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Algebra

  • Quandles: An Introduction to the Algebra of Knots, by Elhamdadi and Nelson, covers topics in topology, knot theory, and algebra that are presented in an easily accessible fashion and often not standard material in undergraduate curricula. It is apart of the AMS Student Mathematical Library series and I had the pleasure of taking linear algebra, topology, and graduate representation theory from the first author. For a fun, video introduction to knot theory, see this talk by Colin Adams.
  • Quantum Theory, Groups and Representations, by Woit, is indeed an introduction to the use of symmetry in mainly quantum mechanics; so, it covers quite a lot of topics but the exposition is very clear. It also contains guides and an extensive list of references for further explorations. The author maintains an interesting blog, which may be seen here.
  • Quantum Groups in Two-Dimensional Physics, by Gómez, Ruiz-Altaba, and Sierra, covers a number of integrable systems in statistical mechanics (such as Yang-Baxter) and conformal field theories via the theory of quantum groups and formal algebraic deformations. I was first introduced to the book out of a directed research course looking at deformations of the Poincaré algebra. I found the book much more accessible after learning lots of representation theory and multilinear algebra; some good supplementary texts may include Baxter's Exactly Solved Models in Statistical Mechanics, Kauffman's Knots and Physics, and the two-volume Braid Group, Knot Theory and Statistical Mechanics by Yang and Ge. Related are these informal notes on Braids and Quantization by John Baez.

Analysis

  • Classical and Stochastic Laplacian Growth by Gustafsson, Teodorescu, and Vasil'ev:
  • Solitons: Differential Equations, Symmetries, and Infinite Dimensional Algebras by Miwa, Jimbo, and Date:
  • Painlevé Transcendents: The Riemann-Hilbert Approach by Fokas, Its, Kapaev, and Novokshenov:

Gauge Theory

  • Loop Groups by Pressley and Segal
  • Krichever-Novikov Type Algebras: Theory and Applications by Schlichenmaier
  • A Mathematical Introduction to Conformal Field Theory by Schottenloher

Geometry

  • Spin Geometry by Lawson and Michelsohn:
  • Two-Dimensional Conformal Geometry and Vertex Operator Algebras by Huang
  • Topological Hydrodynamics by V.I. Arnold