Scientific Series

I will review some recent progress in derived differential geometry, in particular pertaining to moduli stacks of solutions of elliptic partial differential equations on manifolds (with boundaries, and also with `logarithmic' boundaries, which include, for instance, manifolds with asymptotically cylindrical ends). In particular, this framework allows one to work efficiently with the compactified moduli spaces of symplectic topology and gauge theory. In another direction, I will explain some work in progress on the derived geometry of jet spaces, which can be used to endow moduli stacks of solutions of EOMs of a classical field theory with shifted symplectic structures.

Random unitaries form the backbone of numerous components of quantum technologies, and serve as indispensable toy models for complex processes in quantum many-body physics. In all of these applications, a crucial consideration is in what circuit depth a random unitary can be generated. I will present recent work, in which we show that local quantum circuits can form random unitaries in exponentially lower circuit depths than previously thought. We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over n qubits in log n depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in poly log n depth, and in all-to-all-connected circuits in poly log log n depth. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.

Galaxies are the medium through which we study the structure of the universe. However, widely applied statistical models of galaxies are generally over-simplified: even recently proposed models cannot capture the dependencies on environment or formation history. To solve this problem, I will introduce Graph Neural Networks (GNNs), a general and ideal tool for physical modelling. Geometrically constrained GNNs vastly improve our models, and allow us to ask detailed questions about the importance of formation history and environment for cosmological galaxy modeling. I will also prove a surprising equivalence between these two aspects of galaxy formation.

In this talk, I will discuss how M2-M5 intersections in a twisted M-theory background yield the R-matrices of the quantum toroidal algebra of gl(1). These R-matrices are identified with the Miura operators for the q-deformed W- and Y-algebras. Additionally, I will show how the M2-M5 intersection (or equivalently, the Miura operator) generates the qq-characters of the 5d N=1 gauge theory, offering new insight into the algebraic meaning of the latter.

To justify the existence of measurements that can not be performed jointly on quantum systems, Heisenberg put forward a heuristic argument, involving the famous gamma-ray microscope Gedankenexperiment, based on the existence of measurements that irreversibly alter the physical system on which they act. Today, the impossibility of jointly measuring some physical quantities, termed measurement incompatibility, and irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act, are understood to be distinct but related features of quantum mechanics. In our work, we formally characterized the relationship between these two properties, showing that measurement incompatibility implies irreversible disturbance, though the converse is false. The counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Our research followed an operational approach exploiting the framework of Operational Probabilistic Theories. In particular, it required the development of two new classes of operational theories: Minimal Operational Probabilistic Theories and Minimal Strongly Causal Operational Probabilistic Theories. These theories are characterized by a restricted set of dynamics, limited to the minimal set consistent with the set of states. In Minimal Strongly Causal Operational Probabilistic Theories, classical conditioning is also allowed.

In my talk I will explain how to extend the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. In particular, this extension allows to fill in the gap in cluster construction of the q-difference Painlev'e equations. Isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in non-obvious sense, cluster structure. These dual mutations cause certain polynomial mutations of dimer partition functions and polygon mutations of the corresponding decorated Newton polygons.