Scientific Series

Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily forbids useful knowledge about the other. Two quantum measurements that do not commute form an example of complementary measurements, and this phenomenon can also be defined for ensembles of states. Although a key quantum feature, it is unclear whether complementarity can be understood more operationally, as a necessary resource in some quantum information task. Here we show this is the case, and relates to a task which we term unambiguous exclusion. As well as giving complementarity a clear operational definition, this also uncovers the foundational underpinning of unambiguous exclusion tasks for the first time.

In this talk, I will present the construction of the Yangian of a cotangent Lie algebra from the geometry of the equivariant affine Grassmannian. Furthermore, I will discuss how this quantum group can be dynamically twisted to a quantum groupoid over a neighborhood in the moduli space of G-bundles over a compact Riemann surface. These constructions are motivated by relations between a certain holomorphic-topological 4d gauge field theory and the geometric Langlands correspondence. Representations of the Yangian are pertubative line operators of said theory, while the dynamical twist of the Yangian controls the action of these operators via Hecke modifications in this setting. This talk is based on the two joint works arXiv:2405.19906 and arXiv:2411.05068 with Wenjun Niu.

Various models of physics beyond the Standard Model predict the existence of ultralight bosons. These particles can be produced through superradiant instabilities, which create boson clouds around rotating black holes, forming so-called "gravitational atoms". In this talk, I review a series of papers that study the interaction between a gravitational atom and a binary companion. The companion can induce transitions between bound states of the cloud (resonances), as well as transitions from bound to unbound states (ionization). These processes back-react on the binary’s dynamics and leave characteristic imprints on the emitted gravitational waves (GWs), providing direct information about the mass of the boson and the state of the cloud. However, some of the resonances may destroy the cloud before the binary enters the frequency band of future gravitational wave detectors. This destruction leaves a mark on the binary’s eccentricity and inclination, which can be identified through a statistical analysis of a population of binary black holes.

Quasi-Einstein equations are generalizations of the Einstein equation. They arise from warped product Einstein metrics (Kaluza-Klein reductions), Ricci solitons, cosmology, near-horizon geometries, and smooth measured Lorentzian length spaces. Despite their apparent generality, they often have a surprising rigidity. I will review some recent developments in the area, focusing on near-horizon geometries, including Dunajski and Lucietti's near-horizon version of the Hawking rigidity theorem. I will discuss an application to 5-dimensional extreme (Myers-Perry type) black holes whose horizons admit the structure of the group SU(2).

Gravity possesses hidden symmetries that emerge upon dimensional reduction. One of the first examples being the Elhers SL(2,R) group revealed when reducing four-dimensional Einstein gravity to three-dimensions. However useful such symmetries are, especially to design solution generating techniques, they act in a highly non-local way on the gravitational data. On the other hand it is known that the solution space of asymptotically Flat spacetimes can be expressed covariantly in terms of an infinite number of tensors defined on the null conformal boundary. Therefore it is expected that hidden symmetries will act on the boundary data. In this talk, focusing on the simpler case of Petrov algebraically special spacetimes, I want to show that at the level of the boundary, the action becomes local, therefore much simpler, and makes explicit gravitational electric/magnetic dualities.

In this talk, we develop a systematic framework for understanding symmetries in topological phases in \(2+1\) dimensions using the string-net model, encompassing both gauge symmetries that preserve anyon types and global symmetries permuting anyon types, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge symmetry of topological orders in \(2+1\) dimensions: the Fibonacci gauge symmetry of the doubled Fibonacci topological order, described by the Fibonacci fusion \(2\)-category. Our approach involves two steps: first, establishing duality between different string-net models with Morita equivalent input UFCs that describe the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input UFCs, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.

We present a tensorization algorithm for constructing tensor train representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function and a small set of sample points defining the domain of interest. Thus, it is particularly well-suited for machine learning models, where the domain of interest is naturally defined by the training dataset. We show that this approach can be used to enhance the privacy and interpretability of neural network models. Specifically, we apply our decomposition to (i) obfuscate neural networks whose parameters encode patterns tied to the training data distribution, and (ii) estimate topological phases of matter that are easily accessible from the tensor train representation. Additionally, we show that this tensorization can serve as an efficient initialization method for optimizing tensor trains in general settings, and that, for model compression, our algorithm achieves a superior trade-off between memory and time complexity compared to conventional tensorization methods of neural networks.