**Prerequisite**

Hamiltonian mechanics, quantum mechanics. Complex analysis. Lie groups, Lie algebras and root systems, supersymmetry, basics of commutative rings (brief recap will be offered). Symmetric polynomials and related combinatorics. Familiarity with algebraic Bethe Ansatz and construction of transfer matrices.

**Introduction**

A typical feature of quantum integrable models is the possibility to construct and/or describe Bethe Algebra—a large family of commuting operators whose simultaneous diagonalisation fully controls the spectrum of the theory. We introduce rich combinatorial and geometrical features of the Bethe Algebra which are encoded through a variety of relations between Baxter Q-functions and which are universal for a variety of different models. The relations themselves turn out to be meaningful also in the situations when their derivation from the commutative algebra is not known or even possible. Being supplemented with the ansatz on the analytical structure of Q-functions, explicit equations can then be derived to encode the spectra of concrete physical models. Two examples are discussed in detail: One is rational spin chains where the obtained techniques turned out to be an extremely powerful computation tool which also, in certain cases, can be used to rigorously prove completeness of Bethe equations. The second one is AdS/CFT integrability - we will show how our general approach allows one to derive Quantum Spectral Curve that encodes the exact spectrum of planar N=4 SYM.

**Syllabus**

**Part 1 - Bethe Algebra for rational GL(M|N) spin chains**

[1] Introduction: computing spectrum of Heisenberg spin chain, with integrability and without

[2] Fused flag structure of GL(M|N) QQ-system

[3] Proof of completeness of Wronskian Bethe equations. Fast analytic Bethe solver. Lazy numerical Bethe solver.

[4] Classification of Unitary representations and non-compact Young diagrams. One-loop spectrum of AdS/CFT

[5] Separation of Variables for rational GL(M) spin chains

**Part 2 - Bethe Algebra and fused flag geometry**

[6] Recap of Lie algebras and root systems. Mutation games with Bethe equations and Weyl group.

[7] ODE/IM correspondence on the example of anharmonic oscillator

[8] ODE/IM correspondence, extended QQ-system, opers, and fused flags

[9] Bethe algebra for D-series using pure spinors. Beyond ADE.

**Part 3 - AdS/CFT integrability**

[10] History and main results.

[11] Quantum Spectral Curve

[12] Monodromy bootstrap

**Lecturer Intro**

Dmytro Volin received PhD degree in theoretical physics in 2009 at Universite Paris XI for his thesis done at C.E.A. Saclay. During 2013-2018, he worked at the institute of mathematics at Trinity College Dublin, and since 2018 he is an associate professor at Uppsala University, affiliated with the department of physics and astronomy; theoretical physics and with the department of mathematics; centre for geomtry and physics. He works on various topics of quantum integrability and mathematical physics, and is known in particular for formulation of quantum spectral curve that solves the AdS/CFT spectral problem.