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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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I will summarize recent progress in uncovering the analytic structure of scattering amplitudes. The overarching theme will be exploiting new intricate ways of analytically continuing time, extending beyond the Wick rotation. I will highlight a broad range of applications: from high-precision calculations in particle physics, through computations of gravitational waves, to formal topics in the scattering of strings.

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In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates. We show that even in the presence of unital sources like the depolarizing channel, the distribution, under the combined noise
channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates — meaning it is never too "flat" — regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both
of which anticoncentrate at sufficiently large depths. As consequences, our results have interesting algorithmic implications on both the hardness and easiness of noisy random circuit sampling, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results. This talk is based on arXiv:2306.16659.

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Recently, there has been much interest in black hole echoes, based on the idea that there may be some mechanism (e.g., from quantum gravity) that waves/fields falling into a black hole could partially reflect off of an interface before reaching the horizon. There does not seem to be a good understanding of how to properly model a reflecting surface in numerical relativity, as the vast majority of the literature avoids the implementation of artificial boundaries, or applies transmitting boundary conditions. Here, we present a framework for reflecting a scalar field in a fully dynamical spherically symmetric spacetime, and implement it numerically. We study the evolution of a wave packet in this situation and its numerical convergence, including when the location of a reflecting boundary is very close to the horizon of a black hole. This opens the door to model exotic near-horizon physics within full numerical relativity.

I introduce a unified framework for interpreting neural network classifiers tailored toward automated scientific discovery. In contrast to neural network-based regression, for classification, it is in general impossible to find a one-to-one mapping from the neural network to a symbolic equation even if the neural network itself bases its classification on a quantity that can be written as a closed-form equation. In this paper, I embed a trained neural network into an equivalence class of classifying functions that base their decisions on the same quantity. I interpret neural networks by finding an intersection between this equivalence class and human-readable equations defined by the search space of symbolic regression. The approach is not limited to classifiers or full neural networks and can be applied to arbitrary neurons in hidden layers or latent spaces or to simplify the process of interpreting neural network regressors.

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Artificial Intelligence and big data are dramatically transforming the way we work, live and connect. Innovators have begun designing AI solutions to advance society at a rapid pace, but often new technologies bring both promise and risk. How can we trust AI and safeguard society from unintended consequences to ensure a safe and human-centred digital future?

Join the University of Waterloo in partnership with the Perimeter Institute for the TRuST Scholarly Network’s Conversations on lecture series where technology leaders from UWaterloo, Google and NASA will discuss how AI is transforming society and if we should trust these technologies.

The recent breakthrough in the detection of gravitational waves (GWs) from merging black hole (BH) and neutron star (NS) binaries by advanced LIGO/Virgo has generated renewed interest in understanding the formation mechanisms of merging compact binaries, from the evolution of massive stellar binaries and triples in the galactic fields, dynamical interactions in dense star clusters to binary mergers in AGN disks. I will review different aspects of the dynamical formation channels, and discuss how observations of spin-orbit misalignments, eccentricities, masses and mass ratios in a sample of merging binaries by aLIGO can constrain these formation channels. The important roles of space-borne gravitational wave detectors will also be discussed.

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Gravitational-wave observatories have established a new field of transient astronomy. The most recent LIGO-Virgo-KAGRA catalog, GWTC-3, identified 90 merging binaries, which range from a double neutron star with a total mass of 2.7 at 40 Mpc (GW170817) to a double black hole with a total mass of 150 at 5.3 Gpc (GW190521). These observations have many connections to nuclear astrophysics. They are revealing the remnants of stellar evolution and supernovae in merging binary systems, they are constraining event rates and astrophysical environments for heavy-element nucleosynthesis, and they are illuminating the dense matter dynamics inside the cores of merging neutron stars. Here, I will describe the imprint of dense matter on gravitational waves, the implications of existing observations for nuclear physics, and some prospects for the coming years including the science potential of proposed next-generation observatories like Cosmic Explorer.

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Zoom link https://pitp.zoom.us/j/99015121355?pwd=NStOc2srbEJXdW9aSTJJbDk4RWZhdz09

Isometric tensor networks (isoTNS) form a subclass of tensor network states that have an additional isometric condition, which implies that they can be efficiently prepared with a linear-depth quantum circuit. In this work, we introduce a procedure to construct isoTNS encoding of certain 2D classical partition functions. By continuously tuning a parameter in the isoTNS, the many-body ground state undergoes quantum phase transitions, exhibiting distinct 2D topological order. We illustrate this by constructing an isoTNS path with bond dimension $D = 2$ interpolating between distinct symmetry-enriched topological (SET) phases. At the transition point, the isoTNS wavefunction is related to a gapless point in the classical six-vertex model. Furthermore, the critical wavefunction supports a power-law correlation along one spatial direction while remains long-range ordered in the other spatial direction. We provide an exact linear-depth parametrized local quantum circuit that realizes the path. The above features can therefore be efficiently realized on a programmable quantum device. In the second part of my talk, I will show how to discover efficiently measurable order parameters for quantum phases using model-independent training of quantum circuit classifiers. The possibility of the efficient realization of phase transition path is useful for benchmarking quantum phase recognition methods in higher than one dimension.

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Zoom link https://pitp.zoom.us/j/93183360141?pwd=RVdYeUxUbE1aZ1dUbzRSL3lBb0lHZz09