The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem. We formulate neural network training as the stochastic evolution of an initially random matrix ensemble, driven by stochastic gradient descent (SGD) updates that reshape the spectral bulk while amplifying signal strength. This induces a Baik-Ben Arous-Péché (BBP) transition during training, where isolated eigenvalues detach from the random bulk distribution, providing a dynamical framework for representation formation in high-dimensional learning dynamics. We demonstrate this in a solvable linear teacher-student model, where spectral evolution is analytically tractable and a phase diagram of trainability governed by the step size (or learning rate) and initial weight variance is obtained, and subsequently extend our formalism beyond the linear regime to nonlinear and stochastic settings. Numerical simulations in realistic settings support this picture, showing robust emergence of spectral alignment during training. Our results suggest that spectral analysis may provide a unified perspective of stochastic learning dynamics, linking trainability, optimisation hyperparameters, spectral phase transitions, and representation learning in neural networks.

This review discusses the recently proposed fuzzy sphere regularization for studying $2+1$D critical phenomena, particularly three-dimensional (3D) conformal field theory (CFT). The fuzzy sphere scheme not only offers remarkable efficiency in extracting extensive CFT data at low computational cost but also reveals unexpected connections among 3D CFT (critical phenomena), noncommutative geometry, and the quantum Hall effect. We introduce the fundamental ideas of fuzzy sphere regularization, emphasizing its role in demonstrating the state-operator correspondence of 3D CFTs on the $S^2 \times \mathbb{R}$ geometry. Additionally, we review key developments in this approach across various directions and outline potential future applications.

We examine the exclusive semileptonic decays $B_s \to D_s^{**} \ell \nu_\ell$, with $D_s^{**} =$ $\bigl\{D_{s0}^*,D_{s1}^*,D_{s1},D_{s2}^*\bigr\}$, within the Standard Model and beyond, using form factors evaluated in the Heavy Quark Effective Theory, including corrections up to $\mathcal{O}(\alpha_s, \Lambda/{m_Q})$. A data-driven approach is employed to extract Heavy Quark Effective Theory parameters, and the resulting synthetic data are used to parameterize the form factors via the $z$-expansion. With the resulting form factor information across the full kinematic region, we compute various observables derived from the two-fold angular decay distribution, and predict precise lepton flavor universality ratios: $R_{D_{s0}^*}= 0.158(20)$, $R_{D_{s1}^*}= 0.045(5)$, $R_{D_{s1}}= 0.073(4)$, $R_{D_{s2}^*} = 0.066(9)$. We also analyse potential new physics effects using the Weak Effective Theory and the Standard Model Effective Field Theory, performing a global analysis considering both real and complex Wilson coefficients. Furthermore, we investigate new physics contributions arising from the general Two Higgs Doublet Model. We evaluate the sensitivity of decay observables to new physics, highlighting their potential to probe deviations from the Standard Model in future measurements. Notably, the scalar and tensor new physics operators induce large sensitivity, with some observables deviating by more than $2 \sigma$ from Standard Model predictions.

In this paper, we explore (2+1)D quantum electrodynamics (QED) at finite density on a quantum computer, including two fermion flavors. Our method employs an efficient gauge-invariant ansatz together with a quantum circuit structure that enforces Gauss's law. As a proof of principle, we benchmark our simulation protocol on a small lattice system, demonstrating the identification of phase transitions in terms of the particle number of the fermion flavors. Classical simulations are used to obtain optimized variational parameters, which are then deployed in inference runs on IBM quantum hardware. We conclude by discussing hardware limitations and prospects for scaling this method to larger systems.

We study the dynamics of a heavy quark in a thermal plasma consisting of non-perturbatively interacting soft momentum gluons at high temperatures, described in terms of an effective theory of QCD. Discretizing this effective field theory on a three-dimensional lattice, we propose a numerical strategy that allows us to simulate the dynamics of a heavy quark for different values of initial momenta and for a wide temperature range, higher than $480$ MeV. This allows us, for the first time, to extract the momentum dependence of the heavy quark drag and diffusion coefficients in a non-perturbatively interacting thermal, non-Abelian plasma.

We provide a detailed analysis of the gravitational wave spectrum of $SU(N)$ pure Yang-Mills theory. The confinement phase transition is described with an effective Polyakov loop model, using the latest lattice data as an input. In particular, recent lattice studies clarified the large-$N$ scaling of the surface tension, which we incorporate through a modification of the kinetic term. We demonstrate that the thin-wall approximation agrees with the Polyakov loop model at small $N$ while it breaks down at large $N$. Furthermore, we include reliable estimates of the bubble wall velocity using a recently developed framework based on a large enthalpy jump at the phase transition. Altogether, this allows us to derive the gravitational wave signals for all $SU(N)$ confinement phase transitions and clarifies the behaviour at large $N$. The strongest signal arises for $N=20$, but overall the predicted signals remain rather weak. Our work paves the way for future studies of other gauge groups and systems with fermions.