In the search for finite temperature quantum memory, people have found an exotic 3D topological model, called the fracton model. A large class of them are known to have foliation structure. However, whether a foliated non-abelian fracton exists or not is still an open question.

We made a step forward by providing an explicit entanglement renormalization quantum circuit for layer decoupling of the quantum double model. We also constructed a different kind of 3D topological model, which allows for non-abelian generalization. However, its relation to the fracton model and traditional 3D topological order is not clear. I’ll present how to use logical/symmetry operators and a pivotal idea of entanglement renormalization circuit to study ground states properties.

A classical question in algebraic geometry asks to count the number of plane curves of degree d meeting a smooth elliptic curve in a single point tangent to order 3d. This question is best reformulated in terms of log Gromov–Witten invariants which I will introduce. By considering the degeneration of the elliptic curve to the toric boundary Navid Nabijou and I provide a localisation formalism to count these curves. We uncover a refined set of enumerative invariants which we believe are related to certain scattering diagram calculations. If time permits I will explain what happens in higher dimension.

Data-intensive technologies such as AI may reshape the modern world. We propose that two features of data interact to shape innovation in data-intensive economies: first, states are key collectors and repositories of data; second, data is a non-rival input in innovation. We document the importance of state-collected data for innovation using comprehensive data on Chinese facial recognition AI firms and government contracts. Firms produce more commercial software and patents, particularly data-intensive ones, after receiving government public security contracts. Moreover, effects are largest when contracts provide more data. We then build a directed technical change model to study the state’s role in three applications: autocracies demanding AI for surveillance purposes, data-driven industrial policy, and data regulation due to privacy concerns. When the degree of non-rivalry is as strong as our empirical evidence suggests, the state’s collection and processing of data can shape the direction of innovation and growth of data-intensive economies.

Symplectic embedding problems are at the core of symplectic topology. Many results have been found involving balls, ellipsoids and polydisks. More recently, there has been progress on problems involving lagrangian products and related domains. In this talk, I explain what is known about symplectic embeddings of these domains. There are rigid and flexible phenomena and for some problems, the transition between the two happen at a surprising place. In order to get to the results, we will use the Arnold-Liouville theorem and billiard dynamics.