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Non-Hermitian systems with parity-time ($\mathcal{PT}$) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the…

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…

The phenomenon of degeneracy of an $N-$plet of bound states is studied in the framework of quantum theory of closed (i.e., unitary) systems. For an underlying Hamiltonian $H=H(\lambda)$ the degeneracy occurs at a Kato's exceptional point…

Classical atomistic simulations based on interatomic potentials resolve lattice instabilities, defect nucleation, and microstructure evolution with high fidelity, but their accessible system sizes remain far below those required for…

We present possible manifestations of octahedral and tetrahedral symmetries in nuclei. These symmetries are associated with the OhD and TdD double point groups. Both of them have very characteristic finger-prints in terms of the nucleonic…

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

Exceptional point degeneracies (EPD) of linear non-Hermitian systems have been recently utilized for hypersensitive sensing. This proposal exploits the sublinear response that the degenerate frequencies experience once the system is…

We investigate non-Hermitian degeneracies, also known as exceptional points, in continous elastic media, and their potential application to the detection of mass and stiffness perturbations. Degenerate states are induced by enforcing…

Manifestly non-Hermitian quantum graphs with real spectra are introduced and shown tractable as a new class of phenomenological models with several appealing descriptive properties. For illustrative purposes, just equilateral star-graphs…

We propose a new method of the $\beta$-$\gamma$ constraint for quadrupole deformation in antisymmetrized molecular dynamics (AMD) to describe various cluster and shell-model structures in the ground and excited states of light nuclei. We…

Manifestations of pronounced shell effects are discovered when adding nonaxial octupole deformations to a harmonic oscillator model. The degeneracies of the quantum spectra are in a good agreement with the corresponding main periodic orbits…

The structural parameters of alpha helix and some forms of DNA-structures are determined by methods of algebraic topology. These structures are locally periodic and correspond to the bifurcation point for minimal surfaces given by…

We show how the modular symmetries that have been found to be consistent with most available scaling data from quantum Hall systems, derive from a rigid family of algebraic curves of the elliptic type. The complicated special functions…

In holography, the IR behavior of a quantum system at nonzero density is described by the near horizon geometry of an extremal charged black hole. It is commonly believed that for systems on $S^3$, this near horizon geometry is $AdS_2\times…

Isomorphs are curves in the thermodynamic phase diagram of invariant excess entropy, structure, and dynamics, while pseudoisomorphs are curves of invariant structure and dynamics, but not of the excess entropy. The latter curves have been…

Shells are ubiquitous thin structures that can undergo large nonlinear elastic deformations while exhibiting combined modes of bending and stretching, and have profound modern applications. In this paper, we have proposed a new Isogeometric…

We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many…

The von Neumann equation with delta self-interaction kernel serves as a statistical model for nonlinear waves, and it exhibits a bifurcation between stable and unstable regimes. In oceanography it is known as the Alber equation, and its…

Topologically ordered states are fundamentally important in theoretical physics, which are also suggested as promising candidates to build fault-tolerant quantum devices. However, it is still elusive how topological orders can be affected…

In this thesis I present numerical studies of entanglement entropy and the quantum null energy condition in strongly coupled far-from-equilibrium quantum states using holography. I give a careful introduction into the numerical methods used…