**Properties of Gromov-Witten invariants defined via global Kuranishi charts ***submitted*

We prove that the Gromov-Witten invariants constructed by Hirschi-Swaminathan satisfy the Kontsevich-Manin axioms. The construction extends to the equivariant setting and we prove a localisation formula for the equivariant Gromov-Witten invariants of certain Hamiltonian torus manifolds.

**On Donaldson’s 4-6 question** *joint with Luya Wang*, *submitted*

We prove that the examples of Smith and McMullen-Taubes provide infinitely many counterexamples to one direction of the Donaldson four-six question and the closely related Stabilising Conjecture. In the other direction we show that manifolds that be-

come deformation equivalent after one stabilisation have the same Gromov-Witten invariants.

**Infinitely many monotone Lagrangian tori in higher projective spaces** *joint with Soham Chanda and Luya Wang*, *submitted*

We construct infinitely many exotic tori in all higher projective spaces by lifting the Vianna tori from the projective plane and showing that the lifts remain non-symplectomorphic.

**Global Kuranishi charts and a product formula in symplectic Gromov-Witten theory** * joint with Mohan Swaminathan*, *accepted for publication in Selecta Mathematica*

We construct a global Kuranishi chart for moduli space of stable pseudoholomorphic maps of arbitrary genus. We use this presentation to prove a product formula in symplectic Gromov-Witten theory.

**Lagrangian intersections and cuplength in generalised cohomology*** **joint with Noah Porcelli*, *accepted for publication in Mathematical Research Letters*

Using generalised cohomology theories we find better lower bounds on the number

of intersection points of two Lagrangians in the absence of transversality conditions.

#### Notes

**Flow categories **Notes taken during the Floer homotopy theory summer school in Vancouver