In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

3d N=2 Chern-Simons-matter theories have a large variety of boundary conditions that preserve 2d N=(0,2) supersymmetry, and support chiral algebras. I'll discuss some examples of how the chiral algebras transform across dualities. I'll then explain how to construct duality interfaces in 3d N=2 theories, and relate dualities *of* duality interfaces to "Pachner moves" in triangulations of 4-manifolds. Based on recent and upcoming work with K. Costello, D. Gaiotto, and N. Paquette.

In this talk, we shall provide a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space.
This variational principle has many applications in partial differential equations while unifies and generalizes several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.