2010 Winter Learning and Research

Here, I blog about math along with my other happenings (not very consistently). I also keep a wishlist for things I'd like to learn about and a summary outlining things I've accomplished.

Definition: MY 2010 WINTER begins on Sunday, December 27, 2009 (after I get back from Costa Rica) and ends on Saturday, May 1, 2010.

Blog

Green Other than math/physics
Red Evalution
Purple In progress
Blue Finished (at least for now)
Magenta In retrospect

• December 29 (Tuesday)
  • Updated my math-related webpages.
• December 30 (Wednesday)
  • Meeting with Joel: some more motivations, updated questions and his recent work/draft on quivers.
• January 1 (Friday)
  • Joel and Chandrika's paper, "Modules with 1-dimensional socles and components of Lusztig quiver varieties in type A": read Section 1 (p1)
  
  
• January 4 (Monday)
  • Joel and Chandrika's paper, "Modules with 1-dimensional socles and components of Lusztig quiver varieties in type A": read part of Section 2 (p2)
• January 5 (Tuesday)
  • Meeting with Joel: discussed about Joel and Chandrika's paper, as well as Kac-Moody Lie algebras.
  • Joel and Chandrika's paper, "Modules with 1-dimensional socles and components of Lusztig quiver varieties in type A": read more of Section 2 (p2)
• January 6 (Wednesday)
  • Humphreys: reread some of Section 10.1 and began Section 10.3 (p47-48, p51)
  • Eisenbud: read the Introduction chapter.
  • We played bridge for the first time in the new year, coming 2nd overall (out of 8 pairs) and 1st in flight B (out of 3 pairs) in the open game.
• January 7 (Thursday)
  • Joel and Chandrika's paper, "Modules with 1-dimensional socles and components of Lusztig quiver varieties in type A": went through the rest of Section 2 without a complete understanding, and read Section 3.1 in some detail (p2-3)
  • Sent in some revision suggestions for Joel and Chandrika's paper.
  
  
• March 22 (Monday)
  • Plan for the week: finish proofreading, think about surfaces and homologies

Wishlist (Go to Longer Wishlist)

• Things I want to understand:
• Current Tasks:
• Math textbooks to read:
  Representation Theory
  • Fulton & Harris, "Representation Theory -- A First Course" (Fulton-Harris)
  • Humphreys, "Introduction to Lie Algebras and Representation Theory"
  Geometry
  • Bredon, "Topology and Geometry"
  Homological Algebra & Category Theory
  • Gelfand and Manin, "Methods of Homological Algebra" (Gelfand-Manin)
  • Mac Lane, "Categories for the Working Mathematician"
  Algebraic Geometry
  • Eisenbud, "Commutative Algebra: with a View Toward Algebraic Geometry"
  • Perrin, "Algebraic Geometry: An Introduction" [Course website]
  Algebraic Knot Theory
  • Dror, "AKT-09" (online vide course on Algebraic Knot Theory, 2009 Fall)
  Quantum Groups & TQFTs
  • Kassel, "Quantum Groups" (should finish)
  • Turaev, "Quantum Invariants of knots and 3-Manifolds"
  • Barkalov and Kirillov, "Tensor Categories and Modular Functors" (Barkalov-Kirillov)
• Physics textbooks to read:
  Quantum Field Theories
  • Sakurai, "Modern Quantum Mechanics"
  • Peskin & Schroeder, "An Introduction to Quantum Field Theory"
  • Wen, "Quantum Field Theory of Many-body Systems"
  • Perimeter Institute, PSI courses 2009-2010
• Math papers to read: (not quite up to date)
  • Brion's paper, "Representations of quivers"
    Lusztig's paper, "Canonical Bases Arising from Quantized Enveloping Algebras"
    Crawley-Boevey and Schroeer's paper, "Irreducible components of varieties of modules"
    Geiss, Leclerc and Schroer's paper, "Semicanonical bases and preprojective algebras"
    Savage's paper, "Geometric and Combinatorial Realization of Crystal Graphs"
    Savage's paper, "Finite-dimensional algebras and quivers"
    Joel and Chandrika's paper, "Modules with 1-dimensional socles and components of Lusztig quiver varieties in type A"
  • Dror's paper, "On Khovanov's categorification of the Jones polynomial"
    Dror's paper, "Khovanov's homology for tangles and cobordisms" (Dror_2)
    Lauda's paper, "A categorification of quantum sl(2)"
    Khovanov's paper, "A categorification of the Jones polynomial"
  • Freedman, Nayak, Walker and Wang's paper, "On picture (2+1)-TQFTs"
  • Dror's paper, "The fundamental theorem of Vassiliev invariants" (Dror_3)
• Physics papers to read: (not quite up to date)
  • Levin and Wen's paper, "String-net condensation: A physical mechanism for topological phases"
    Kitaev's paper, "Fault-tolerant quantum computation by anyons"
    Buerschaper and Aguado's paper, "Mapping Kitaev's quantum double lattice models to Levin and Wen's string-net models"
    Lahtinen's expository paper, "Anyons, Quantum Double Symmetry and Topological Quantum Computation"
    Hector's paper, "A Family of Non-Abelian Kitaev Models on a Lattice: Topological Condensation and Confinement" (Hector_2)
  • Hector's paper, "Topological subsystem codes"
    Poulin's paper, "Stabilizer Formalism for Operator Quantum Error Correction"

Summary