In this talk I will describe joint work with Yuki Maehara in which we give a model-independent (i.e. a purely $\infty$-categorical) construction of the (non-symmetric) Gray monoidal structure on the $\infty$-category of $(\infty,2)$-categories. Our construction is a generalisation to the $\infty$-categorical setting of a construction of the Gray monoidal structure for 2-categories due to Ross Street, which uses the techniques from Brian Day's PhD thesis for extending a monoidal structure along a dense functor. The proof of our construction uses, among other things, the results from Yuki's PhD thesis on the Gray tensor product for 2-quasi-categories. I will also mention a few of the open problems concerning the Gray monoidal structure for $(\infty,2)$-categories, and explain how our results can be used to simplify (though not yet solve) one of these problems.

In this talk, I'll present a recent proposal of a Fourier-Mukai kernel for VGIT problems associated to flops due to Ballard-Diemer-Favero. When the GIT problem is that of $\mathbb G_m $ acting on a smooth scheme, they check that these kernels produce the `grade-restriction windows' that have been well studied in the literature. In the non-smooth case, we show that this proposal produces windows by working on "smooth" dg-resolutions. If time permits, we will also discuss some work in progress concerning generalizations to general groups.