### The families of Hamiltonians sharing common symmetry structure

We describe a general procedure which allows to construct, starting from a given Hamiltonian, the whole family of new ones sharing the same set of unparameterized trajectories in phase space. The symmetry structure of this family can be completely characterized provided the symmetries of initial Hamiltonian are known. Our approach covers numerous examples considered in literature. It provides a simple proof of Darboux theorem and Hietarinta et al. coupling-constant metamorphosis method.

### Typical Macroscopic Long-Time Behavior for Random Hamiltonians

We consider a closed macroscopic quantum system in a pure state $\psi_t$ evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces $\mathcal{H}_\nu$ (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of $\psi_t$ looks like macroscopically, specifically on how much of $\psi_t$ lies in each $\mathcal{H}_\nu$. Previous bounds concerned the \emph{absolute} error for typical $\psi_0$ and/or $t$ and are valid for arbitrary Hamiltonians $H$; now, we provide bounds on the \emph{relative} error, which means much tighter bounds, with probability close to 1 by modeling $H$ as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of $H$ are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of $\psi_0$ from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin.

### Integrable systems of finite type from F-cohomological field theories without unit

One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology classes on the moduli spaces of curves satisfying certain natural splitting properties. Typically, these PDEs have an infinite expansion in the dispersive parameter, which happens because they involve contributions from the moduli spaces of curves of arbitrarily large genus. In this paper, for each rank $N\ge 2$, we present a family of F-CohFTs without unit, for which the equations of the associated integrable system have a finite expansion in the dispersive parameter. For $N=2$, we explicitly compute the primary flows of this integrable system.

### Algorithmic Threshold for Multi-Species Spherical Spin Glasses

We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational principle that we study in detail. We rely on the branching overlap gap property introduced in our previous work and develop a new method to establish it that does not require the interpolation method. Consequently our results apply even for models with non-convex covariance, where the Parisi formula for the true ground state remains open. As a special case, we obtain the algorithmic threshold for all single-species spherical spin glasses, which was previously known only for even models. We also obtain closed-form formulas for pure models which coincide with the $E_{\infty}$ value previously determined by the Kac-Rice formula.

### Tau function and moduli of meromorphic forms on algebraic curves

We study the moduli space of meromorphic 1-forms on complex algebraic curves with at most simple poles and fixed residues. We interpret the Bergman tau function on this moduli space as a section of a line bundle and study its asymptotic behavior near the boundary. As an application, we decompose its pushforward to the moduli space of curves into a linear combination of standard generators of the rational Picard group with explicit coefficients that depend on the residues.

### Asymptotics of Helmholtz-Kirchhoff Point-Vortices in the Phase Space

A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz-Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier-Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.

### Determining coefficients of thermoelastic system from boundary information

Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for full symbol of the thermoelastic Dirichlet-to-Neumann map $\Lambda_g$ with variable coefficients $\lambda,\mu,\alpha,\beta \in C^{\infty}(\bar{M})$. We prove that $\Lambda_g$ uniquely determines partial derivatives of all orders of the coefficients on the boundary. Moreover, for a nonempty open subset $\Gamma\subset\partial M$, suppose that the manifold and the coefficients are real analytic up to $\Gamma$, we show that $\Lambda_g$ uniquely determines the coefficients on the whole manifold $\bar{M}$.

### Quantum non-Markovianity: Overview and recent developments

In the current era of noisy intermediate-scale quantum (NISQ) devices, research in the theory of open system dynamics has a crucial role to play. In particular, understanding and quantifying memory effects in quantum systems is critical to gain a better handle on the effects of noise in quantum devices. The main focus of this review is to address the fundamental question of defining and characterizing such memory effects -- broadly referred to as quantum non-Markovianity -- from various approaches. We first discuss the two-time-parameter maps approach to open system dynamics and review the various notions of quantum non-Markovianity that arise in this paradigm. We then discuss an alternate approach to quantum stochastic processes based on the quantum combs framework, which accounts for multi-time correlations. We discuss the interconnections and differences between these two paradigms, and conclude with a discussion on necessary and sufficient conditions for quantum non-Markovianity.

### Uniqueness for the Camassa-Holm equation with non-homogeneous boundary conditions

We establish the uniqueness of solutions of the Camassa-Holm equation on a finite interval with non-homogeneous boundary conditions in the case of bounded momentum. A similar result for the higher-order Camassa-Holm system is also given. Our proofs rely on energy-type methods, with some multipliers given as solutions of some auxiliary elliptic systems.

### Integrable Mechanical Billiards in Higher Dimensional Space Forms

In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combination of quadrics having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and the hyperbolic space by means of central projection. With the same approach, we also extend these results to the $n$-dimensional cases.

### BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories whose stack of objects admits a good moduli space. We prove that this algebra is isomorphic to the positive part of the enveloping algebra of a generalised Kac-Moody Lie algebra generated by the intersection cohomology of certain connected components (corresponding to roots) of the good moduli space. Some major examples include the BPS algebras of (1) the category of semistable coherent sheaves of given slope on a K3 or more generally quasiprojective symplectic surface (2) preprojective algebras of quivers, (3) multiplicative preprojective algebras and (4) fundamental groups of (quiver) Riemann surfaces. We define the BPS Lie algebras of 2-Calabi-Yau categories and prove that they coincide with the ones obtained by dimensional reduction and the critical cohomological Hall algebra. Consequences include (1) A proof in full generality of the positivity conjecture for absolutely cuspidal polynomials of Bozec-Schiffmann, a strengthening of the Kac positivity conjecture (2) A proof of the cohomological integrality conjecture for local K3 surfaces (3) A lowest weight vector description for the cohomology (in all degrees) of Nakajima quiver varieties.

### Moduli of Higgs bundles over the five punctured sphere

We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $\mu\in[0,1]^5$. The moduli space corresponding to the central weight $\mu_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus $n$ points is false.

### Machine Learning in Physics and Geometry

We survey some recent applications of machine learning to problems in geometry and theoretical physics. Pure mathematical data has been compiled over the last few decades by the community and experiments in supervised, semi-supervised and unsupervised machine learning have found surprising success. We thus advocate the programme of machine learning mathematical structures, and formulating conjectures via pattern recognition, in other words using artificial intelligence to help one do mathematics.

### Current fluctuations in a semi-infinite line

We present an application of a fluctuating hydrodynamic theory to study current fluctuations in diffusive systems on a semi-infinite line in contact with a reservoir with a slow coupling. Within this hydrodynamic framework, we show that the distribution of time-integrated current across the boundary at large times follows a large deviation principle with a rate function that depends on the coupling strength with the reservoir. The system exhibits a long-term memory of its initial state which was earlier reported on an infinite line and can be described using quenched and annealed averaging on the initial state. We present an explicit expression of the rate function for independent particles, which we verify using an exact solution of the microscopic dynamics. For the symmetric exclusion process we present expression of the first three cumulants for both quenched and annealed ensemble.

### Affine Nijenhuis operators and Hochschild cohomology of trusses

The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced in [{\sc J.F.\ Cari\~nena, J.\ Grabowski, G.\ Marmo}, {\it Quantum Bi-Hamiltonian Systems}. Int.\ J.\ Mod.\ Phys.\ A {\bf 15}, 4797--4810, 2000.]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative {\em affgebras}. It is shown that this construction leads to compatible Lie brackets on an affine space.

### Topological classification of non-Hermitian Hamiltonians with frequency dependence

We develop a topological classification of non-Hermitian effective Hamiltonians that depend on momentum and frequency. Such effective Hamiltonians are in one-to-one correspondence to single-particle Green's functions of systems that satisfy translational invariance in space and time but may be interacting or open. We employ K-theory, which for the special case of noninteracting systems leads to the well-known tenfold-way topological classification of insulators and fully gapped superconductors. Relevant theorems for K-groups are reformulated and proven in the more transparent language of Hamiltonians instead of vector bundles. We obtain 54 symmetry classes for frequency-dependent non-Hermitian Hamiltonians satisfying anti-unitary symmetries. Employing dimensional reduction, the group structure for all these classes is calculated. This classification leads to a group structure with one component from the momentum dependence, which corresponds to the non-Hermitian generalization of topological insulators and superconductors, and two additional parts resulting from the frequency dependence. These parts describe winding of the effective Hamiltonian in the frequency direction and in combined momentum-frequency space.

### Kullback-Leibler divergence for the Fréchet extreme-value distribution

We derive a closed-form solution for the Kullback-Leibler divergence between two Fr\'echet extreme-value distributions. The resulting expression is rather simple and involves the Euler-Mascheroni constant.

### Lie 2-groups from loop group extensions

We give a very simple construction of the string 2-group as a strict Fr\'echet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.

### Numerical Bifurcation Analysis of Turing and Symmetry Broken Patterns of a Vegetation PDE Model

We study the mechanisms of pattern formation for vegetation dynamics in water-limited regions. Our analysis is based on a set of two partial differential equations (PDEs) of reaction-diffusion type for the biomass and water and one ordinary differential equation (ODE) describing the dependence of the toxicity on the biomass. We perform a linear stability analysis in the one-dimensional finite space, we derive analytically the conditions for the appearance of Turing instability that gives rise to spatio-temporal patterns emanating from the homogeneous solution, and provide its dependence with respect to the size of the domain. Furthermore, we perform a numerical bifurcation analysis in order to study the pattern formation of the inhomogeneous solution, with respect to the precipitation rate, thus analyzing the stability and symmetry properties of the emanating patterns. Based on the numerical bifurcation analysis, we have found new patterns, which form due to the onset of secondary bifurcations from the primary Turing instability, thus giving rise to a multistability of asymmetric solutions.

### On a Torsion/Curvature Analogue of Dual Connections and Statistical Manifolds

In analogy to the concept of a non-metric dual connection, which is essential in defining statistical manifolds, we develop that of a torsion dual connection. Consequently, we illustrate the geometrical meaning of such a torsion dual connection and show how the use of both connections preserves the cracking of parallelograms in spaces equipped with a connection and its torsion dual. The coefficients of such a torsion dual connection are essentially computed by demanding a vanishing mutual torsion among the two connections. For this manifold we then prove two basic Theorems. In particular, if both connections are metric-compatible we show that there exists a specific $3$-form measuring how the connection and its torsion dual deviate away from the Levi-Civita one. Furthermore, we prove that for these torsion dual manifolds flatness of one connection does not necessary impose flatness on the other but rather that the curvature tensor of the latter is given by a specific divergence. Finally, we give a self-consistent definition of the mutual curvature tensor of two connections and subsequently define the notion of a curvature dual connection.

### Extremal Black Holes as Relativistic Systems with Kepler Dynamics

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrization to the Kepler problem, and that all Hamiltonians conserving a Laplace-Runge-Lenz-like vector are related in this way to Kepler.

### The Ternary Structure of Lie Algebroid Connections

We examine the abelian heap of linear connections on anchored vector bundles and Lie algebroids. We show how the ternary structure on the set of linear connections interacts' with the torsion and curvature tensors. The endomorphism truss of linear connections is constructed.

### Hirota type Bazhanov--Stroganov maps

Bazhanov--Stroganov maps are set theoretical solutions to the 4-simplex equation, namely the fourth member of the family of $n$-simplex equations, which are fundamental equations of mathematical physics. In this letter, we develop a method for constructing Bazhanov--Stroganov 4-simplex maps as extensions of solutions to the Zamolodchikov tetrahedron equation. We employ this method to construct birarional, noninvolutive 4-simplex maps which boil down to the famous Hirota tetrahedron map at a certain limit.

### Approximations of the modified Bessel functions of the second kind $K_ν$. Applications in random field generation

We propose an analytical approximation for the modified Bessel function of the second kind $K_\nu$. The approximation is derived from an exponential ansatz imposing global constrains. It yields local and global errors of less than one percent and a speed-up in the computing time of $3$ orders in magnitude in comparison with traditional approaches. We demonstrate the validity of our approximation for the task of generating long-range correlated random fields.

### On the stability of solutions to Schrödinger's equation short of the adiabatic limit

We prove an adiabatic theorem that applies at timescales short of the adiabatic limit. Our proof analyzes the stability of solutions to Schrodinger's equation under perturbation. We directly characterize cross-subspace effects of perturbation, which are typically significantly less than suggested by the perturbation's operator norm. This stability has numerous consequences: we can (1) find timescales where the solution of Schrodinger's equation converges to the ground state of a block, (2) lower bound the convergence to the global ground state by demonstrating convergence to some other known quantum state, (3) guarantee faster convergence than the standard adiabatic theorem when the ground state of the perturbed Hamiltonian ($H$) is close to that of the unperturbed $H$, and (4) bound tunneling effects in terms of the global spectral gap when $H$ is `stoquastic'' (a $Z$-matrix). Our results apply to quantum annealing protocols with faster convergence than usually guaranteed by a standard adiabatic theorem. Our upper and lower bounds demonstrate that at timescales short of the adiabatic limit, subspace dynamics can dominate over global dynamics. Thus, we see that convergence to particular target states can be understood as the result of otherwise local dynamics.