2-Chern-Simons theory, or more commonly known as 4d BF-BB theory with gauged shift symmetry, is a natural generalization of Chern-Simons theory to 4-dimensional manifolds. It is part of the bestiary of higher-homotopy Maurer-Cartan theories. In this article, we present a framework towards the combinatorial quantization of 2-Chern-Simons theory on the lattice, taking inspiration from the work of Aleskeev-Grosse-Schomerus three decades ago. The central geometric input is the 2-truncation $\Gamma^2$ of the $\infty$-groupoid of simplices formed by the underlying lattice $\Gamma$. On such a "2-graph", we model states of 2-Chern-Simons holonomies as Crane-Yetter's \textit{measureable fields}. We show that the 2-Chern-Simons action endows the 2-graph states -- as well as their quantum 2-gauge symmetries -- the structure of a Hopf category, and that their associated higher $R$-matriex gives it a comonoidal {\it cobraiding} structure. This is an explicit realization of the categorical ladder proposal of Baez-Dolan, in the context of Lie group lattice 2-gauge theories. Moreover, we will also analyze the lattice 2-algebra on the graph $\Gamma$, and extract the observables of discrete 2-Chern-Simons theory from it.

The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature $\beta_{\mathrm{c}}(q)$, where $q$ is the strength of the transverse field. Bj\"ornberg [Commun. Math. Phys., 232 (2013)] investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing $q$ and the spin-spin coupling $J$ in a proper manner, with fixed temperature. In this paper, we prove that the susceptibility diverges as $(\beta_{\mathrm{c}}(q)-\beta)^{-1}$ as $\beta\uparrow\beta_{\mathrm{c}}(q)$ for $d>4$ assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Bj\"ornberg & Grimmett [J. Stat. Phys., 136 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., $q=0$).

We consider an ensemble of $2\times 2$ normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining probability distributions based on symmetry and statistical independence. The probability densities turn out to be power law with exponents that depend on the boundedness of the domain. For small spacings, $\sigma$, the probability density varies as $\sigma^{\nu}$, $\nu \geq 2$. The degree of level repulsion is a parameter of great interest as it makes a connection to quantum chaos; the lower bound of $\nu$ for our ensemble coincides with the Gaussian Unitary Ensemble. We believe that the systematic development presented here paves the way for further generalizations in the field of random matrix theory for PT-symmetric quantum systems.

These are the notes on two-dimensional conformal field theory, based on a lecture course for graduate math students, given by P.M. in fall 2022 at the University of Notre Dame. These notes are intended to be substantially reworked and expanded in coauthorship with Nicolai Reshetikhin.

Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the characteristic polynomial for particular ensemble of $N \times N$ matrices is that it is expressed as the average of the power $N$ of the characteristic polynomial of some other ensemble of random matrices, now of size $k \times k$. With only a few exceptions, such dualities involve (the $\beta$ generalised) classical Gaussian, Laguerre and Jacobi ensembles Hermitian ensembles, the circular Jacobi ensemble, or the various non-Hermitian ensembles relating to Ginibre random matrices. In the case of unitary symmetry in the Hermitian case, they can be studied using the determinantal structure. The $\beta$ generalised case requires the use of Jack polynomial theory, and in particular Jack polynomial based hypergeometric functions. Applications to the computation of the scaling limit of various $\beta$ ensemble correlation and distribution functions are also reviewed. The non-Hermitian case relies on the particular cases of Jack polynomials corresponding to zonal polynomials, and their integration properties when their arguments are eigenvalues of certain matrices. The main tool to study dualities for moments of the spectral density, and generalisations, is the loop equations.

Meixner~(1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials which are orthogonal with respect to Gaussian distribution, Charlier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind, orthogonal with respect to negative binomial distribution, and Meixner polynomials of the second kind, orthogonal with respect to Meixner distribution. The Segal--Bargmann transform provides a unitary isomorphism between the $L^2$-space of the Gaussian distribution and the Fock or Segal--Bargmann space of entire funcitons. This construction was also extended to the case of the Poisson distribution. The present paper deals with the latter three classes of orthogonal Sheffer sequences. By using a set of nonlinear coherent states, we construct a generalized Segal--Bargmann transform which is a unitary isomorphism between the $L^2$-space of the orthogonality measure and a certain Fock space of entire functions. In a special case, such a Fock space was already studied by Alpay--J\o rgensen--Seager--Volok~(2013) and Alpay--Porat~(2018). To derive our results, we use normal ordering in generalized Weyl algebras that are naturally associated with the orthogonal Sheffer sequences.

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.

A new method, dual-space cluster expansion, is proposed to study classical phases transitions in the continuum. It relies on replacing the particle positions as integration variables by the momenta of the relative displacements of particle pairs. Due to the requirement that the particles must be static, coupling via the momenta partitions the set of particles into a set of clusters, and transforms the partition function into a sum over the different cluster decompositions. This allows us to derive a formula for the density that finite clusters can carry in the infinite system. In a simplified example, we then demonstrate that in two and higher dimensions this density has a threshold, beyond which the particles form infinite clusters. The transition is accompanied by a singularity in the free energy. We also show that infinite clusters are always present in condensed phases, most likely submacroscopic in liquids and macroscopic in crystals.

Throughout this paper, we comprehensively study instantons with every kind of continuous conformal symmetry. Examples of these objects are hard to come by due to non-linear constraints. However, by applying previous work on moduli spaces, we introduce a linear constraint, whose solution greatly simplifies these non-linear constraints. This simplification not only allows us to easily find a plethora of novel instantons with various continuous conformal symmetries and higher rank structure groups, it also provides a framework for classifying such symmetric objects. We also prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we discuss the connections between instantons with continuous symmetries and other gauge-theoretic objects: hyperbolic and singular monopoles as well as hyperbolic analogues to Higgs bundles and Nahm data.

We study a particular system of partial differential equations in which the harmonic, the divergence and the gradient operators of the unknown functions appear (harmonic-divgrad system). Using the Killing Hopf theorem and leveraging the properties of Riemannian manifolds with constant sectional curvature we establish the conditions under which these equations admit only the trivial solutions proving their trivialization on positive curvature space forms. The analysis of this particular system is motivated by its occurrence in the study of asymptotic symmetries in $p$-form gauge theories and in mixed symmetry tensor gauge theories.

The orthogonal momentum amplituhedron O_k was introduced in 2021 by Huang, Kojima, Wen, and Zhang in the study of scattering amplitudes of ABJM theory. It was conjectured that it admits a decomposition into BCFW cells. We prove this conjecture.

Mirzakhani-Zograf proved the large genus asymptotic expansions of Weil-Petersson volumes and showed that the asymptotic coefficients are polynomials in $\mathbb Q[\pi^{-2},\pi^2]$. They also conjectured that these are actually polynomials in $\mathbb Q[\pi^{-2}]$. In this paper, we prove Mirzakhani-Zograf's conjecture.

It is a folklore belief in the theory of spin glasses and disordered systems that out-of-equilibrium dynamics fail to find stable local optima exhibiting e.g. local strict convexity on physical time-scales. In the context of the Sherrington--Kirkpatrick spin glass, Behrens-Arpino-Kivva-Zdeborov\'a and Minzer-Sah-Sawhney have recently conjectured that this obstruction may be inherent to all efficient algorithms, despite the existence of exponentially many such optima throughout the landscape. We prove this search problem exhibits strong low degree hardness for polynomial algorithms of degree $D\leq o(N)$: any such algorithm has probability $o(1)$ to output a stable local optimum. To the best of our knowledge, this is the first result to prove that even constant-degree polynomials have probability $o(1)$ to solve a random search problem without planted structure. To prove this, we develop a general-purpose enhancement of the ensemble overlap gap property, and as a byproduct improve previous results on spin glass optimization, maximum independent set, random $k$-SAT, and the Ising perceptron to strong low degree hardness. Finally for spherical spin glasses with no external field, we prove that Langevin dynamics does not find stable local optima within dimension-free time.

We show that the fermion, in the context of a system that comprises many such entities - which, by virtue of the Pauli exclusion principle, possesses a Fermi surface at zero temperature - may itself be thought of as a collection of non-local particle-hole excitations across this Fermi surface. This result is purely kinematical and completely general - not being restricted to any specific dimension, applicable to both continuum and lattice systems. There is also no implication that it is applicable only to low-energy phenomena close to the Fermi surface. We are able to derive the full single-particle dynamical Green function of this fermion at finite temperature by viewing it as a collection of these non-local particle-hole excitations. The Green function of the fermion then manifests itself as a solution to a first-order differential equation in a parameter that controls the number of particle-hole pairs across the Fermi surface, and this equation itself reveals variable coefficients that may be identified with a Bose-Einstein distribution - implying that there is a sense in which the non-local particle-hole excitations have bosonic qualities while not being exact bosons at the level of operators. We also recall the definition of the non-local particle-hole operator that may be used to diagonalize the kinetic energy of free fermions of the sort mentioned above. Number-conserving products of creation and annihilation operators of fermions are expressible as a (rather complicated) combination of these non-local particle-hole operators.

I present an exact solution for the convex roof of the square root threetangle for all states within the Bloch sphere. The working hypothesis is that optimal decompositions contain as many states from the zero-polytope as possible which can be called zero-state locking. The footprint of the measure of entanglement consists in a characteristic pattern for the fixed pure states on the surface which form the optimal solution. The solution is subject to transformation properties due to the SL-invariance of the entanglement measure.

We theoretically investigate the many-body dynamics of a tight-binding chain with dephasing noise on the infinite interval. We obtain the exact solution of an average particle-density profile for the domain wall and the alternating initial conditions via the Bethe ansatz, analytically deriving the asymptotic expressions for the long time dynamics. For the domain wall initial condition, we obtain the scaling form of the average density, elucidating that the diffusive transport always emerges in the long time dynamics if the strength of the dephasing, no matter how small, is positive. For the alternating initial condition, our exact solution leads to the fact that the average density displays oscillatory decay or over-damped decay depending on the strength of the dissipation. Furthermore, we demonstrate that the asymptotic forms approach those of the symmetric simple exclusion process, identifying corrections from it.

The collective ground state for a spherical symmetric dust ball has been investigated recently in [R. Casadio, Phys. Lett. B 843 (2023) 138055]. In this study, we refine that model by obtaining a mass distribution that accounts for the superposition of wavefunctions across different layers. The refined mass distribution shows significant deviations from the approximation without quantum superpositions. Specifically, the new nearly parabolic distribution replaces the linear mass profile of the original work, featuring an overall downward concavity, which leads to a non-vanishing tension. Notably, the regularity of the metric and causal structure are preserved in the refined analysis.

Negative refractive index materials have garnered widespread attention due to their anomalous electromagnetic properties. In this paper, we utilize complementing boundary conditions to conduct a priori estimates for Cauchy problems and derive the limiting absorption principle. Consequently, we establish the well-posedness of the transmission problem involving conventional materials and negative refractive index materials within a simulated two-dimensional periodic structure.

A compilation of new results on the asymptotic behaviour of the Humbert functions $\Psi_1$ and $\Psi_2$, and also on the Appell function $F_2$, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the $1D$ Glauber-Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.

Given a countable discrete amenable group, we investigate the conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and single contractions. Additionally, we develop a relative version of this characterization, identifying conditions under which the additive realization can be chosen within a specified target set. As an application, our results generalize key aspects of thermodynamic formalism, bridging the gap between additive and non-additive frameworks.

In this paper, we explore the $\mathbb{Z}_2^n$-graded Lie (super)algebras as novel possible generators of symmetries of $S$-matrix. As the results, we demonstrate that a $\mathbb{Z}_2^n$-graded extension of the supersymmetric algebra can be a symmetry of $S$-matrix. Furthermore, it turns out that a $\mathbb{Z}_2^n$-graded Lie algebra appears as internal symmetries. They are natural extensions of Coleman-Mandula theorem and Haag-Lopszanski-Sohnius theorem, which are the no-go theorems for generators of symmetries of $S$-matrix.

We develop a conformally invariant (CI) framework in $(1+3)$-dimensional de Sitter (dS) spacetime, that unifies the descriptions of graviton, ``partially massless'' graviton, and photon fields. This framework is grounded in a rigorous group-theoretical analysis in the Wigner sense and employs Dirac's six-cone formalism. Originally introduced by Dirac, the concept of conformal space and the six-cone formalism were used to derive the field equations for spinor and vector fields in $(1+3)$-dimensional Minkowski spacetime in a manifestly CI form. Our framework extends this approach to dS spacetime, unifying the treatment of massless and ``partially massless'' fields with integer spin $s>0$ under conformal symmetry. This unification enhances the understanding of fundamental aspects of gravitational theories in curved backgrounds.

We adapt boundary deformation techniques to solve a Neumann problem for the Helmholtz equation with rough electric potentials in bounded domains. In particular, we study the dependance of Neumann eigenvalues of the perturbed Laplacian with respect to boundary deformation, and we illustrate how to find a domain in which the Neumann problem can be solved for any energy, if there is some freedom in the choice of the domain. This work is motivated by a Runge approximation result in the context of an inverse problem in point-source scattering with partial data.

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podle\'s sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.

There is increased interest in time-dependent (non-autonomous) Hamiltonians, stemming in part from the active field of Floquet quantum materials. Despite this, dispersive time-decay bounds, which reflect energy transport in such systems, have received little attention. We study the dynamics of non-autonomous, time-periodically forced, Dirac Hamiltonians: $i\partial_t\alpha =D(t)\alpha$, where $D(t)=i\sigma_3\partial_x+ \nu(t)$ is time-periodic but not spatially localized. For the special case $\nu(t)=m\sigma_1$, which models a relativistic particle of constant mass $m$, one has a dispersive decay bound: $\|\alpha(t,x)\|_{L^\infty_x}\lesssim t^{-\frac12}$. Previous analyses of Schr\"odinger Hamiltonians suggest that this decay bound persists for small, spatially-localized and time-periodic $\nu(t)$. However, we show that this is not necessarily the case if $\nu(t)$ is not spatially localized. Specifically, we study two non-autonomous Dirac models whose time-evolution (and monodromy operator) is constructed via Fourier analysis. In a rotating mass model, the dispersive decay bound is of the same type as for the constant mass model. However, in a model with a periodically alternating sign of the mass, the results are quite different. By stationary-phase analysis of the associated Fourier representation, we display initial data for which the $L^\infty_x$ time-decay rate are considerably slower: $\mathcal{O}(t^{-1/3})$ or even $\mathcal{O}(t^{-1/5})$ as $t\to\infty$.

Despite their simplicity, quantum harmonic oscillators are ubiquitous in the modeling of physical systems. They are able to capture universal properties that serve as reference for the more complex systems found in nature. In this spirit, we apply a model of a Hamiltonian for open quantum systems in equilibrium with a particle reservoir to ensembles of quantum oscillators. By treating (i) a dilute gas of vibrating particles and (ii) a chain of coupled oscillators as showcases, we demonstrate that the property of varying number of particles leads to a mandatory condition on the energy of the system. In particular, the chemical potential plays the role of a parameter of control that can externally manipulate the spectrum of a system and the corresponding accessible quantum states.