The quantum geometric tensor packs the Berry curvature and the quantum metric into one object, and it has become a standard tool for Hall responses, Wannier localization, flat-band superconductivity and optical sum rules. But the usual version has four limitations. It is a linear-order object. It lives at zero temperature. And it is tied either to Bloch bands of noninteracting electrons or to twist angles of an interacting ground state. The talk replaces the pure state by the density matrix, and the quantum distance by the Bures distance between the thermal state and the state driven by an arbitrary time-dependent field. Expanding that distance in the field strength defines a Bures metric at second order and a Bures connection at third order, and both come out as explicit spectral functions. Specific limits of the metric recover earlier results: a finite-temperature Souza–Wilkens–Martin (SWM) sum rule, the quantum Fisher information, and Fermi’s golden rule recast as geometry. At zero temperature, the connection reduces to the band-theory Christoffel symbol. The second half follows the connection into nonlinear optics. The sum rule for the shift current contains a genuinely multi-state term, built from transitions between two excited states. That term is the non-metricity of the mixture/exponential connections of quantum information geometry.

How do you draw a fresh sample from the distribution of natural images, when the only distributions you can sample directly are the uniform and the Gaussian? Both answers in this session start the same way. Draw a latent variable $z$ from a simple distribution, then learn a map that sends it to data. Training by maximum likelihood then runs into one obstacle: the posterior over the latent variable is intractable. The variational autoencoder handles it with one learned Gaussian posterior and a single decoding step, trained on the evidence lower bound (ELBO) with the reparameterization trick. The denoising diffusion probabilistic model (DDPM) has $T$ latent layers, and it fixes the approximate posterior to be a known noising process, so no encoder is learned at all. Its ELBO then collapses to a remarkably simple objective: predict the noise that was added. The handwritten notes carry every derivation, from the Gaussian-mixture M-step, through the closed-form Gaussian KL, to the DDPM posterior mean and sampling loop.

Landau’s paradigm says a phase is labelled by a symmetry and how it breaks, and that the critical modes are order-parameter fluctuations. Three familiar things appear to escape it: topological order, confinement/deconfinement, and the exact masslessness of the photon — none of them has a local order parameter. The claim of this talk is that all three are symmetry-breaking phenomena, once you allow the symmetry to act on loops and surfaces rather than on points. The route there is a single conceptual pivot: a symmetry is not a transformation of the fields but a topological operator supported on a submanifold, and the only thing being generalized is the dimension of that support. Two sessions took the argument from differential forms and linking numbers, through the recasting of ordinary Noether symmetry as a codimension-1 defect, to the definition of $p$-form symmetry and its realizations in Maxwell theory, the toric code, $\mathbb{Z}_N$ gauge theory, Chern–Simons theory and the $\mathbb{Z}_N$ center of Yang–Mills — ending with generalized order parameters, the photon as a Goldstone boson, a higher-form Mermin–Wagner theorem, and ’t Hooft anomalies.

Current algebra asks a deceptively simple question: instead of building a many-body theory out of particles $c_{\mathbf k}, c^\dagger_{\mathbf k}$, can we build it out of the fluid variables — the densities and currents that experiments actually measure? For free fermions in one dimension the answer is the familiar $U(1)$ Kac–Moody algebra, but the standard derivation leans hard on a filled Fermi sea, a linearized dispersion and a momentum cutoff. This talk replaces that derivation with the Bjorken–Johnson–Low prescription, which extracts the equal-time commutator from the high-frequency tail of a correlation function and therefore never has to assume what the ground state looks like. Applied to the Hatsugai–Kohmoto (HK) model — an exactly solvable non-Fermi liquid that violates Luttinger’s theorem — the method shows that the natural low-energy objects are not bare currents but parton (holon/doublon) currents, that they close into an affine $\mathfrak{su}(2)$ algebra, and that a manifestly local Sugawara-type Hamiltonian built from them reproduces the HK equations of motion and two-body correlators in the infrared. The suggested moral: the notorious non-locality of the HK model may be an artifact of writing local degrees of freedom in non-local variables.

Lecture I of a two-part introduction to conformal field theory, told from the condensed-matter side. The organising question: a critical $\phi^4$ theory is strongly interacting and we cannot solve it — so what can symmetry alone tell us? The answer runs from the emergent scale invariance at a fixed point, through the conformal group and the correlators it fixes, to the operator product expansion and the bootstrap, where crossing symmetry plus unitarity pin the 3D Ising critical exponents to six digits without ever evaluating a path integral. A preview of radial quantization closes the session.

A three-part pedagogical tour: what a neural network actually is and how it is trained, what attention adds once the data are sequences, and how both are being used right now in many-body physics — neural-network wavefunction ansätze optimised by variational Monte Carlo, and reduced density matrices learned without the wavefunction at all. The organising claim: a network is a very flexible fitting function, and the physics lives entirely in the details of how much of it you build in by hand.