Graduate Special Topics

Department: 
MATH
Course Number: 
8803
Hours - Lecture: 
3
Hours - Total Credit: 
3
Typical Scheduling: 
Every Fall and Spring Semester

The following table contains a list of all graduate special topics courses offered by the School of Math within the last 5 years. More information on courses offered in the current/upcoming semester follows below. 

 

Semester Instructor               Title                                                  
Spring 2027 Hannah Choi Neuronal Dynamics and Networks
 

Alex Dunn

 Algebraic Number theory 
  Jacob Erickson Cartan Geometries
  Xiaoyu He Topics in Mathematical AI Safety
  Jennifer Hom Heegaard Floer homology
Fall 2026 Leonardo Abbrescia Formation of shocks in multidimensional nonlinear waves and fluids
  John Etnyre Haken manifolds and algorithms in 3-manifold topology
  Adam Kanigowski Selected Topics in Dynamical Systems and Ergodic Theory
  Tobias Ried Optimal Transport: Theory and Applications
Spring 2026 Hannah Choi Neuronal Dynamics and Networks
  Christian Houdré From Longest Increasing Subsequences in Random Words to Quantum Statistics, II
  Thang Le Introduction to Quantum Topology
  Anton Zeitlin Representation Theory II
Fall 2025 Matt Baker Matroid Theory
  John Etnyre Manifolds and Handlebodies
  Xiaoyu He Ramsey and Turan Theory
  Christian Houdré From Longest Increasing Subsequences in Random Words to Quantum Statistics
  Tobias Ried Optimal Transport: Theory and Applications
  Anton Zeitlin Representation Theory
  Wei Zhu Mathematical Foundations of Machine Learning
Spring 2025 Gong Chen Nonlinear Dispersive Equations
  Alex Dunn Analytic Number Theory II
  Jen Hom Knot Concordance and Homology Cobordism
  Heinrich Matzinger AI, Transformers, and Machine Learning Methods: Theory and Applications
Fall 2024 Alex Blumenthal Big and Noisy: Ergodic Theory for Stochastic and Infinite-Dimensional Dynamical Systems
  Mohammad Ghomi Geometric Inequalities
  Michael Lacey Discrete Harmonic Analysis
  Rose McCarty Structure for Dense Graphs
  John McCuan Mathematical Capillarity
  Haomin Zhou Machine Learning Methods for Numerical PDEs
Spring 2024 Anton Bernshteyn Set Theory
  Greg Blekherman Convex Geometry
  Hannah Choi Mathematical Neuroscience
  Alex Dunn Analytic Number Theory I
  John Etnyre 3-Dimensional Contact Topology
  Chongchun Zeng Topics in PDE Dynamics II
Fall 2023 Jen Hom Knots, 3-Manifolds, and 4-Manifolds
  Tom Kelly Absorption Methods for Hypergraph Embeddings and Decompositions
  Zhiwu Lin Topics in PDE Dynamics I
  Galyna Livshyts Concentration of Measure and Convexity
  Cheng Mao Statistical Inference in Networks
Spring 2023        Igor Belagradek Diffeomorphism Groups
Fall 2022 Hannah Choi Neuronal Dynamics and Networks
  John Etnyre Topics in Algebraic Topology
  Christopher Heil     Measure Theory for Engineers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In the lists below, Math 8803-XXX refers to the special topics course taught by the instructor whose last name begins with XXX. 

 

 

Prerequisites: 

Spring 2027: 

  • Math 8803-CHO: MATH 2552 (Differential Equations) or equivalent*. Familiarity with Python, MATLAB, or other programming languages
  • Math 8803-DUN: Math 4150 (Number Theory) or equivalent. 
  • Math 8803-ERI: Math 6455
  • Math 8803-HOM: Math 6441
  • Math 8803-XHE: None

 

Fall 2026: 

  • MATH 8803-ABB: Math 6341 or equivalent
  • MATH 8803-ETN: Math 6441
  • MATH 8803-KAN Solid background in analysis is strongly recommended. Knowledge in probability and measure theory will be very helpful.
  • MATH 8803-RIE: You will need a good understanding of real analysis (in particular measure theory) in this course, as provided in MATH 6337 and MATH 6338. Knowledge of basic PDE theory and advanced analysis is helpful. Undergraduate students interested in this course should discuss their background with the instructor prior to applying for a permit. 

*Note that undergraduate-level prerequisites for graduate courses are not checked by the registration system. Undergraduate students must meet all posted prerequisites, regardless of their major: this includes undergraduate BS/MS students. 

 

 

Course Text: 

Spring 2027: 

  • Math 8803-CHO: See syllabus
  • Math 8803-DUN: See syllabus
  • Math 8803-ERI: See syllabus
  • Math 8803-HOM: See syllabus
  • Math 8803-XHE: None

 

Fall 2026:

  • MATH 8803-ABB: See syllabus
  • MATH 8803-ETN: See syllabus
  • MATH 8803-KAN: See syllabus
  • MATH 8803-RIE: See syllabus

 

Topic Outline: 

Spring 2027: 

  • Math 8803-CHO: This course covers various topics in neural dynamics at the level of single cell and population, and  their connections to biological neural networks. 
  • Math 8803-DUN: An introduction to Algebraic Number Theory.
  • Math 8803-ERI: The plan for the course is to cover the fundamentals of the modern Erlangen program and the Cartan machinery, with primary focus on the parabolic case.
  • Math 8803-HOM: We will define Heegaard Floer homology and related algebraic structures, and discuss applications to low-dimensional topology.
  • Math 8803-XHE: Rapid progress in AI capabilities has raised extreme concerns both about near-term prosaic risks (cognitive surrender, bio- and cyber-terrorism, misuse, economic disruption) as well as long-term existential risks to human civilization. This course is an introduction to the burgeoning field of Mathematical AI Safety. Tentative topics include: anatomy of LLMs, open-source game theory, agent foundations, singular learning theory, mechanistic interpretability, scalable oversight.

 

Fall 2026: 

  • MATH 8803-ABB: We will give an overview of Christodoulou's groundbreaking monograph on the formation of shocks in 3D relativistic fluids.
  • MATH 8803-ETN: We will discuss several approaches to understanding three dimensional manifolds. Specifically we will discuss Haken manifolds, algorithms to identify 3 manifolds, and other topics.
  • MATH 8803-KAN: The course will focus on topics in dynamics and ergodic theory. We plan to cover:

    1) Elements of topological dynamics -- symbolic systems, basic notions of abstract chaos, entropy.

    2) Abstract ergodic theory -- properties of measure preserving transformations, mixing and ergodic properties, classical constructions.

    3) Smooth dynamics 

    a) Hyperbolic systems: Anosov systems and their statistical properties, Markov partitions, Lyapunov exponents

    b) Parabolic dynamics: ergodic properties of horocycle flows, flows on surfaces.

    c) Elliptic dynamics: elements of diophantine approximation, elliptic constructions.

  • MATH 8803-RIE: 

    This is a graduate level special topics course in the theory and selected applications of optimal transport. The goal of this course is to give you a solid introduction to the theoretical foundations of optimal transport and its diverse applications in the sciences. The plan is to cover the following topics: 

    • Multimarginal optimal transport: existence and examples

    • Convex analysis and Kantorovich duality

    • Brenier's theorem and Monge solutions of the optimal transport problem

    • Wasserstein barycenters and interpolation of measures

    • Dynamical formulation: Benamou-Brenier formula

    • Applications: Wasserstein gradient flows and JKO scheme; HWI inequality; Wasserstein GANs