Papers

Portfolio optimization under uncertainty is inherently a multi-objective decision problem involving complex interactions among return, risk, market dynamics, and practical investment constraints. Existing reliability based portfolio optimization approaches primarily rely on static optimization frameworks and often fail to capture sequential decision making, tail risk, and market frictions such as transaction costs. To address these limitations, we propose a deep reinforcement learning framework for multi-objective reliability based portfolio optimization (MORP-DRL). The proposed framework jointly optimizes expected return and downside risk using three complementary risk measures: variance, Conditional Value-at-Risk (CVaR), and Entropic Value-at-Risk (EVaR). To model uncertainty and heavy-tailed market behavior, asset returns are represented using GARCH(1,1), Extreme Value Theory, and a t-copula dependence structure, while realistic scenarios are generated through quasi-Monte Carlo simulation. A Proximal Policy Optimization (PPO) based strategy is developed under practical constraints including transaction costs and portfolio bounds, and is benchmarked against NSGA-II. Experiments on ten global equity indices across pre-COVID, COVID, and post-COVID market regimes demonstrate that MORP-DRL achieves competitive risk-return performance, reduced downside risk during periods of market stress, and scalability to high-dimensional portfolio settings.

The classical Vieta formulas relate the coefficients of a complex scalar polynomial to the elementary symmetric polynomials of its roots. In this paper, we establish analogous spectral identities for complex matrix polynomials. For a monic matrix polynomial, we prove that the sum of all its eigenvalues equals the sum of the roots of the product of its diagonal scalar polynomials and is also equal to a constant multiple of the sum of the roots of the scalar polynomial obtained by summing its diagonal entries. We further show that the product of the eigenvalues of a monic matrix polynomial is determined by the determinant of its constant coefficient matrix. As a consequence, we recover the matrix Vieta formulas of Fuchs and Schwarz \cite{Fuchs-Schwarz}, for independent solutions of matrix algebraic equations via a density argument. We also derive corresponding identities for non-monic matrix polynomials with nonsingular leading coefficients and show, by means of an example, that the identities established for monic matrix polynomials do not extend directly to the non-monic case.

We prove that the thick part of the $\mathrm{PSL}_n(\mathbb{R})$-Hitchin-Riemann moduli space has infinite total Atiyah–Bott–Goldman volume for $n>2$. This result stands in contrast to Mumford's compactness criterion. To achieve this result, we employ Goldman flows and internal sequences to find an infinite series of subsets of identical volume, the images of which in the Hitchin-Riemann moduli space are all mutually disjoint and sit in the thick part.

Let $\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Draganić, Keevash and Müyesser recently proved that every regular Dirac graph has $\Omega(2^n)$ cyclic subsets, resolving a problem of Erdős and Faudree. We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\Omega(n)$ and $d<n/2$, then $$ \operatorname{Cyc}(G)\ge (q-o(1))2^{n/q}, \quad \text{where } \quad q=\left\lfloor \frac{n}{d+1}\right\rfloor \ge 2. $$ This bound is asymptotically best possible, including the leading coefficient $q$, as witnessed at the staircase levels by the disjoint union of $q$ equal cliques. Consequently, the optimal exponential rate changes by discrete jumps as $d$ crosses the thresholds $n/k$, rather than varying smoothly with $d$. We also prove the optimal exponential rate at the Dirac boundary: every $n$-vertex $n/2$-regular graph satisfies $\operatorname{Cyc}(G)\ge 2^{(1-o(1))n},$ which is sharp up to a subexponential factor by $K_{n/2,n/2}$.

Reiher's clique density theorem determines the sharp lower envelope for the density of $K_r$ at fixed edge density. We prove a higher-order version in which the prescribed quantity is itself a clique density. For every $3\le s<r$, we determine the minimum possible $K_r$-density among graphons with prescribed $K_s$-density. For $s\ge3$ the constraint is genuinely nonlinear and leaves the edge density undetermined; nevertheless, on the positive range the sharp lower boundary is the classical multipartite edge-to-clique profile, reparametrised by $K_s$-density. We also prove stability on the positive branches of this profile: at every interior point, near extremality forces cut-distance closeness to the corresponding extremal family at the induced edge density.

In this paper, we introduce various types of inner products and norms on the bicomplex number system $\mathbb{C}_2$, the bicomplex vector space ${\mathbb{C}_2}^{n}$, the space of bicomplex matrices ${C_2}^{m \times n}$, and the space of bicomplex polynomials $\mathbb{C}_2[\xi]_n$. We investigate the relationships among these inner products and norms, and establish several results. Furthermore, we prove that $\mathbb{C}_2$ and ${\mathbb{C}_2}^{n}$ are Banach algebras and Hilbert spaces. These results provide a unified framework for the study of inner product structures and normed linear spaces over bicomplex numbers and their associated spaces.

Among many (equivalent, via invertible transformations) representations of the evolution of a dynamical system, which one is to be preferred? Here we show how the use of infinite-dimensional Lie group theory (and its numerical implementation) allows us to single out one representation, by selecting an element of the group of diffeomorphisms acting on the dynamical system. We present and discuss several types of “phase conditions" defining the selected representation, and illustrate their computational implementation. Study of dynamics modulo diffeomorphisms “liberates" mathematical modeling of physical phenomena from a user's preferred coordinates, and spontaneously selects a “right latent space" for the system.

In recent years, quantum linear system algorithms have been applied to partial differential equations (PDEs), particularly in high-dimensional settings, demonstrating an exponential speedup in dimension. Concurrently, randomized and quantum-inspired classical linear solvers have emerged, showing computational complexity comparable to their quantum counterparts in many application areas. In this paper, we investigate the applicability of these quantum-inspired classical algorithms to PDEs. We provide both upper and lower bounds on their computational complexity, proving that these methods cannot achieve exponential speedup in dimension for discretizations of high-dimensional Poisson problems. Our theoretical findings definitively demonstrate that quantum-inspired classical algorithms are not competitive with quantum algorithms for solving PDEs, confirming that quantum methods retain a significant advantage for high-dimensional problems.

Mixed-integer linear programming (MILP) instances used for solver development are hard to obtain when models come from private or application-specific pipelines. A generator must keep the structure that solvers and learned policies rely on. Existing general generators usually choose their generation unit from a formulation template, summary statistics, local graph edits, or blocks found after recombination. These units do not explicitly record how a local part of the MILP is coupled to the rest of the instance. We propose GraphBU, a graph-native generator whose basic unit is a local subproblem plus its interface. The method promotes coupling nodes into master constraints or boundary variables and uses the resulting block units for compatibility-checked replacement. The analysis focuses on the properties needed by this construction: promotion separates interfaces, replacement can preserve feasibility under an interface-slack condition, and the graph construction is invariant to row-column permutations. On MILP instances generation, this unit keeps graph statistics close to the source family, preserves feasibility on most datasets, and improves downstream Predict-and-Search training. Genrated by GraphBU, The average graph-statistical similarity was approximately 0.934, the average feasibility was approximately 96.7%, and the average increase in the main index of downstream PS was approximately 8.0%.

We say that a symmetric function space $X$ has the $(IR)$ property whenever all sets of $N$ independent mean zero functions $f_1,\ldots,f_N\in X$, $\|f_k\|_X\ge 1$, are poorly approximated by any linear combinations of arbitrary $n$ functions, if $n$ is sufficienly smaller that $N$; namely, for some $\gamma=\gamma(X)>0$ we have $d_n(\{f_1,\ldots,f_N\},X)\ge \gamma$, $n\le \gamma N$, where $d_n(K,X)$ is the Kolmogorov $n$-width of the set $K\subset X$. The spaces $X=L_p$ satisfy this property if and only if $1\le p\le2$ or $p=\infty$. The goal of this paper is to move from $L_p$ scale to a larger class of symmetric spaces. We obtain rather broad conditions, under which such a space $X$ has the $(IR)$ property and prove precise statements for particular scales of Lorentz $L_{p,q}$ spaces and Orlicz spaces.

The Vietoris-Rips filtration $\mathcal{VR}(-)$ is a standard tool for analyzing the shape of data within topological data analysis. Beginning with seminal work of Sheehy, a substantial amount of research has centered on constructing linear-size sparse approximations to $\mathcal{VR}(-)$ and related filtrations for metric spaces of bounded doubling dimension. We show that this geometric assumption is necessary in a precise sense. Working in the framework of homotopy interleavings, we show that for any fixed $c \in [1, \sqrt{2})$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has exponential size. We also show that for any fixed $c \geq 1$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has superlinear size, yielding an obstruction to linear-size approximations for any fixed approximation factor. Both results extend to the intrinsic Čech filtration and to any bifiltration containing $\mathcal{VR}(-)$ as a $1$-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.

Quantum sensing is a promising technology capable of demonstrating clear advantage over comparable classical techniques for precise measurement. One application of quantum sensing is in function estimation, which can be done using a network of entangled quantum sensors, allowing for measurements with greater optimal sensitivity than unentangled sensing protocols. In cases where quantum sensor networks will be used to measure data that should remain private (e.g., biomedical data), it is imperative that these protocols include a privacy mechanism to hide sensitive information. In this work, we show that entangled sensor networks are vulnerable to certain privacy-violating attacks. To mitigate these attacks, we introduce secure sensing protocols endowed with differential privacy. We reconcile differential privacy with retaining Heisenberg-limited scaling, and introduce several protocols achieving varying balances between the two. We show that our main protocol, an $n$-node network sensing protocol that injects noise directly into the sensing Hamiltonian, exhibits a tradeoff between the desirable $O(1/n^2)$ Heisenberg scaling of the mean-squared error of the function estimate and the level of privacy attainable. Under assumptions on the network (a common source of randomness and a constant fraction of honest parties), we show that this protocol is locally implementable and achieves $(O(1), \delta)$-differential privacy for arbitrarily small $\delta$ while retaining Heisenberg scaling of the mean-squared error. We prove that our protocols are resilient to attacks by broad classes of classical and quantum adversaries, and find advantages in the privacy-utility tradeoff when using quantum techniques.

For integers $1\le \ell<k$, let $\Pi^k_\ell$ denote the single-forbidden $\ell$-degree Turán spectrum of $k$-uniform hypergraphs. We introduce transfer functions for this spectrum: explicit functions $f$ such that, for every $F$, there is another single $k$-graph $F^*$ with $\pi_\ell(F^*)=f(\pi_\ell(F))$. This gives a mechanism for producing new single-forbidden densities while retaining full control of the resulting value. Our transfer functions are realized by a new family of suspension-type operations, called tree suspensions. From these operations we obtain three explicit maps: one acting on $\Pi^k_\ell$ for every $1\le\ell<k$, a second acting when $\ell\ge k/2$, and a third acting in the ordinary Turán case $\ell=1$. The common feature is a robust tree structure which gives the lower bound by a two-part construction and, in the regimes above, admits a matching embedding or Lagrangian upper bound. As a first application, the universal transfer function propagates accumulation points. Using the recent zero-accumulation results for $\ell\ge2$ together with the ordinary Turán accumulation result of Conlon and Schülke, we prove that $\Pi^k_\ell$ has infinitely many accumulation points for every $k\ge3$ and every $1\le\ell<k$. This recovers, in particular, the known infinitude of accumulation points in the ordinary and codegree spectra. As a second application, combining two independent transfer functions forces algebraic degrees to grow. For every $k\ge3$ and every $\ell\in\{1,\lceil k/2\rceil,\ldots,k-2\}$, the spectrum $\Pi^k_\ell$ contains algebraic numbers of arbitrarily large degree over $\mathbb Q$. Thus the arithmetic complexity previously known for finite forbidden families already occurs in the single-forbidden spectrum, both for ordinary Turán density and for a broad range of degree Turán densities.

A method is proposed for solving the two-dimensional problem of electromagnetic wave scattering by a cluster of an arbitrary number of parallel, infinitely long, homogeneous, non-overlapping right circular cylinders. The cylinders may have arbitrary radii and complex permittivities, and their axes, while remaining parallel, may occupy arbitrary positions in the transverse plane. The solution is constructed using an analytical expansion of the electromagnetic field in cylindrical harmonics. Multiple scattering is taken into account by Graf's addition theorem, which leads to a system of linear equations for the expansion coefficients. This system is solved numerically with condition number monitoring and, when necessary, extended-precision arithmetic, followed by a multistage verification of convergence. The method provides numerically verified solutions with controlled accuracy over a wide range of parameters, including densely packed subwavelength configurations. As an example, scattering of a normally incident, linearly polarized monochromatic plane wave by a subwavelength cluster of three identical aluminum nanocylinders (nanowires) is studied. The scattering, absorption, and extinction cross sections, as well as the scattering indicatrix, are computed and analyzed. Streamlines of the Poynting vector field are constructed, demonstrating redistribution of the energy flux between the cylinders of the cluster and the formation of localized regions of field enhancement near their surfaces.

The red-blue vertex partitions now known as crumby colorings originate in a conjecture of Thomassen related to Wegner's conjecture on squares of planar graphs. In such a coloring, the blue vertices induce a graph of maximum degree at most one, while the red vertices induce a graph with no isolated vertices and no simple path with three edges. Barát, Blázsik and Damásdi proved that every 2-connected outerplanar graph of maximum degree at most three admits a crumby coloring, and conjectured that the 2-connectivity assumption can be removed. We prove this conjecture: every finite simple subcubic outerplanar graph admits a crumby coloring. To prove the conjecture, we introduce a rooted grammar for subcubic outerplanar graphs. The grammar describes such graphs recursively using rooted branches and two-terminal path fragments; cyclic blocks are handled by deleting the root to obtain a path fragment. We correspondingly extend crumby colorings to crumby-admissible colorings: in a rooted branch the root, and in a path fragment the two terminals, are allowed to be temporary isolated red vertices. This relaxation makes induction along the grammar possible while retaining only finite boundary information. The induction reduces to verifying an explicit finite family of lower certificates, namely nonempty sets of boundary types and root states. The required verification has two parts: the family must be closed under all steps of the decomposition, and every certified completed branch must contain a final-legal root state, so that the temporary defect disappears and the resulting coloring is a genuine crumby coloring. This final step is computer-assisted: a stand-alone certificate checker, supplied with the paper, verifies the stated closure and crumby conditions for the supplied certificate. All structural reductions and the certificate-induction principle are proved by hand.

In this paper, we prove that both the seminilpotent and the fully nilpotent CoHAs of $\mathbf{A}^2$ are commutative. This result is in strong contrast with the CoHA of $\mathbf{A}^2$ without nilpotency conditions, previously studied by Davison, which is related to the Lie algebra $W_{1+\infty}$ of differential operators on $\mathbf{C}^*$. The latter is highly noncommutative. Our proof combines two constraints on the Lie bracket on the affinized BPS Lie algebra: it is filtered with respect to the perverse filtration and it is graded with respect to the cohomological degree. In the case of the Jordan quiver and nilpotent CoHAs, these constraints force the Lie bracket to vanish. We also describe the equivariant nilpotent CoHAs in the presence of the action of a one-dimensional torus rescaling the first coordinate of $\mathbf{A}^2$ with weight $1$ and the second with weight $-1$. In this case, one obtains enveloping algebras of Rees Lie algebras associated with the nilpotent and the seminilpotent filtrations on the Lie algebra $W_{1+\infty}^+$, reminiscent of the description of the equivariant non-nilpotent CoHA given by Davison.

Under a necessary topological assumption, two global results are established for complete three dimensional manifolds. The first one provides a sharp upper bound for the bottom spectrum in terms of the scalar curvature lower bound. The second one shows that such manifolds do not admit any positive Green's function if the scalar curvature is bounded from below by a positive constant.

This paper studies the constrained-capacity for precoded faster-than-Nyquist (FTN) signaling with finite-alphabet inputs. Despite the promise of accelerated transmission, the fundamental rate limit of precoded FTN signaling under practical finite-alphabet constraints remains unclear. By introducing cyclic prefix (CP) and cyclic suffix (CS), the FTN channel is decomposed into a set of parallel eigenchannels by the discrete Fourier transform (DFT) matrix, based on which the constrained capacity is derived. The results demonstrate that time acceleration can improve spectral efficiency over Nyquist signaling even when a fixed modulation order is employed. Moreover, in the low and moderate signal-to-noise ratio (SNR) regimes, a smaller constellation combined with stronger time acceleration can outperform a larger constellation with weaker acceleration. Next, the asymptotic behavior of the constrained capacity is analyzed as the acceleration factor tends to zero under both fixed transmit-SNR and fixed receive-SNR definitions. It is shown that the constrained capacity for DFT-precoded FTN is fundamentally limited by the constellation size. In addition, the constrained capacity under channel mismatch is studied and a mismatched achievable information rate (AIR) formulation is developed to show the effects of practical constraints on the performance degradation. Finally, adaptive bit loading across eigenchannels is investigated to exploit the higher-quality eigenchannels.

Let the ground field be perfect of positive characteristic. Using Altmann-Hausen-Süss theory, we obtain a combinatorial classification of equivariantly normal curves with prescribed quotient in both the affine and projective settings. As a consequence, we derive an explicit upper bound on the number of isomorphism classes of equivariantly normal curves over a fixed base curve with prescribed branch locus. Furthermore, assuming that the ground field is algebraically closed, we determine, for a fixed cardinality of the branch locus, precisely when the set of isomorphism classes of equivariantly normal projective curves with prescribed quotient is finite and when it is infinite.

It has been known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension of an arbitrary real or complex Lie algebra can be realized as subalgebras of formal vector fields over formal power series. In this note, we characterize the Lie subalgebras which admit a convergent realization in the sense of locally analytic vector fields. We give generalizations of these properties for the problem of output realization. We give reformulations and applications of these algebraic results in the context of control theory. In particular, we recover and clarify previous results on the realization of Chen–Fliess series for control-affine systems, the equivalence of control systems, and the existence of embedded or canonical systems.

We show that the quantum Frobenius morphism and its splitting are not fully compatible with the canonical basis for any finite-dimensional simple Lie algebra if the rank is sufficiently large. The incompatibility occurs at same place where Leclerc found his imaginary vectors, and where there are monomials in the tight monomial cone which do not belong to the canonical basis.

Let $J_{\ge s}(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]$, with two distinct vertices adjacent whenever their intersection has size at least $s$. Equivalently, $J_{\ge s}(n,k)$ is the complement of a threshold Kneser graph. We determine both the symmetric minimum rank over an arbitrary infinite field and the real positive semidefinite minimum rank of this family. Specifically, for $k\ge2$, $1\le s\le k-1$, and $n\ge2k-s$, we prove $$ \operatorname{mr}^{\mathbb F}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s} $$ for every infinite field $\mathbb F$, and $$ \operatorname{mr}_{+}^{\mathbb R}\left(J_{\ge s}(n,k)\right) = \binom{n-2(k-s)}{s}. $$ The lower bound follows from a diagonal submatrix indexed by two carefully chosen families of $k$-subsets. For the upper bound, we construct a symmetric matrix using an exterior power of a bilinear form, a Lagrange interpolation identity, and a generic nonvanishing argument. Over $\mathbb R$, an interlacing choice of parameters makes the bilinear form positive definite and yields a positive semidefinite matrix attaining the required upper bound. As consequences, we answer a question from an American Institute of Mathematics workshop, determine the real faithful orthogonality dimension of all graphs $J_{\ge s}(n,k)$ in the stated range, and recover the known minimum-rank formula for Johnson graphs.

Physics-informed neural networks (PINNs) provide a promising framework for solving partial differential equations while embedding the underlying physical laws directly into the learning process. This study presents a PINN-based framework for modeling transient elastodynamic wave propagation in bimaterial systems governed by the axisymmetric equations of linear elasticity. A steel-aluminum specimen representative of a Split Hopkinson Pressure Bar configuration is considered, and the governing elastodynamic equations, together with the corresponding initial, boundary, and interface conditions, are incorporated directly into the network through a physics-informed loss function. High-fidelity finite-element simulations performed using ANSYS Workbench Explicit Dynamics are used for validation and as supplementary data constraints during training. The proposed framework accurately predicts wave transmission and reflection across the bimaterial interface and reproduces axial and radial displacement histories, face-averaged responses, and the dominant stress and strain evolution with close agreement to the finite-element solutions. The trained network further demonstrates the ability to predict wave responses at previously unseen time instants and for modified material properties without requiring additional finite-element simulations, providing a continuous surrogate model for elastodynamic analysis. Mesh-sensitivity studies confirm numerical robustness, while additional material combinations demonstrate the generality of the proposed methodology. The results show that integrating physics-informed neural networks with explicit finite-element analysis provides an accurate and computationally efficient framework for elastodynamic wave propagation in heterogeneous solids, offering an effective surrogate modeling approach for high-rate solid mechanics and impact engineering applications.

We generalise the standard cross-ratio of four points on a projective line to a cross-ratio of a configuration of four planes in projective $n$-space, the first pair $A_1,\,A_2$ being $k$-dimensional and the second pair $B_1,\,B_2$ being $(n-k-1)$-dimensional, with $A_i \cap B_j = \emptyset$. Over the complex numbers, we show that this cross-ratio equals the augmented height pairing of the corresponding cycles $A_1-A_2, \, B_1-B_2$. Over a discretely valued field, we show that the valuation of the cross-ratio equals the intersection degree of the cycles once they are spread out over the valuation ring. Putting the two together, we conclude that the asymptotics of the Archimedean height pairing of a holomorphic family of configurations are governed by this intersection degree. We also define a degenerate cross-ratio for when $A_i \cap B_j \neq \emptyset$ and interpret the "limit height" of a degenerating holomorphic family of planes as the degenerate cross-ratio of the central plane configuration.

Based on a conjecture of Loughran and the second author, we give an explicit prediction for the leading constant in Malle's conjecture for Galois $\mathcal{H}$-extensions of $\mathbb{Q}$ ordered by discriminant, where $\mathcal{H}$ is the $3\times 3$ Heisenberg group over $\mathbb{F}_4$. The predicted leading constant is not a single Euler product, but rather a sum of two distinct Euler products. Our methods also give an efficient algorithm for computing the conjectural Loughran-Santens leading constant for many $2$-groups of nilpotency class $2$.

We study the scattering of time-harmonic electromagnetic waves by periodic layered gratings, modelled by the 2D Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions, and consists of a finite number of stacked layers. The boundary value problem is formulated on a single periodic cell using quasi-periodic boundary conditions. The radiation condition in the vertical directions is imposed through Dirichlet-to-Neumann (DtN) operators. To efficiently treat multilayer configurations, we adopt a formulation based on the T-matrix method. The global scattering problem is decomposed into boundary value problems posed on individual layers. On the layer boundaries, the field is expressed in terms of quasi-periodic modal expansions, and the layer T-matrix describes the map between incoming and outgoing wave modes. Each local T-matrix is approximated numerically using a plane-wave based Trefftz Discontinuous Galerkin (TDG) method, which provides an efficient discretization of the layer scattering response. The T-matrix technique leads to linear computational complexity in the number of layers in the grating.

The problem of classifying geometric structures on manifolds is very much related to the discussion of the automorphism groups actions on character varieties, which are spaces of equivalence classes of representations. In this chapter we survey some results on this topic, mostly focusing on representations of surface groups (both in the orientable and non-orientable cases) and free groups. An important principle in the study of the dynamics on character varieties $X=X(\pi_1(S),G)$ for surface groups $\pi_1(S)$ is the following dichotomy: when the target group $G$ is compact, $X$ has nontrivial homotopy type, and the action of the mapping class group is chaotic; whereas when the target group $G$ is non-compact, $X$ contains contractible sets on which the mapping class group acts properly. We will expand on this dichotomy in various cases. After introducing the necessary background, we will discuss representations into $\mathsf{PSL}_2(\mathbb{R})$ and $\mathsf{PGL}_2(\mathbb{R})$, discussing the number of connected components, the geometric properties (Bowditch question), the dynamics (Goldman conjecture) and some components with an `exotic' behaviour (Deroin-Tholozan representations). We will also underline how the theory for representations of fundamental groups of orientable closed hyperbolizable surfaces needs to be adapted when one considers surfaces with punctures or non-orientable surfaces. We will then discuss representations into compact groups, where we will discuss mostly ergodicity results in various settings, and some non-ergodicity results at the end. Thirdly, we will consider representations in $\mathsf{PSL}_2(\mathbb{C})$. We will discuss convex-cocompact representations, primitive-stable and Bowditch representations and their relationship. Finally, we will describe how some of the results mentioned can be generalized for representations into higher-rank Lie groups.

A conjecture of Alon and Wei states that, for any $d$-regular graph $G$ with $n$ vertices, there exists a spanning subgraph $H$ such that for all $0\le i\le d$, we have $m(H, i)$, the number of vertices in $H$ with degree $i$, is between $\frac{n}{d+1}-2$ and $\frac{n}{d+1}+2$. We prove the conjecture for all fixed $d$ when $n$ is sufficiently large. More precisely, if $q=(q_0,\ldots,q_d)$ satisfies $$ \sum_{i=0}^d q_i=n,\qquad \sum_{i=0}^d i q_i\equiv 0\pmod 2,\qquad \left|q_i-\frac{n}{d+1}\right|\le 1 \quad (0\le i\le d), $$ then there is a spanning subgraph $H\subseteq G$ such that $$ m(H,i)=q_i \qquad (0\le i\le d). $$

In this paper, we investigate the generalized spectral closedness of graph classes defined by a family $\mathcal{F}$ of forbidden induced subgraphs. To systematically study this property, we introduce a novel combinatorial concept of patterned closed walks (or $\beta$-closed walks), which naturally interlaces the edges of a graph with those of its complement. By establishing the induced-subgraph expansion of these $\beta$-closed walk counts, we obtain an algebraic sufficient condition for generalized spectral closedness based on the existence of a walk-realizable $\mathcal{F}$-supporter. Crucially, the search for such a walk-realizable supporter is reduced to a linear programming feasibility problem. As primary applications of this computational framework, we prove that the classes of threshold graphs and chain graphs are generalized spectrally closed.

A categorical approach is proposed for the construction of 8-periodic chain complexes of Witt groups of categories with duality. Exactness of the resulting sequences is proved under the hypothesis that the categories are abelian and artinian or when the endomorphism algebra of each object is semilocal and rad-adically complete.