Geordie Williamson

Research Interests:

I work in geometric representation theory, and love all things vaguely geometric. I spend most of my time thinking about:

• perverse sheaves and the decomposition theorem,
  
  
• modular representation theory, particularly from a geometric perspective, torsion in cohomology,
  
  
• weights (e.g. Frobenius action and Weil conjectures or Hodge theory) and especially their interactions with perverse sheaves,
  
  
• Soergel bimodules, their Hodge theory and relations to representation theory,
  
  
• combinatorial models for perverse sheaves (e.g. Soergel bimodules, moment graphs, "intersection cohomology" of polytopes etc.),
  
  
• algebra that looks geometric (e.g. the coinvariant algebra of H3),
  
  
• categorification, diagrammatic algebra, "generators and relations",


• higher categories (usually stopping at 2 or 3!), especially monoidal categories and their module categories,


• Kazhdan-Lusztig theory and its modular versions (p-canonical basis etc.)
  
  
• braid group actions and link homology,


• microlocal approaches to perverse sheaves, characteristic cycles.

I am also interested (as an innocent bystander) in the following:

• derived equivalence (in representation theory, mirror symmetry, ...),


• Deligne-Lusztig theory and Broué's conjecture,


• Langlands philosophy, particularly geometric Langlands and mod p versions,
  
  
• real reductive Lie groups and their representation theory,
  
  
• character sheaves, characters of finite groups of Lie type,
  
  
• topology of algebraic varieties, singularity theory, stratification theory,
  
  
• algebraic geometry, especially interactions with complex geometry and Hodge theory, period domains etc.,
  
  
• knot theory, low dimensional topology.
  
  

Notes from some lectures I found interesting:

• The decomposition theorem and the topology of algebraic maps
  Notes from lectures given by Luca Migliorini in Freiburg in February, 2010.
  
  
• Mirror symmetry, Langlands duality and the Hitchin system
  Notes from lectures given by Tamas Hausel in Oxford, Hilary Term, 2010. (Notes by Greg Berczi, Michael Groechenig and myself.)
  
  
• Six lectures on Deligne-Lusztig theory
  Notes from lectures given by Raphael Rouquier in Oxford, Hilary Term, 2010. (Notes from Lecture 5 by David Craven.)
  
  
• Character sheaves, tensor categories and non-abelian Fourier transform
  Notes from lectures given by Victor Ostrik in Luminy, September 2010. 2010.
  
  
• Koszul duality and applications in representation theory
  Notes from lectures given by Wolfgang Soergel in Luminy, September 2010.
  
  

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Here are some passages I enjoy.