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CohFT and Integrable hierarchies

External Lecturer: 
Danilo Lewanski (UNITS)
Course Type: 
PhD Course
Academic Year: 
20 h
Course brief description: 
Cohomological Field theories were introduced by Kontsevich and Manin as natural collections of cohomology classes on the moduli spaces of curves. Integrable hierarchies are infinite lists of PDEs with structure. There exist two recipes to associate an integrable hierarchy to a Cohomological Field theory: the first was constructed by Dubrovin and Zhang around 20 years ago, the second by Buryak in 2015. A natural and open conjecture is the equivalence between the two constructions. The equivalence is a certain polynomial transformation, which has as of now mainly been found case by case.
Tentative Syllabus:
  • Moduli spaces of smooth and stable curves: stratification and cohomology
  • Exercises via the Sage admcycles ptackage to compute their intersection theory
  • Cohomological field theories: examples and properties
  • Buryak construction of an Hamiltonian system of PDEs via the double ramification cycle
  • Buryak-Rossi quantisation
  • Relation with Dubrovin-Zhang construction: the main Miura equivalence conjecture
  • Examples
  • A. Buryak, Double ramification cycles and integrable hierarchies, Communications in Mathematical Physics, 2015.
  • A. Buryak, B. Dubrovin, J. Guéré, P. Rossi, Tau-Structure for the Double Ramification Hierarchies, Communications in Mathematical Physics, 2018.
  • A. Buryak, P. Rossi, Double Ramification Cycles and Quantum Integrable Systems, Letters in Mathematical Physics, 2016.
  • B. A. Dubrovin, Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, a new 2005 version of arXiv:math/0108160v1, 295 pp.
Background required:
Standard Bachelor in Mathematics/Physics background, the course is meant to be mainly self-contained. Elements of Algebraic Geometry and in PDEs are useful but not necessary.
Next Lectures: 

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