This week

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In previous work with Norris and Silvestri, we have shown that the global fluctuations present in these models exhibit behaviour that can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks. In this talk I will discuss work in progress with Larissa Richards in which we explore how the correlation structure of local fluctuations near the cluster boundary changes at the point of phase transition.

Nonequilibrium systems are ubiquitous, from swarms of living organisms to machine learning algorithms. While much of statistical physics has focused on predicting emergent behavior from microscopic rules, a growing question is the inverse problem: how can we guide a nonequilibrium system toward a desired state? This challenge becomes particularly daunting in high-dimensional or complex systems, where classical control approaches often break down. In this talk, I will integrate methods from optimal control theory with techniques from soft matter and statistical physics to tackle this problem in two broad classes of nonequilibrium systems: active matter—focusing on multimodal strategies in animal navigation and mechanical confinement of active fluids—and learning systems, where I will apply control theory to identify optimal learning principles for neural networks. Together, these approaches point toward a general framework for controlling nonequilibrium dynamics across systems and scales.

Title: Hasse principle for intersections of two quadrics via Kummer surfaces

Abstract: I will discuss recent work with Skorobogatov establishing the Hasse principle for a broad class of degree 4 del Pezzo surfaces, conditional on finiteness of Tate--Shafarevich groups of abelian surfaces. A corollary of this work is that the Hasse principle holds for smooth complete intersections of two quadrics in P^n for n\geq 5, conditional on the same conjecture. This was previously known by work of Wittenberg assuming both finiteness of Tate--Shafarevich groups of elliptic curves and Schinzel's hypothesis (H).
I will also discuss forthcoming work with Lyczak which, again under the Tate--Shafarevich conjecture, shows that the Brauer--Manin obstruction explains all failures of the Hasse principle for certain degree 4 del Pezzo surfaces about which nothing was known previously.

In this talk, I will revisit results on the construction of Hamiltonian surface charges in general relativity in the presence of a finite timelike boundary, with an emphasis on how different boundary conditions influence the definition of conserved quantities. The analysis, originally published a few years ago [2109.02883], focuses on Dirichlet, Neumann, and York's mixed boundary conditions, and demonstrates how each leads to consistent, integrable charges using canonical methods. These results are shown to match those obtained via a covariant phase space formalism enhanced by a boundary Lagrangian. A key outcome of the study is the identification of an integrable charge for the Einstein-Hilbert action that differs from Komar's and remains well-defined even without Killing symmetries. We also analyze how the charge depends on the choice of boundary conditions, demonstrating that both quasi-local and asymptotic expressions are affected. These findings are relevant to current efforts to understand gravitational dynamics in finite regions and may have implications for the thermodynamics of black holes.

Active filaments, such as kinesin propelled microtubules in gliding assay experiments, give rise to a plethora of active phases. In order to better understand which features of these phases are emergent and which exist at even the single filament level, we investigate the dynamics of individual active elastic filaments with chiral self-propulsion. To this end, we study the fully general time evolution of an overdamped plane curve and derive equations for the evolution of the curve’s shape and orientational characteristics. Applying this formalism to the specific case of an active elastic filament with chiral self-propulsion, we determine that sufficiently flexible filaments can exhibit stationary states with shape multi-stability which in turn gives rise to rotational dynamics. Further, the time-dependent evolution towards such steady states is highly nontrivial with both wave-like and diffusive characteristics available depending on the elastic properties of the system.

We propose that topological order can replace sterile neutrinos as dark matter candidates to cancel the Standard Model global gravitational anomalies. Standard Model (SM) with 15 Weyl fermions per family (lacking the 16th, the sterile right-handed neutrino nuR) suffers from mixed gauge-gravitational anomalies tied to baryon number plus or minus lepton number B+(-)L symmetry. Including nuR per family can cancel these anomalies, but when B+(-)L symmetry is preserved as discrete finite subgroups rather than a continuous U(1), the perturbative local anomalies become nonperturbative global anomalies. We systematically enumerate these gauge-gravitational global anomalies involving discrete B+(-)L that are enhanced from the fermion parity Z2F to Z2NF. The discreteness of B+(-)L is constrained by multi-fermion deformations beyond-the-SM and the family number Nf. Unlike the free quadratic nuR Majorana mass gap preserving the minimal Z2F, we explore novel scenarios canceling (B+(-)L)-gravitational anomalies while preserving the Z2NF discrete symmetries, featuring 4-dimensional interacting gapped topological orders or gapless sectors (e.g., conformal field theories). We propose symmetric anomalous sectors as quantum dark matter to cancel SM global anomalies. We find the uniqueness of the family number at Nf = 3, such that when the representation of Z2NF from the faithful B+L for baryons at both Nf and N equal to 3 is extended to the faithful Q + NcL for quarks at N = NcNf = 9, this symmetry extension ZNc=3 to ZNcNf =9 to ZNf =3 matches with the topological order dark matter construction. Key implications include: (1) a 5th force mediating between SM and dark matter via discrete B+(-)L gauge fields, (2) dark matter as topological order quantum matter with gapped anyon excitations at ends of extended defects, and (3) Ultra Unification and topological leptogenesis. [Based on arXiv:2502.21319, arXiv:2501.00607, arXiv:2412.21196, arXiv:2411.05786, arXiv:2012.15860, arXiv:2112.14765, arXiv:2204.08393, arXiv:2302.14862, arXiv:2312.14928].