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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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Rich clusters of galaxies are the largest gravitational magnifiers in the Universe. One of the most interesting gravitational lensing phenomena arises when background galaxies overlap with the lensing caustics cast by the cluster lens, such that a portion of it is tremendously magnified by hundreds to even thousands fold. As a result, Nature’s most luminous classes of stars have been individually or collectively detected by space telescopes from cosmological distances. Quantitatively studying their behavior will enable us to probe an impressive hierarchy of fine mass structures inside the lens: from star-free sub-galactic cold dark matter halos, to intracluster stars, and to even minuscule dark matter clumps predicted in many of the particle physics models of the dark matter. I will talk about what we have theoretically understood about the extremely magnified stars, what latest observational advances there are, and what unique constraints on dark matter micro-structures can be derived.

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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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