CyberSec Research

We construct generalized symmetries in two-dimensional symmetric product orbifold CFTs $\text{Sym}^N(\mathcal{T}),$ for a generic seed CFT $\mathcal{T}$. These symmetries are more general than the universal and maximally symmetric ones previously constructed. We show that, up to one fine-tuned example when the number of copies $N$ equals four, the only symmetries that can be preserved under twisted sector marginal deformations are invertible and maximally symmetric. The results are obtained in two ways. First, using the mathematical machinery of $G$-equivariantization of fusion categories, and second, via the projector construction of topological defect lines. As an application, we classify all preserved symmetries in symmetric product orbifold CFTs with the seed CFT given by any $A$-series $\mathcal{N}=(2,2)$ minimal model. We comment on the implications of our results for holography.

We study asymptotics of the $d=4$, $\mathcal{N}=1$ superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-$N$ index in terms of the $R$-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell $R$-charges, we determine the asymptotic degeneracy in the univariate specialization for $\hat{A}_{m}$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and $\hat{A}_{3}$. In these cases we find $\ln |c_{n}| \sim \gamma n^{\frac{1}{2}}+ \beta \ln n + \alpha$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $\gamma$ in further $Y^{p,p}$ examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the discontinuity surface.

A key challenge in quantum complexity is how entanglement structure emerges from dynamics, highlighted by advances in simulators and information processing. The Lieb--Robinson bound sets a locality-based speed limit on information propagation, while the Small-Incremental-Entangling (SIE) theorem gives a universal constraint on entanglement growth. Yet, SIE bounds only total entanglement, leaving open the fine entanglement structure. In this work, we introduce Spectral-Entangling Strength, measuring the structural entangling power of an operator, and prove a Spectral SIE theorem: a universal limit for R\'enyi entanglement growth at $\alpha \ge 1/2$, revealing a robust $1/s^2$ tail in the entanglement spectrum. At $\alpha=1/2$ the bound is qualitatively and quantitatively optimal, identifying the universal threshold beyond which growth is unbounded. This exposes the detailed structure of Schmidt coefficients, enabling rigorous truncation-based error control and linking entanglement to computational complexity. Our framework further establishes a generalized entanglement area law under adiabatic paths, extending a central principle of many-body physics to general interactions. Practically, we show that 1D long-range interacting systems admit polynomial bond-dimension approximations for ground, time-evolved, and thermal states. This closes the quasi-polynomial gap and proves such systems are simulable with polynomial complexity comparable to short-range models. By controlling R\'enyi entanglement, we also derive the first rigorous precision-guarantee bound for the time-dependent density-matrix-renormalization-group algorithm. Overall, our results extend SIE and provide a unified framework that reveals the detailed structure of quantum complexity.

This paper is focused on the development of the notions of canonical and canonoid transformations within the framework of Hamiltonian Mechanics on locally conformal symplectic manifolds. Both, time-independent and time-dependent dynamics are considered. Noether-like theorems relating one-parameter groups of transformations with canonical and noncanonical symmetries, are formulated, proved as well as illustrated with elementary examples.

We derive and analyze a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. Our system consists of a Navier--Stokes--Cahn--Hilliard model in the bulk that is coupled to a surface Navier--Stokes--Cahn--Hilliard model on the boundary. Compared with previous models, the inclusion of an additional surface Navier--Stokes equation is motivated, for example, by biological applications such as the seminal \textit{fluid mosaic model} (Singer \& Nicolson, \textit{Science}, 1972) in which the surface of biological cells is interpreted as a thin layer of viscous fluids. We derive our new model by means of local mass balance laws, local energy dissipation laws, and the Lagrange multiplier approach. Moreover, we prove the existence of global weak solutions via a semi-Galerkin discretization. The core part of the mathematical analysis is the study of a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator. Its eigenfunctions are used as the Galerkin basis to discretize the bulk-surface Navier--Stokes subsystem.

We compare the notions of metric-compatibility and torsion of a connection in the frameworks of Beggs-Majid and Mesland-Rennie. It follows that for $\ast$-preserving connections, compatibility with a real metric in the sense of Beggs-Majid corresponds to Hermitian connections in the sense of Mesland-Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland-Rennie for unitary cocycle deformations of a large class of Riemannian manifolds as well as the Heckenberger-Kolb calculi on all quantized irreducible flag manifolds.

We provide non-asymptotic error bounds in the path Wasserstein distance with quadratic integral cost between suitable functionals of the telegraph process and the corresponding functional of Brownian motion with explicit diffusivity constant. These results cover, in particular, the well-known example of the exponential integral functional of the geometric Brownian motion. The non-asymptotic error bounds tend to zero in the so-called Kac regime. Moreover, the error bounds remain valid when the flip rate for the telegraph process is small. We assess the sharpness of the error bounds through numerical experiments.

The $l$-hypermaps, $l\ge2$, which generalize (dual of) ribbon graphs ($l=2$ case), are interesting enumerative objects. In this paper, based on a theorem of Carlet--van de Leur--Posthuma--Shadrin and the matrix-resolvent method, we derive an explicit formula for $k$-point generating series of enumeration of $l$-hypermaps, which generalizes the one obtained in [30] for the $l=2$ case. We also generalize a theorem of Dubrovin [29].

In this work, we investigate new solutions to the decoration transformation in terms of various special functions, including the hyperbolic gamma function, the basic hypergeometric function, and the Euler gamma function. These solutions to the symmetry transformation are important to decorate Ising-like integrable lattice spin models obtained via the gauge/YBE correspondence. The integral identities represented as the solution of the decoration transformation are derived from the three-dimensional partition functions and superconformal index for the dual supersymmetric gauge theories.

We continue the work of Belliard, Pimenta and Slavnov (2024) studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.

Quantum instruments are mathematical devices introduced to describe the conditional state change during a quantum process. They are completely positive map valued measures on measurable spaces. We may also view them as non-commutative analogues of joint probability measures. We analyze the $C^*$-convexity structure of spaces of quantum instruments. A complete description of the $C^*$-extreme instruments in finite dimensions has been established. Further, the implications of $C^*$-extremity between quantum instruments and their marginals has been explored.

The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative.

The atmospheric structure of gas giants, especially those of Jupiter and Saturn, has been an object of scientific studies for a long time. The measurement of the gravitational fields by the Juno mission for Jupiter and the Cassini mission for Saturn offered new possibilities to study the interior structure of these planets. Accordingly, the reconstruction of the wind velocities from gravitational data on gas giants has been the subject of many research papers over the years, yet the mathematical foundations of this inverse problem and its numerical resolution have not been studied in detail. This article suggests a rigorous mathematical theory for inferring the wind fields of gas giants. In particular, an orthonormal basis is derived which can be associated to models of the gravitational potential and the interior wind velocity field. Moreover, this approach provides the foundations for existing resolution concepts of the inverse problem.

We perform a Lie symmetry analysis on the tempered-fractional Keller Segel (TFKS) system, a chemo-taxis model incorporating anomalous diffusion. A novel approach is used to handle the nonlocal nature of tempered fractional operators. By deriving the full set of Lie point symmetries and identifying the optimal one-dimensional subalgebras, we reduce the TFKS PDEs to ordinary differential equations (ODEs), yielding new exact solutions. These results offer insights into the long-term behavior and aggregation dynamics of the TFKS model and present a methodology applicable to other tempered fractional differential equations.

Exact single-time and two-time correlations and the two-time response function are found for the order-parameter in the voter model with nearest-neighbour interactions. Their explicit dynamical scaling functions are shown to be continuous functions of the space dimension $d>0$. Their form reproduces the predictions of non-equilibrium representations of the Schr\"odinger algebra for models with dynamical exponent $\mathpzc{z}=2$ and with the dominant noise-source coming from the heat bath. Hence the ageing in the voter model is a paradigm for relaxations in non-equilibrium critical dynamics, without detailed balance, and with the upper critical dimension $d^*=2$.

We study Poincar{\'e} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{\'e} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{\'e} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest.

We discuss and formalize a topological means by which the initial singularity might be mollified, at the level of the spacetime manifold's structure, in classical cosmological models of a homogeneous expanding universe. The construction, dubbed a "reflective" topological big bang, generalizes Schrodinger's elliptic de Sitter space and is built to be compatible with the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) picture of the large-scale universe, only minimally modifying it via some nontrivial topology at an earliest "moment" in the universe's history. We establish a mathematical characterization of the admissible topological structures of reflective topological big bangs, and we discuss implications for a standard concern in cosmology, the horizon problem. We present a nonreflective example that we've christened the Itty-Bitty Blender spacetime: this spacetime and its universal cover, the Eternal Trumpet spacetime, exhibit interesting potential structures of spacetimes avoiding the Hawking and Penrose singularity theorems.