Mathematical Interests

1. Lie Theory

Short-term interests (ready to tackle!):

• Clebsch–Gordan coefficients for SU(2)
• Weights, root systems and the Weyl group, especially in the context of SU(3)
• Curvature of Lie groups

Long-term interests (curious, but need a stronger foundation first):

• Peter–Weyl theorem
• Applications to analysis on manifolds
• Kähler manifolds

2. Representation Theory

Short-term interests:

• Combinatorial aspects of representation theory of the symmetric group
• The Haar measure and its role in the representation theory of compact Lie groups
• Applications to point group determination in chemistry
• Quivers

Long-term interests:

• Representations of Hopf algebras and quantum groups

3. Knot Theory

Short-term interests:

• Seifert surfaces
• Keis and quandles

Long-term interests:

• Khovanov homology
• The connection between the Jones polynomial and quantum groups

4. Topology

Short-term interests:

• Cohomology
• CW-complexes
• Differential topology and fixed-point theorems
• The topology of order complexes
• Topological data analysis

Long-term interests:

• Morse theory
• Higher homotopy groups and fibrations

5. Geometry

Short-term interests:

• Differential forms
• More on the geometry of Lie groups, for example their curvature or geodesics
• Geometric group theory

Long-term interests:

• De Rham cohomology
• Exotic spheres
• Symplectic geometry