Papers

Topological $K$-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary $K$-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface $X$, we prove nonfactorization in three settings: fibrewise $K^0(X)$, occupied-state $K^0(B)$, and operator-algebraic $K$-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani–Sunada bottom-band Hessian is the Hodge inner product on $H^1(X;\mathbb R)$. Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichmüller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as $O(|G_n|^{-\eta})$ for some $\eta>0$, and $C^s$ observables as $O((\log |G_n|)^{-s})$, while established coherent large-rank limits recover bulk density-of-states moments.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce LieStoNet, an end-to-end, template-free framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$, $0<\alpha\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $\delta$-shock, we assume $\alpha\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $\delta$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.