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After discussing some of the difficulties and progress in defining real and open
counts, I will describe a generalisation of the higher-genus Welschinger invariants
defined by E. Shustin to the symplectic setting. I will then outline a recursive
formula allowing for reduction of the genus and the degree for computing these
invariants. This is a joint work in progress with E. Brugallé, Y. Ding, and A.
Renaudineau.
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