Mathematical Interests

(currently just a giant list of things that I want to learn more about)

Applied and computational algebra

• Gröbner bases
• Toric and tropical geometry
• Connections to combinatorics and polyhedral geometry
• Matroids
• Homotopy continuation
• Intersection theory and Schubert calculus
• Cluster algebras
• Chemical reaction networks
• Algebraic statistics (especially connections to phylogenetics)
• Algebraic methods in image analysis
• Sampling of varieties
• Algebro-geometric perspectives on persistent homology

Knot theory

• Khovanov homology (and variations thereof: odd, unified, annular, Lee homology...)
• In particular: detection results, algorithms, spectral sequences, homology operations...
• Relations between Khovanov homology and Floer theory.
• Connections between the Jones polynomial (and Khovanov homology) and quantum groups
• Link concordance
• Hyperbolic knots
• Keis, quandles and racks

Representation theory

• Quantum groups and Hopf algebras
• The Haar measure and its role in the representation theory of compact Lie groups
• The Peter–Weyl theorem
• Quiver representations and Gabriel's theorem
• Applications in algebraic combinatorics
• Applications to point group determination in chemistry

Topology

Algebraic topology:

• Cobordism groups
• Higher-dimensional TQFTs
• K-theory
• Morse theory
• Characteristic classes
• Surgery
• Exotic spheres
• Teichmüller theory and the mapping class group