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Gravity creates space-time boundaries that limit observables without limiting the flow of energy and information. This means that quantum systems in cosmology are often open systems: to describe them we must include the effects of interaction with an unobservable environment. In many cosmological settings, different observers see different parts of the spacetime; the ensemble of the open systems for each observer makes up the full cosmology. In this talk I will introduce a class of out-of-equilibrium quantum systems constructed to mimic some key features of cosmology and demonstrate the utility of treating the full system as an ensemble of open systems. I will use these models to illustrate the possible connections between cosmological open quantum systems and thermodynamics, as well as open-systems-inspired ways to think about locality.

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Ultra-cold Fermi gases exhibit a rich array of quantum mechanical properties, including the transition from a fermionic superfluid Bardeen-Cooper-Schrieffer (BCS) state to a bosonic superfluid Bose-Einstein condensate (BEC), which can be precisely probed experimentally. However, accurately describing these properties poses significant theoretical challenges due to strong pairing correlations and non-perturbative interactions. In this talk, I will discuss our recent development—a Pfaffian-Jastrow neural-network quantum state equipped with a message-passing architecture, designed to efficiently capture pairing and backflow correlations. We benchmark our approach against existing Slater-Jastrow frameworks and state-of-the-art diffusion Monte Carlo methods. Analysis of pair distribution functions and pairing gaps reveals the emergence of strong pairing correlations around unitarity. We demonstrate that transfer learning stabilizes the training process in the presence of strong, short-ranged interactions, allowing for an effective exploration of the BCS-BEC crossover region. Our findings highlight the potential of neural-network quantum states as a promising strategy for investigating ultra-cold Fermi gases.

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Significant advancements in our understanding of the physical world have been driven by increasingly precise atomic spectroscopy. The level of accessible precision entered a new realm with the advent of laser cooling and trapping. Now we can extend the ultrahigh spectroscopic precision, or atomic clock technology, to more complex quantum particles like diatomic molecules. The ability to quantify molecular degrees of freedom, such as nuclear vibrations, with nearly atomic-clock precision illuminates their previously hidden properties. Moreover, it suggests possibilities to leverage this precision for probing fundamental aspects of physical interactions, including enhanced tests of Newtonian gravity at the nanometer scale.

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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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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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Light bosons, including light axions, dark photons, and dilaton/moduli, are well motivated extensions to the Standard Model of particle physics and intriguing dark matter candidates. Much progress has been made in recent years in both astrophysical and lab searches for these light bosons with the understanding that these light bosons act like weak classical waves which permeate the space we occupy. 

In this talk, I will discuss some novel phenomenon of light bosons in the dark sector, based on important in medium effects of these particles. I will show how to take advantage of medium effects of the photon to optimize searches for these light bosons with data coming from cosmic microwave background (CMB) experiments, and improve sensitivity to light dark photons and axions by orders of magnitude. I will also show how thinking of dark photon as weak classical waves breaks down during production of light bosons in both early and late universe, when gauged strings are produced in an event we call the string Bosenova, as well as the resulting observable consequences.

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Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here we explore the surprising connection between symmetry-protected topology (SPT) and separability of their boundary mixed states. More specifically, we consider lattice systems in d space dimensions with anomalous symmetry G, where the anomaly is characterized by a bulk SPT invariant in the group cohomology Hd+2(G,U(1)). We show that any mixed state ρ that is strongly symmetric under G, in the sense that G ρ ∝ ρ, is necessarily (d+2)-nonseparable, i.e. is not the mixture of tensor products of d+2 states in the Hilbert space. Furthermore, such states cannot be prepared from any (d+2)-separable states using finite-depth local quantum channels, so the non-separability is long-ranged in nature. The anomaly-nonseparability connection also allows us to generate simple examples of mixed states with nontrivial multi-partite entanglement.

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Feynman diagrams and the associated integrals are crucial in perturbative quantum field theory for translating theory input into concrete, observable quantities. I will talk about the fascinating interplay of physics and mathematics that emerges from the ensemble of these diagrams. From linear algebra to quantum mechanics or wave phenomena to Fourier analysis - physics and mathematics tend to regularly exchange ideas that often lead to breakthroughs on the other side. I will illustrate two new examples of such exchanges I contributed to. One exchange is from the mathematical theory of tropical geometry to evaluating intricate, physically relevant Feynman integrals that have been inaccessible before. In the second exchange, we used ensembles of Feynman diagrams and their renormalization to prove a long-standing conjecture in geometric group theory. The results give new insights into the 'dark matter'-problem in the moduli spaces of graphs and curves' cohomology.  Both these moduli spaces' cohomologies are of fundamental interest in algebraic geometry, topology and geometric group theory.

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Neural Network Field Theories (NNFTs) are field theories defined via output ensembles of initialized Neural Network (NN) architectures, the backbones of current state-of-the-art Deep Learning techniques. Different limits of NN architectures correspond to free, weakly interacting, and non-perturbative regimes of NNFTs, via central limit theorem and its violations. Nature of field interactions in NNFTs can be controlled by tuning architecture parameters and hyperparameters, systematically, at initialization. I will present a systematic construction of scalar NNFT actions, using various attributes of NN architectures, via a new set of Feynman rules and techniques from statistical physics. Conversely, I will present the construction of a class of NN architectures exactly corresponding to some interacting scalar field theories, via a systematic deformation of NN parameter distributions. As an example of the latter method, I will present the construction of an architecture for $\lambda \phi^4$ scalar NNFT. Lastly, I will introduce Grassmann NNFTs, their free and interacting regimes via central limit theorem for Grassmanns, and construction of an architecture corresponding to free Dirac NNFT. This approach provides us a way to initialize NN architectures exactly representing certain field configurations, and are useful for computing attributes, e.g. correlators, of field theories on lattice.

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The Langlands program is a grand organizing vision for a large slice of number theory and representation theory. A shockingly accurate metaphor for the Langlands program has emerged as electric-magnetic duality in four-dimensional gauge theory, but where the role of spacetime is played by objects from arithmetic. I will describe recent work with Yiannis Sakellaridis and Akshay Venkatesh, in which we apply ideas from QFT (the Gaiotto-Witten electric-magnetic duality for boundary theories) to a fundamental problem in number theory, predicting the relation between L-functions of Galois representations and integrals of automorphic forms.

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What happened in the earliest moments of the Universe? Despite decades of theoretical and observational work, a detailed understanding of the primordial Universe has remained elusive. In this talk, I will show how surveys of millions of galaxies can shed new light on these issues by constraining the statistics of the Universe just after inflation. By combining robust perturbative models with novel numerical techniques, I have placed some of the first (non-Gaussian) constraints on the dynamics and field content of inflation from galaxy surveys; these will sharpen considerably in the next decade as the volume of observational data grows. Through a combination of traditional perturbative approaches and new symmetry-based techniques, I will demonstrate how current and future galaxy surveys can be used to search for a wealth of new physics, including primordial scattering processes and parity violation, opening the door to constraining ultra-high-energy physics with low-energy data.

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