This course is a few related mini-courses glued together which can be attended separately, each will take ~6-12 hours of class time. It's a continuation of the topics course from last time, and like before there will be a focus on open problems you might be interested in.

Website: https://alyoshalatyntsev.github.io/georepcourse/
Recordings: https://drive.google.com/drive/folders/1T9CpkaPHgTMHg1EZX7Fv4w4dFXwFK1R9?dmr=1&ec=wgc-drive-%5Bmodule%5D-goto (request access)
Lectures on Tuesdays, with a few extra lessons throughout the term to be announced

Offline attendance strongly encouraged!

Tentative plan of the first three mini-courses:
1. Geometry and represenations of Yangians and quantum affine algebras. We define Yangians of Kac-Moody Lie algebras and study the category O of their representations.
2. Quantum field theory for mathematicians. The Atiyah--Segal axioms for quantum field theories, examples coming from deformation theory and geometry, operads and $\Eb_n$ algebras. We will go on by explaning to a mathematician the following subjects and give examples: Topological, conformal and holomorphic quantum field theories; Twists and superconformal field theories; Categories of line operators; Local observables and factorisation algebras.
3. Vertex algebras. What is a vertex algebra? After answering this, we go on to study examples (for instance, certain vertex quantisations of Poisson varieties), their categories of modules, and some beautiful results from the 90s such as the Feigin--Frenkel isomorphism relating certain ``W'' vertex algebras and Langlands duality. We finish by discussing $q$-vertex algebras and relating the definition of vertex algebra to the structures introduced in (2).

Suggested reading:
1. Chari--Pressley, A Guide to Quantum Groups (1994), the chapters on Yangians and affine quantum groups; Molev, Yangians and Classical Lie Algebras (2007); Nakajima, Quiver varieties and finite dimensional representations of quantum affine algebras (2001); Hernandez--Jimbo, Asymptotic representations and Drinfeld rational fractions (2012).
2. Atiyah, Topological quantum field theories (1988); Segal, The definition of conformal field theory (2004); Lurie, On the classification of topological field theories (2009); Costello--Gwilliam, Factorization Algebras in Quantum Field Theory, Vol. 1 (2017); Elliott--Safronov, Topological twists of supersymmetric algebras of observables (2019); Dimofte--Niu, Tannakian QFT: from spark algebras to quantum groups (2024).
3. Frenkel--Ben-Zvi, Vertex Algebras and Algebraic Curves (2004); Frenkel--Reshetikhin, Deformations of W-algebras associated to simple Lie algebras (1998); Borcherds, Quantum vertex algebras (2001); Bruegmann, Vertex algebras and Costello--Gwilliam factorization algebras (2021).