health: Northwestern University is committed to supporting the wellness of our students. Student Affairs has multiple resources to support student wellness and mental health. If you are feeling distressed or overwhelmed, please reach out for help. Students can access confidential resources through the Counseling and Psychological Services (CAPS), Religious and Spiritual Life (RSL) and the Center for Awareness, Response and Education (CARE). Additional information on all of the resources mentioned above can be found at the following links: • https://www.northwestern.edu/counseling/ • https://www.northwestern.edu/religious-life/ • https://www.northwestern.edu/care/

Accessibility: Northwestern University is committed to providing the most accessible learning environment as possible for students with disabilities. Should you anticipate or experience disability-related barriers in the academic setting, please contact AccessibleNU to move forward with the university’s established accommodation process (email: accessiblenu@northwestern.edu; p: 847-467-5530). If you already have established accommodations with AccessibleNU, please let me know as soon as possible, preferably within the first two weeks of the term, so we can work together to implement your disability accommodations. Disability information, including academic accommodations, is confidential under the Family Educational Rights and Privacy Act.

FQ25 Math 445-1 ("Differential Geometry"): Prof. Jared Wunsch
This class will be an introduction to Riemannian geometry.  It will be at the second-year graduate level: the assumption is that you will have completed the first-year Geometry and Topology sequence.  In particular, you should be familiar with the definitions and basic properties of smooth manifolds, the tangent and cotangent bundles, vector fields, and flows.  I'll give some quick review of the basic concepts as we go, but it will be too fast to assimilate if you have not seen this material before.  The class will essentially begin with "what is a Riemannian metric on a smooth manifold?" and go from there.   The first year course deals almost exclusively with the topology of manifolds, which by definition all look locally the same; the point is that now we introduce extra structure (a metric) and obtain a rich class of objects that have interesting local, as well as global, invariants (e.g., curvature).  This is the big difference between doing topology and doing geometry.
A very loose list of topics we'll cover (as time permits)
(This is probably too ambitious, but we'll try!)
The text wil be John Lee's "Introduction to Riemannian Manifolds," although this is a long book and we can only cover parts of it.  You should have free access to this through NU's SpringerLink subscription.
Some (modest amount of) written work and an oral presentation will be required for students enrolled in the course.  I will hand out one or more problem sets in the course of the quarter.  Every student is expected to complete six problems and tell me which they have done. Additionally, each student will be responsible for carefully writing up two problems from the list and uploading their solutions to Canvas, where all students will be asked to read and (politely) critique the writeups.  After any necessary corrections, each student will give a short talk on each of the problems they have completed in one or more in-class problems sessions in the latter half of the quarter.
[Post-qualifying-exam students working on a thesis problem can be exempted from this, and should talk to me.]
Class will meet at 2pm MWF in Lunt 101.