Georgia Institute of Technology College of Sciences

Spring 2026: 

• 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-TLE: Topics include the Jones polynomial and quantum link invariants,  Topological Quantum Field Theory (ribbon and modular tensor categories), quantum representations of mapping class groups, and skein modules/algebras. 

• Math 8803-HOU: The course will cover the following sequence of topics: 
  • Introduction
  • Weak Invariance Principles
  • Longest Increasing Subsequences:  The One Sequence Case
  • Maximal Eigenvalues of Matrices from the Gaussian Unitary Ensemble
  • Young Diagrams and the RSK Algorithm 
  • Convergence of the Shape of RSK Young Diagrams
  • Spectra of GUE and Related Gaussian Matrices
  • A Conjecture for Markov Random Words
  • Two or More Random Words, Longest Common (and Increasing) Subsequences 
  • Strong Invariance Principles and Rates of Convergence
  • Quantum Tomography
  • Quantum Estimation
• Math 8803-ZEI: In the first part, we will continue studying representations of classical complex Lie algebras, followed by representations of compact Lie groups and the Peter-Weyl theorem. We will explore applications, including an explicit study of the hydrogen atom from a representation-theoretic perspective. The course then covers the structure of Lie groups, including the classification of compact and complex reductive groups, the topology of Lie groups and homogeneous spaces, and the Bruhat decomposition.
  
  The second part is devoted to advanced topics in modern representation theory: representations of affine Lie algebras, elements of vertex algebra theory, and quantum groups. Topics include highest-weight representations of affine Lie algebras, constructions of vertex algebras with applications to conformal field theory, and quantum groups in the context of knot invariants and quantum topology.

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-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