We discuss techniques to calculate symplectic invariants of CY 3-folds $M$, namely Gromov-Witten (GW) invariants, Pandharipande-Thomas (PT) invariants, and Donaldson-Thomas (DT)  invariants. Physicallythe latter are closely related to BPS brane bound states in type IIA string compactifications on $M$. We focus on the rank $r_{\bar 6}=1$  DT invariants  that count $\bar D6-D2-D0$ brane bound states related to PT- and  high genus GW invariants, which are calculable by mirror symmetry and topological string B-model methods modulo certain boundary conditions, and the rank zero DT invariants that count rank $r_4=1$   $D4-D2-D0$ brane bound states. It has been conjectured  by Maldacena, Strominger, Witten and  Yin that  the latter are governed by an index that has modularity properties due to $S$-duality in string theory and extends to a mock modularity index of higher depth for $r_4>1$. Again the modularity allows to fix at least the $r_4=1$ index up to boundary conditions fixing their polar terms.  Using Wall crossing formulas obtained by Feyzbakhsh certain PT invariants or Katz-Klemm-Vafa (KVV) invariants close to the Castelnuovo bound can be related to the $r_4=1$  $D4-D2-D0$ invariants. This provides further boundary conditions for the topological string B-model approach as well as for the $D4-D2-D0$ brane indices.  The approach allows to prove the Castelnuovo bound and  calculate the  $r_{\bar 6}=1$  DT- invariants or the GW invariants to higher genus than hitherto possible, as was pointed out in https://arxiv.org/abs/2301.08066 

by Alexandrov, Feyzbakhsh, Pioline, Schimannek and me. See also  http://www.th.physik.uni-bonn.de/Groups/Klemm/data.php for concrete evaluations. 

Lecture I: ``Recursive solution of the perturbative topological String''

In this lecture we explain how to solve the topological string recursively in terms of non holomorphic modular objects and discuss the integer invariants that it calculates.

Lecture II:  S-duality and the index of $D4-D2-D0$ bound states''

We review the approach of  Maldacena, Strominger, Witten and Yin  to the calculate the abelian ($r_4=1$)   $D4-D2-D0$ bound state degneracies from a modular index and the relation of these invariants to the PT  and DT invariants discussed above ( if time permits we comment on the mock  modularity of higher depth for $r_4>1$).            

 Lecture III: Castelnuovo bound and Wall crossing''

 In this lecture we define the above  mentioned  symplectic invariants  more mathematically  and discuss their stability and their Wall crossing behaviour and summarize the implication on concrete  calculations of them.