In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian
rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around $k$ points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy,  using an outer-inner solution gluing approach. The asymptotically singular profile  around each point resembles a scaled finite mass solution of Liouville's equation.
We also discuss the {\em vortex filament conjecture} for the three-dimensional case. This is joint work with Juan D\'avila, Monica Musso and Juncheng Wei.

This talk is based on joint work with Dominic Joyce and Markus Upmeier. Issues we'd like to talk about are a) the orientability of moduli spaces that
appear in various gauge-theoretic problems; and b) how to orient those moduli spaces if they are orientable. We begin with briefly mentioning backgrounds and motivation, and recall basics in gauge theory such as the Atiyah-Hitchin-Singer complex and the Kuranishi model by taking the anti-self-dual instanton moduli space as an example. We then describe the orientability and canonical orientations of the anti-self-dual instanton moduli space, and other
gauge-theoretic moduli spaces which turn up in current research interests.

Amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed as a geometric construction encoding scattering amplitudes in the four-dimensional maximally supersymmetric Yang-Mills theory, they are mathematically interesting objects on their own. In my talk I strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. I focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional
conjugates. I show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron canonical differential form in all these cases. Notably, this also naturally exposes the rich combinatorial structures of amplituhedra, such as their regular triangulations