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Large neural networks are often studied analytically through scaling limits: regimes in which some structural network parameters (e.g. depth, width, number of training datapoints, and so on) tend to infinity. Such limits are challenging to identify and study in part because the limits as these structural parameters diverge typically do not commute. I will present some recent and ongoing work with Alexander Zlokapa (MIT), in which we provide the first solvable models of learning – in this case by Bayesian inference – with neural networks where the depth, width, and number of datapoints can all be large.

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The interplay of quantum fluctuations and interactions can yield novel quantum phases of matter with fascinating properties. Understanding the physics of such systems is a very challenging problem as it requires to solve quantum many body problems—which are generically exponentially hard to solve on classical computers. In this context, universal quantum computers are potentially an ideal setting for simulating the emergent quantum many-body physics. In this talk, I will discuss two different classes of quantum phases: First, we consider symmetry protected topological (SPT) phases and show that a topological phase transitions can be simulated using shallow circuits. We then utilize quantum convolutional neural networks (QCNNs) as classifiers and introduce an efficient framework to train them. Second, we focus on the realization of topological ordered phases and simulate the braiding of anyons. Taking into account additional symmetries, we then investigate phase transitions between different symmetry enriched topological (SET) phases.

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Both the social and natural world are replete with complex structure that often has a probabilistic interpretation. In the former, we may seek to model, for example, the distribution of natural images or language, for which there are copious amounts of real world data. In the latter, we are given the probabilistic rule describing a physical process, but no procedure for generating samples under it necessary to perform simulation. In this talk, I will discuss a generative modeling paradigm based on maps between probability distributions that is applicable to both of these circumstances. I will describe a means for learning these maps in the context of problems in statistical physics, how to impose symmetries on them to facilitate learning, and how to use the resultant generative models in a statistically unbiased fashion. I will then describe a paradigm that unifies flow-based and diffusion based generative models by recasting generative modeling as a problem of regression. I will demonstrate the efficacy of doing this in computer vision problems and end with some future challenges and applications.

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A fundamental result in solid-state physics asserts that a crystalline material cannot be insulating unless the number of electrons per unit cell is an integer. Statements of this nature are immensely powerful because they are sensitive only to the general structure of the system and not to the microscopic details of the interactions. Such "kinematic constraints" have been extensively generalized in contemporary times, commonly under the term "quantum anomaly”. In this colloquium, I will first review some basic aspects of anomaly constraints in many-body quantum physics. Subsequently, I will demonstrate, through several recent examples, the significant role of quantum anomaly in constraining, understanding, and even unveiling novel quantum phases of matter.

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We investigate the recently found \cite{migdal2023exact} reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a  Number Theory problem of finding the statistical limit of the Euler ensemble.
We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the quantum statistical theory of free fermions on a ring, with an external field related to the random fractions of $\pi$. 

We find the solution of this system in the statistical limit $N\to \infty$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the charge density of these fermions.

This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without assumptions, approximations, or fitted parameters.
We compute resulting integrals in \Mathematica{} and present effective indexes for the energy decay as a function of time Fig.\ref{fig::NPlot} and the energy spectrum as a function of the wavevector at fixed time Fig.\ref{fig::SPIndex}. 

In particular, the asymptotic value of the effective index in energy decay $n(\infty) = \frac{7}{4}$, but the universal function $n(t)$ is neither constant nor linear.

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