Scientific Series

Recently, there has been a growing interest in the relations between homotopy theory in mathematics and theoretical physics. Homotopy theory is used to classify and study physical systems. Also, physically motivated conjectures have led to many interesting developments in homotopy theory. I have been studying this subject as a mathematician. 

My recent works have been motivated by the Segal-Stolz-Teichner program, which is one of the most deep and important subjects relating homotopy theory and physics. They propose a geometric model, in terms of supersymmetric quantum field theories, of a homotopy-theoretic object "Topological Modular Forms". Based on this, we show the absence of anomaly in heterotic string theory (joint work with Yuji Tachikawa), and found a new physical and geometric understanding of duality (with Y.Tachikawa) and periodicity (with Theo Jonhson-Freyd) in homotopy theory. These works lead us to further interesting conjectures to explore. I would like to illustrate this exciting interplay between mathematics and physics. 

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Spatial symmetries can enrich the topological classification of interacting quantum matter and endow additional ``weak” topological indices upon systems with non-trivial strong topological invariants (protected by internal symmetries). In this talk, I will discuss the boundary physics of charge conserving systems with a non-zero shift invariant, which is protected by either a continuous U(1) or discrete CN rotation symmetry. In particular, I will discuss an interface between two systems with the same Chern number but distinct shift invariants and show that the interface hosts protected gapless edge modes. For general Abelian topological orders in 2D, I will prove sufficient conditions for gapless edge states protected by continuous rotation symmetry. For the case of discrete rotation symmetries, I will show that the Chern-Simons field theory for systems with gappable edges predicts fractional corner charges. These can also be computed when the system is placed on the two-dimensional surface of a Platonic solid, which relates to the fractional charge bound at disclination defects. Time permitting, I will discuss recent results regarding the relation between the shift and many-body real-space invariants for 2D systems with crystalline symmetry.

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Neutrinos and other light relic particles leave a number of imprints in the cosmic microwave background anisotropies and on maps of the large-scale structure of our Universe. These imprints can not only demonstrate the presence of these particles and constrain their masses, but can provide insight into their nature via signatures of interactions or other behavior that changes during the history of the Universe. I will describe this physics, present constraints on non-standard neutrino self-interactions and other forms of dark radiation from cosmic datasets. I will also discuss prospects for detecting neutrino mass and highlight how relic neutrinos force us to adopt new technology when modeling structures in the Universe today.

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The implementation of optical astronomical interferometry presents several technical challenges, such as establishing a shared reference frame and conducting nonlocal measurements. This presentation aims to establish a connection between the theoretical framework developed in quantum information science and the schemes employed in astronomical interferometry. Our discussion will focus on the trade-offs between required resource and operational performance. We will categorize these methods into three types: nonlocal schemes, local schemes with a reference frame, and local schemes operating without a reference frame. By using this interdisciplinary connection, we also explore a generalized intensity interferometer. This approach achieves performance levels comparable to traditional intensity interferometers but introduces interesting fundamental distinctions.

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When the spacetime metric is regarded as a quantum field, the classical trajectories of freely falling objects are subject to random fluctuations, or “noise”. This fundamental noise might even be observable at gravitational wave detectors: if detected, it would provide experimental evidence for the quantization of gravity. The effect of the quantum-gravitational noise is to turn the classical geodesic deviation equation into a stochastic, Langevin-like equation. Moreover, when these results are extended to congruences of geodesics, the quantum fluctuations of spacetime give rise to an additional term in the Raychaudhuri equation with the same sign as the other terms.

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Gravitational waves detected by advanced ground-based interferometers have given us an unprecedented look at the universe. Yet, a central mystery remains: How are sources of gravitational waves — in particular, black-hole and neutron-star mergers — formed in nature? The measured masses, spins, and redshifts of these compact objects offer some clues to solving this mystery. In this talk, I will describe state-of-the art techniques for characterizing and simulating pairs of black holes and neutron stars. To conclude, I will argue that a multi-pronged approach combining both model-agnostic data analysis and detailed simulations will help unlock the histories of these exotic objects.

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Local tomography (or tomographic locality) is the principle that the state of a composite systems is determined by the probabilities it assigns to outcomes of experiments performed separately on the component systems. It's well known that complex quantum theory enjoys, and real quantum theory lacks, this feature. This means that a composite of two real quantum systems has additional "global" degrees of freedom. What if we could simply factor these out? In this talk, I'll describe how this can be done, not only for real quantum theory, but for essentially any probabilistic theory. The result is a locally tomographic theory we call the "locally tomographic shadow" of the original. I will also discuss what this shadow theory looks like in the case of real quantum theory. (This is joint work with Howard Barnum and Matthew Graydon).

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This talk explains the theory of "factorisation" or "vertex algebra" analogues of the theory of quantum groups, (2312.07274).

We will first recap the theory of quantum groups and their connections to wider mathematics, due to Drinfeld, Etingof, Kazhdan, Jimbo, Reshetikhin, Turaev, and many others. We will then explain how to give a natural definition of "factorisation" version of these objects, give the basic structure theory about their categories of representations, and derive an explicit model for them. This relates to previous work of Etingof-Kazhdan and Frenkel-Reshetikhin.

We will sketch some physics heuristics for the above theory; briefly, our structures should appear on the category of line operators in 4d theories due to Costello, Witten and Yamazaki, e.g. on the representations Rep U_q(g^) of affine quantum groups.

We will then give some examples. The last part discusses the conjectures that naturally suggest themselves, for instance by analogy to the connections referred to above.

Prerequisites on chiral and vertex algebras will be given at a talk earlier in the week. (1-2pm Tuesday, Bob Room)

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Historically, the synergy between physics and geometry, from the times of Archimedes and Newton to the era of Einstein, has repeatedly been the catalyst for pivotal breakthroughs in physics and mathematics. In this talk, I will introduce a new narrative demonstrating how physics and geometry intertwine, leading to unexpected and significant results in critical phenomena in physics. Specifically, I will elucidate how non-commutative geometry—a mathematical framework born from the insights of physicists—offers fresh perspectives on conformal field theory, a subject with profound applications across various physics domains, from condensed matter to quantum gravity, and string theory.

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Current and future weak lensing surveys contain significant information about our universe. However, their optimal cosmological analysis is challenging, with traditional analyses often resulting in information loss due to reliance on summary statistics like two-point correlation functions. While deep learning methods offer promise in capturing the complex non-linear features of these cosmological fields, they often suffer from issues such as inadequate uncertainty quantification, susceptibility to distribution shifts, and interpretability limitations, which hinder their scientific applicability. In this talk, I propose a novel approach leveraging generative probabilistic modeling with Normalizing Flows to learn the data likelihood function at the field level, facilitating more effective cosmological information extraction. This framework not only enables anomaly detection of distribution shifts to improve the robustness of the analysis, but also fostering interpretability via generated samples. I will also discuss incorporating physical prior knowledge, such as symmetries and multiscale structure, into the model architectures to improve their generalization capabilities. Finally, I will explore the broader implications of deep probabilistic models in physics, highlighting their potential applications in diverse areas ranging from astronomical observations to high-energy physics and lattice field theory.

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In this talk, I will argue and practically illustrate that insights in quantum information, concretely coming from the tensor network representations of quantum many-body states, can help in devising better privacy-preserving machine learning algorithms. In the first part, I will show that standard neural networks are vulnerable to a type of privacy leak that involves global properties of the data used for training, thus being a priori resistant to standard protection mechanisms. In the second, I will show that tensor networks, when used as machine learning architectures, are invulnerable to this vulnerability. The proof of the resilience is based on the existence of canonical forms for such architectures. Given the growing expertise in training tensor networks and the recent interest in tensor-based reformulations of popular machine learning architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed when using machine learning on sensitive data.

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We explore the physical consequences of embedding Einstein gravity with negative cosmological constant in Conformal Gravity in four dimensions. In the bulk, the procedure is equivalent to Holographic Renormalization, as the Einstein-AdS action appears augmented by the correct boundary counterterms. In codimension-2, 4D Conformal Gravity induces a 2D conformal invariant, which leads to a renormalized area for minimal surfaces attached to the conformal boundary. For arbitrary surfaces, this proposal reproduces other energy functionals as Willmore Energy and Reduced Hawking Mass. In particular, this procedure provides a more geometric approach to the computation of Holographic Entanglement Entropy for CFTs dual to 4D Einstein gravity.

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