DEPARTMENT OF MATHEMATICS

Topics courses – syllabi

• Statement on the use of generative AI. The use of generative AI as supplementary material to aid with understanding is permitted. However, all homeworks must be fully written in your own words, and checked by yourself. In my opinion, the best way to learn a material is to grapple with the homework problems yourself and try different approaches, without asking for a full answer from an LLM.
• Other policies and resources. This course follows the NU Syllabus Standards. Students are responsible for familiarizing themselves with this information.

SQ26 MATH 511, Concentration of Measure: Prof. Reza Gheissari

• Books. Primarily Concentration Inequalities by Boucheron, Lugosi, Massart, supplemented by High-Dimensional Probability by Vershynin and The Concentration of Measure Phenomenon by Ledoux.
• Prerequisites. Graduate level (measure-theoretic) probability and/or analysis.
• Course aims. This is a course on the concentration of measure phenomenon for probability mea- sures in high dimensional spaces. We will also explore connections to isoperimetric problems, and applications to high-dimensional statistics and mixing rates of Markov processes. Core topics will include concentration and isoperimetric inequalities in product spaces and the high-dimensional sphere, concentration beyond product measures through Poincare and log-Sobolev inequalities, and concentration with martingales. Throughout, we will provide applications to questions from dif- ferent areas, including geometric measure theory, discrepancy theory, probabilistic combinatorics, and statistical physics.
• Expectations. Your regular attendance and active participation in class are expected. There will be exercises worked into the lectures that I encourage you to try as homework.
• Other policies and resources. This course follows the NU Syllabus Standards. Students are responsible for familiarizing themselves with this information.

SQ26 Math 515-2, Topics in Geometry and Topology: Prof. Ben Antieau

• The focus of the course with be the categories Sh(X,D(Z)) of sheaves of complexes of abelian groups on topological spaces in general and manifolds specifically. We will in particular study the continuous K-theory of these categories and how they are related by the original 6-functor formalism.
• Some familiarity with higher categories will be helpful. We will largely follow the notes "Sheaves on Manifolds" of Krause—Nikolaus--Pützstück [Sheaves on Manifolds].

SQ26 Math 517, Cyclic Operads: Prof. Ezra Getzler

• This course is an introduction to a particular class of operads that are of importance in applications to mathematical physics (and possibly also geometric topology): cyclic operads. The focus will be on algebraic operads: we will work in the symmetric monoidal category of complexes of vector spaces, usually over a field of characteristic zero.
• An operad is a mathematical gadget that governs a certain type of algebraic structure (for example, associative algebras, commutative algebras, Lie algebras, Gerstenhaber algebras, Batalin-Vilkovisky algebras, E_k-algebras,... ; but not Jordan algebras, coalgebras, Hopf algebras, fields,...). The course will begin by giving a survey of the theory of operads. We will focus on Koszul operads: for these, the deformation theory of their algebras is more easily studied. All of the operads listed above are Koszul operads.
• The next topic is homological perturbation theory. This is a formalism that allows the transportation of algebraic structures between quasi-isomorphic complexes. It gives explicit formulas. As an application, we will explain how the higher holonomy of superconnections can be viewed as an application of homological perturbation theory to Lie algebras.
• Cyclic operads are operads whose algebras admit a notion of invariant bilinear form. For unital associative algebras, this is the same thing as a trace; for unital commutative algebras, this is the same thing as a linear form; for Lie algebras, it is sometimes called a Killing form. For algebras over a cyclic operad, there is a cyclic cohomology theory. The goal of the course is to explain Kontsevich's interpretation of cyclic cochains as closed two-forms on a formal noncommutative stack. This point of view allows for the development of a Cartan calculus, and the extension of homological perturbation theory to algebras with nondegenerate invariant bilinear forms (symplectic forms), also known as cyclic algebras.
• Time permitting, we will discuss generalizations such as modular operads, colored cyclic operads, .... 

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

Topic: derived commutative rings.

• This course will develop the higher categorical machinery necessary to rigorously define, understand, and work with several flavors of derived commutatve rings. These objects will be studied through the lens of cohomology operations and then used to understand the cotangent complex, Hochschild homology, and derived de Rham cohomology.
• There will be some overlap with weeks 2-6 of my topics course from Fall 2020. See the attached course notes for a bit of the flavor.
• Evaluation will be based on oral presentations and/or examinations.

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

• Title: Quantum field theory for mathematicians
• 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)
  • Riemannian metrics, and examples, esp. symmetric spaces
  • Connections; Levi-Civita connection
  • Geodesics
  • Exponential map
  • Tubular neighborhoods
  • Hopf-Rinow theorem
  • Curvature
  • Submanifolds; second fundamental form.
  • Jacobi fields, conjugate points
  • Comparison theorems
• 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.]