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I will review some recent developments that attempt to put modern mathematical tools and structural insights to effective use in physics. The focus will be on two such tools. The first is a perspective on superspace geometry that leads to powerful structural analogies between twisted theories (which encode the dynamics of particular BPS subsectors) and their untwisted parents. The second deals with generalizations of typical field theories akin to the generalization from symplectic to Poisson phase spaces. As an application, one obtains an object with N=(2,0) superconformal symmetry in six dimensions. Working at the holomorphic level on flat space, this object becomes the current algebra of the exceptional simple superalgebra E(3|6), which localizes to the W_2 algebra and dimensionally reduces to sl(2) super Yang-Mills theory. This is a mixture of joint work with numerous collaborators, in particular Hahner, Raghavendran, and Williams.

I will explain a new 2d gravity/matrix model duality. The bulk theory can be defined as a minimal string theory whose matter sector is given by timelike Liouville theory instead of a minimal model. The theory admits a dual description in terms of a double-scaled matrix integral whose leading density of states is given by the universal Cardy density of states in a 2d CFT of central charge c, thus motivating us to call the bulk theory the Virasoro minimal string. The duality can be derived by exploiting a certain relation to 3d gravity compactified on a circle. For large central charge, it reduces to the duality between JT-gravity and the corresponding double-scaled matrix integral. Based on work in collaboration with Scott Collier, Beatrix Mühlmann and Victor Rodriguez.

Richard Feynman's last blackboard at Caltech contains a number of tantalizing inscriptions about Bethe Ansatz in quantum integrable models, which fascinated him in the last years of his life. After reviewing some basic examples, I will present a modern perspective on the subject, linking it to dualities in QFT and String Theory, as well as the Langlands duality in mathematics. I will then discuss some recent developments that lend support to Feynman's intuition that these ideas could be useful in the study of 4d gauge theory, and formulate some open questions.

What can be measured asymptotically in quantum field theory? Among the answers to this question are scattering amplitudes, but also a whole compendium of inclusive measurements, such as expectation values of gravitational radiation and out-of-time-ordered amplitudes. We show that these asymptotic observables can be related to one another through new versions of crossing symmetry. Assuming analyticity, we propose generalized crossing relations and corresponding paths of analytic continuation. Throughout the talk, we show how to apply crossing in practice, using various tree- and loop-level examples.