Papers

The principal commutation formula underlying the paper by Deift [24], is of the type \[ - z (AB - z I)^{-1} + A (BA - z I)^{-1} B = I, \quad z \in [\rho(AB) \cap \rho(BA)]\backslash\{0\}, \] where, temporarily, and for simplicity only, $A,B$ represent bounded operators in a Banach space. A fundamental consequence of this commutation formula is the fact that $AB$ and $BA$ are essentially isospecral in the sense that \[ \sigma(AB) \backslash \{0\} = \sigma(BA) \backslash \{0\}. \] Moreover, under appropriate trace class hypotheses, this implies the fundamental identity for (Fredholm) determinants \[ \det(I - AB) = \det(I - BA), \] which extends to modified (or regularized) Fredholm determinants. In this paper we briefly recall some of the applications of these commutation formulas to isospectrality, integrable systems, regularization of PDEs, and estimates in Statistical Mechanics and Quantum Field Theory discussed in [24]. Subsequently, we review some of the ramifications of commutation formulas that were established over the past fifty years, including determinants and dimension reduction, random matrix theory and dimension expansion, integrable operators, Poncelet's porism, interpolation, eigenvalue computation, KPZ, Pfaffians, the focusing Gross-Pitaevskii equation, one-dimensional scattering theory, Floquet theory, and the Miura transformation connecting the KdV and mKdV hierarchies.

The classical $A_2$-frame construction of the Coxeter–Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring $S=\mathbb{Z}[i]/(1+i)^3$. For odd $\ell$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4)^{2\ell}$-frame to an even strongly $6$-modular lattice $L_\ell$ of rank $4\ell$. In the real frame ordering the same glue is a one-generator negacyclic code. The cases $L_5$ and $L_7$ have minima $6$ and $8$ and give examples for the type-$6\mathrm{a}$ entries in dimensions $20$ and $28$, which are listed as open in the Nebe–Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for $L_7$ the key relation is the boundary map of a $7$-cycle. For $\ell=3$, $L_3$ has an index-$8$ overlattice isometric to the Coxeter–Todd lattice.

The stochastic block model is a paradigmatic model of networks with community structure. Yet percolation in the model has been studied primarily in cases with a fixed block structure, even though in real networks, the community structure may evolve as the network grows. Here we study percolation in sequences of stochastic block models in which the numbers and sizes of communities, as well as the intra- and intercommunity connection probabilities may all change with the network size. We analyze five such sequences using two methods: linearized self-consistent equations for the locally tree-like models and a branching process at the community scale for the models with nonvanishing clustering. We find that the critical average degree is not generally equal to $1$, even in locally tree-like sequences, because the transition depends on how connections are distributed across the evolving community structure. We also show that the community-scale branching process accurately predicts the transition when intercommunity connections are sufficiently sparse, even in the presence of nonvanishing clustering, while a geometric stochastic block model sequence demonstrates the limitations of this method when correlations between intercommunity connections cannot be neglected. These results extend percolation studies in the stochastic block model to more realistic scenarios with evolving community structure, and may provide new methods to derive the upper and lower bounds for the percolation threshold in geometric long-range percolation.

Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $\chi$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$\chi(\overline{M_R})=\chi(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.

We give a detailed implementation of the mass-splitting construction of Nazarov, Volberg and Yuditskii for reflectionless measures with singular continuous components. The finite approximants are allowed to have isolated atoms. An exact local operation replaces each active atom by a short permanent interval and two child atoms, thereby eliminating the dependence of a future splitting scale on an already prescribed band width. Summable estimates for completed generations and fixed cylinders retain positive mass on a null Cantor set and exclude atoms in the limit. Uniform analytic estimates near the permanent intervals establish reflectionlessness and exclude endpoint singularities. A separate Green function estimate for thin pairs of intervals gives the Widom condition and regularity of the limiting compact set. The resulting probability measure has both a nonzero absolutely continuous component and a nonzero singular continuous component.

Let $\Gamma<\mathsf{SL}(d,\mathbb R)$ be a torsion-free, Zariski-dense Borel-Anosov subgroup, and let $\varphi$ be a linear functional positive on the limit cone of $\Gamma$. For every $\chi \in \operatorname{Hom}(\Gamma, \mathbb R)$, we construct twisted $(\varphi,\chi)$-conformal densities and prove their existence, uniqueness and ergodicity. For every non-trivial normal subgroup $\Gamma_0\lhd\Gamma$ with nilpotent quotient, we classify the ergodic $\varphi$-conformal densities of $\Gamma_0$ in terms of characters of $\Gamma/\Gamma_0$. Up to scaling, they are naturally parametrized by $\operatorname{Hom}(\Gamma/\Gamma_0,\mathbb R)$, and their dimensions are precisely the critical exponents of the corresponding twisted Poincaré series. We also prove a shadow lemma for twisted densities of normal subgroups, with estimates uniform over densities with fixed parameters, and a twisted Hopf-Tsuji-Sullivan theorem.

We obtain a complete characterization of the quasi-stationary distributions of subcritical stable continuous-state branching processes in Brownian random environments, together with the domain of attraction of each member. These distributions form a one-parameter family whose modified Laplace transforms are given explicitly in terms of Tricomi's confluent hypergeometric function. For every initial distribution, we determine whether the conditional law converges to a QSD and identify that QSD whenever it does. The necessary and sufficient criteria are formulated in terms of the tail probabilities and moments of the initial law.

Many inference problems are constrained from two sides: forward information and a second constraint on outcomes. Dual Boundary Condition Inference (DBCI) multiplies them and renormalises. The rule is not new: it is the product-of-experts form and the unit-exponent member of the logarithmic-pooling family. The paper contributes a classification and reducibility characterisation. The algebra does not fix the exponent at which the inputs combine: given the statistic log p_fwd + log p_bwd, which already builds in the product form, maximum entropy supplies the family (p_fwd p_bwd)^lambda and selects no member. If the inputs are read as two equally weighted opinions, three requirements select lambda = 1/2: unanimity preservation, consistency under a common Bayesian update, and minimal symmetric Kullback-Leibler divergence. If they are read as separately applied factors, two requirements select lambda = 1: a neutral second input must leave the first unchanged, and an update to one input must pass through unchanged. For a rank-one intermediate projective measurement, the Aharonov-Bergmann-Lebowitz (ABL) rule realises this form with no extra parameter. Every positive exponent orders outcomes identically, so the choice is invisible to picking the likeliest outcome, but not to scoring a class by total mass. Under the relevant identifications, the same product form gives Bayes' rule, while a final effect proportional to the identity returns the ordinary forward Born probabilities. The second result is a Two-Boundary Reducibility Criterion: DBCI factors through the forward boundary exactly when the effective backward boundary, up to positive rescaling, does. So a fixed prior or model-fixed structural constraint adds no distinctions between cases sharing the same forward boundary, though it may encode substantial information and still materially affect the result.

A complete presentation for the diagonal reduction algebra of $\mathfrak{gl}_n$ was found by Khoroshkin and Ogievetsky. In this paper we present a complete set of ordering relations for a reduction algebra $\mathcal{B}_n$ associated to $\mathfrak{gl}_n$ in a parabolic subalgebra of $\mathfrak{sp}_{2n}$. We also study its connection to the diagonal reduction algebra of $\mathfrak{sp}_{2n}$. We use braid group action by Zhelobenko automorphisms to reduce the computations required. Another important tool we employ is the ABRR equation for the extremal projector. As an application we prove that all homogeneous ordering relations for the diagonal reduction algebra of $\mathfrak{sp}_{2n}$ are obtained from the relations in $\mathcal{B}_n$ by applying type $C_n$ Zhelobenko automorphisms.

Given an even-dimensional smooth manifold and a (co)(closed) bundle-valued $n$-form $a$ that "resembles" a scalar $(n,0)$-form, we may (or may not) recover a "$\mathbb{Z}_2$-(almost) complex structure" under certain assumptions. In the case of a $4$-manifold or a quaternion-Kähler manifold, we can recover a compatible $\mathbb{Z}_2$-complex structure on the whole manifold including the zero locus of $a$. In general, we can recover a $\mathbb{Z}_2$-(almost) complex structure wherever $a$ satisfies a certain nondegeneracy condition, but not necessarily on the whole manifold, with a counterexample identified. The above condition of "resembling" a scalar $(2,0)$-form is satisfied by any $\mathrm{U}(1)$-invariant solution to the Vafa-Witten equation on a compact Kähler surface. This hints at a possibility to generalise the notion of $\mathrm{U}(1)$-invariant solutions to oriented Riemannian $4$-manifolds. However, our results above imply any solution satisfying the condition, and thus seemingly any potential "generalised $\mathrm{U}(1)$-invariant solution", must be trivial if the $4$-manifold admits no compatible $\mathbb{Z}_2$-complex structure.

The equal-weighted portfolio is a passive, rule-based strategy that has historically been difficult to outperform, delivering higher returns than the capitalization-weighted "market" benchmark across many markets and periods. Stochastic portfolio theory (SPT) reveals that this relative performance is regime dependent, with the equal-weighted portfolio underperforming during periods of increasing market concentration and high correlations, particularly market bubbles. These observations have motivated us to formulate and solve a stochastic control problem in which an investor actively allocates between the equal-weighted and market portfolios. The investor bases their allocation decisions on forecasts made under a flexible stochastic diversity–dispersion (SDD) model. Using a quadratic surrogate for implementation frictions, we characterize the optimal allocation through a linear forward–backward SDE and obtain an explicit "aiming in front of a moving target" representation of the optimal trading rate, in the spirit of Gârleanu and Pedersen. The penalty parameters are calibrated in sample to match the cumulative wealth effect of proportional transaction costs, while out-of-sample performance is evaluated with those costs deducted directly from portfolio wealth. Using historical S&P 500 data, we show that a mean-reverting SDD specification reproduces several empirical features of market diversity and dispersion. In out-of-sample backtests from 1995 to 2024, the resulting strategies deliver higher cumulative net returns than both the equal-weighted and market portfolios, and higher information ratios than the equal-weighted portfolio after 15-basis-point proportional transaction costs.

We study equilibrium and risk evaluation in large sparse economic networks with heterogeneous responses, shocks, and bilateral exposures. Our approach approximates the distribution of equilibrium outcomes through local computations on a limiting rooted network. Under marked local weak convergence, we prove convergence in probability of the empirical equilibrium distribution in two settings: uniformly contractive responses and bounded monotone nonexpansive responses whose lower and upper root iterations coalesce. In the latter setting, the limit is independent of the measurable finite-network equilibrium selection, even when equilibria are not unique. Under bounded domain and continuity assumptions, this convergence extends to law-invariant risk measures, including conditional value-at-risk, with recursion-depth error bounds under additional regularity. Numerical experiments on production networks and payment clearing systems illustrate the accuracy and computational benefits of the local approximation.

We consider the dynamic Schrödinger bridge problem on a finite state space and exploit its Markov structure to decompose the problem into an endpoint coupling and a collection of conditional Markov bridges. To sample from the conditional bridges, we develop a quantum Gibbs sampler based on quantum walks on the bridge path space. Under uniformly positive transition probabilities, we derive a spectral-gap bound and provide an explicit procedure for preparing the initial state. Compared with the update bounds of the corresponding classical Gibbs sampler, our quantum walk-query complexity improves the dependence on the time horizon $T$ from quadratic to linear. For the endpoint coupling, we adapt a quantum box-constrained Newton method to compute the Schrödinger potentials, improving the matrix-scaling complexity, at fixed accuracy, from quadratic to $N^{3/2}$ in the state-space size $N$. Finally, we show that exponential reweighting extends both the Schrödinger bridge formulation and the Gibbs spectral-gap estimate to models with additive running costs.

In this paper we consider area-preserving twist maps $f:\mathbf{T}^2\to\mathbf{T}^2$ with vertical rotation sets reduced to a single irrational number $\alpha$. Our first result states that under these hypotheses, there exists a unique $f$-invariant foliation $\mathcal{F}_G=\{\operatorname{Graph}(\phi_t)\}_{t\in\mathbf{T}^1}$, where each function $\phi_t:\mathbf{T}^1\to\mathbf{T}^1$ is $K$-Lipschitz for some constant $K=K(f)$. Moreover, we also prove that if $\mathcal{F}_G$ is a Lipschitz foliation, then $f$ is minimal.

Motzkin paths of length $n$ and standard Young tableaux (SYT) with $n$ cells and at most three rows are both counted by the Motzkin numbers, and many bijections between them are known. The Riordan numbers count the subfamilies of Riordan paths (Motzkin paths with no horizontal step on the $x$-axis) and of tableaux whose three row lengths have the same parity, but none of the known bijections restricts to these subfamilies. We introduce the set of $321$-avoiding permutations in which every left-to-right maximum is either a descent or a fixed point. This family is counted by the Motzkin numbers, and its fixed-point-free elements are the “Riordan permutations” of Callan. We give a bijection from Motzkin paths to these permutations under which Riordan paths correspond to Riordan permutations. We then give a bijection from these permutations to SYT of height at most three, obtained from Robinson–Schensted insertion followed by a parity correction, under which Riordan permutations correspond to tableaux with rows of equal parity and the number of left-to-right maxima becomes a simple tableau statistic. Via Dyck paths, we connect these objects to further families counted by the Riordan numbers, including derangements of genus zero and SYT of shape $(k,k,1^{n-2k})$.

We establish a Harder-Narasimhan theory for torsion coherent $F$-gauges over a perfect field $k$ of positive characteristic and completely classify its stable objects. Our construction uses Ekedahl's derived equivalence with coherent complexes over the Raynaud ring, where the theory refines his half-integral type filtration on diagonal dominoes. At each finite rational slope $d/q$ in lowest terms, there is a unique stable diagonal domino up to Breuil-Kisin twist, constructed explicitly from the corresponding Christoffel word. The semistable category of this slope is equivalent to a product of $q$ copies of the zero-slope category. The classification of stable objects corrects Ekedahl's classification of weakly simple diagonal dominoes of type $\frac{1}{2}$. The theory upgrades to a locally finite Bridgeland stability condition on $\mathrm{Perf}_{\mathrm{tor}}(k^{\mathrm{Syn}})$.

We study cyclotomic cosine evaluations beyond the fully symmetric setting by combining Galois-orbit structure with finite-frequency expansions. First, for truncation-compatible poly?nomial families over Q, we distinguish invariance of the evaluated value from invariance of the polynomial formula. Summability by a rational function of the level forces eventual rationality of the evaluated values. Conversely, every rational-valued sequence can be realized by a compatible family whose total degree is uniformly bounded by two, showing that value-level Galois invariance alone carries essentially no stable-range rigidity. We then introduce explicit orbit-structured pattern sums indexed by multiplicative configurations. For every fixed pattern, at each sufficiently large admissible level n, its cyclotomic cosine evaluation collapses to an affine function n times S0(r,h) minus 2 to the abs(h) power, where S0 is the zero-frequency coefficient of the associated Laurent polynomial. Consequently, every family uniformly generated from finitely many such patterns by a single polynomial recipe has eventually polynomial evaluation. Finally, decomposing the punctured index set into gcd-orbits yields an exact orbitwise formula in terms of Ramanujan sums. The semiprime and prime-power cases become transparent specializations of this general identity. These results identify both the obstruction to extending symmetric rigidity from value-level Galois invariance and a concrete non-symmetric class exhibiting uniform stable-range collapse

We systematically investigate breather solutions in the modified Korteweg-de Vries (mKdV) equation via numerical simulations. We show that the breather solutions are stable under a variety of perturbations, including amplitude changes, modifications of internal parameters, and perturbations of the nonlinearity of the equation. Our results are not only consistent with the known analytical studies on the stability of breathers in integrable systems, extending them further, but also provide numerical evidence that such breather-type structures can persist and remain stable over the simulated time scales in some non-integrable settings with symmetric potentials.

We extend Fine's selection method to the setting of intuitionistic modal logic and use it to provide a model-theoretic proof that Fischer Servi-style intuitionistic $\sf K4$ has the finite model property.

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle \phi, \varphi^2 \rangle$ uniformly in $t_\phi\to \infty$ and $t_\varphi \to \infty$, where $t_\phi$ and $t_\varphi$ are the eigenparameters corresponding to Maass forms $\phi$ and $\varphi$ on $X$.

We study adjoints of $R(T)$-linear, $T$-strongly bounded operators on the conditional space $L^2(T)$ and the interaction between its Hilbert-type geometry and its Riesz-space order. This produces two natural operator moduli: the Riesz–Kantorovich order modulus and the spectral modulus $(S^*S)^{1/2}$. After isolating the $T$-regular operators, we show that they form a Dedekind-complete order ideal in the natural lattice of order-bounded $R(T)$-module homomorphisms. On this operator lattice the order operations remain $T$-strongly bounded and the adjoint is an order automorphism. We then compare the two moduli and explain why they may differ.

A conjecture of Lewis asserts that two reflection factorizations of the same element in a finite Coxeter group $W$ belong to the same Hurwitz orbit if and only if they generate the same subgroup $H$ of $W$ and have the same multiset of $H$-conjugacy classes. We prove this conjecture for every finite Coxeter group all of whose irreducible components are of type $A$, $B$ or $D$. Furthermore we prove a reduction, which reduces the conjecture to a finite problem for the exceptional types.

Many results in categorical algebra rely fundamentally on pointedness, yet numerous categories of mathematical interest are not pointed. Building on ideas from the theory of ideally exact categories, recently introduced by G. Janelidze, we investigate the extent to which constructions and results from pointed contexts can be extended to categories admitting suitable forgetful functors to pointed categories. We introduce the notion of a propointed category, described as a category admitting a conservative right-adjoint functor to a pointed lex category, and give an intrinsic characterisation of this notion. We then investigate and characterise the cases in which the target of the given functor is pointed protomodular, homological, or normal, and establish within these settings generalisations of classical results - including the short five lemma, the nine lemma, and Noether's isomorphism theorems - as well as a well-behaved notion of ideal of an object. We also prove that the 2-category of pointed lex categories is 2-reflective in the 2-category of lex categories with an initial object, the reflection being given by the slice over the initial object, which thus provides a universal 'pointification'.

A substantial body of results is known for curves of degree $d=3$, namely elliptic curves, including the theorems of Siegel, Mazur, and Mordell, among others. More generally, Faltings established comprehensive results for all algebraic curves of degree $d \ge 2$. However, these results do not, in general, furnish an explicit or systematic procedure for classifying algebraic curves that fail to admit solutions over a prescribed set such as $\mathbb{Z}^{+}$. The principal aim of the present work is accordingly to develop a method for classifying families of algebraic curves having no solutions in $\mathbb{Z}^{+}$. We begin by establishing a correspondence between algebraic geometry and linear algebra, from which we develop a classification procedure grounded in linear-algebraic constructions and tools. Specifically, let $S:Ax=b$ be a linear system, where $A \in M_{3k \times m}(\mathbb{Z})$ satisfies suitable conditions on its entries, and let $S=\{s_{1},s_{2},\dots,s_{N}\} \subset \mathbb{Z}^{m}$ denote the solution set of the system, with $A=(a_{ij})$, and $s_{i}=(c_{1},c_{2},\dots,c_{n+1})$. From such a solution we construct algebraic curves of degree $n \le m$, nonsingular curves $C_{js_i}$ of genus $g \ge 0$, of the form $C_{js_{i}}:Y^{2}=a_{3j1}c_{1}X^{n}+a_{3j2}c_{2}X^{n-1}+\dots+a_{3jn}c_{n}$. If $Y \ge 1, X > 1$ and $\forall(X,Y) \in C_{js_{i}}(\mathbb{Z}^{+})$ holds then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{N}=\emptyset$ and if the system $S$ admits infinitely many solutions, then $[C_{js_{i}}(\mathbb{Z}^{+})]_{1 \le j \le k}^{\infty}=\emptyset$ as $N \longrightarrow \infty$.

We develop a unified defect theory for embedded algebraic vector bundles which places several classical constructions within a common framework. The theory recovers tangential, dual, join, and secant defects. We first prove an effective numerical criterion characterizing defectivity by the vanishing of a bidegree. This also allows us to recover the exact defect from the vanishing pattern of the bidegrees, extending a theorem of Holme. Using van der Waerden's theorem on bidegrees, we obtain a formula that decomposes the geometric degree of an embedded vector bundle into contributions from the direction varieties of its general linear restrictions. This leads to a characterization of embedded algebraic vector bundles of minimal degree. As a main application, we establish the sharp universal bound $\operatorname{deg}(TV) \leq \operatorname{deg}(V)^2$ for every smooth irreducible affine variety $V \subseteq \mathbb{A}^n$ and $TV\subseteq \mathbb{A}^{2n}$ its tangent bundle. Finally, under suitable regularity assumptions, we apply the quadratic bound to prolongation varieties of differential algebraic systems, obtaining uniform degree estimates and thereby providing a partial answer to an open problem in differential algebra posed by Pogudin.

An eigenvalue of a signed graph is called main if there exists a corresponding eigenvector non-orthogonal to the all-ones vector. An important result of O'Rourke and Touri (2016) states that almost all (unsigned) graphs have all main eigenvalues. Akbari, França, Ghasemian, Javarsineh, and de Lima (2021) considered main eigenvalues of signed graphs and conjectured that for any unsigned connected graph $G \notin\{ K_2, K_4 - e\}$, there is a switching $\mathbf{s}$ such that all eigenvalues of the signed graph $G^{\mathbf{s}}$ are main. We prove two incomparable asymptotic versions of this conjecture. We show that for any graph $G$ of order $n$, there exists a switching $\mathbf{s}\in\{\pm1\}^n$ such that $G^{\mathbf{s}}$ has $n - O\!\left(\frac{n}{(\log n)^{1/4}}\right)$ main eigenvalues counted with multiplicity. Using a similar proof strategy, we also show that if $G$ has $d$ distinct eigenvalues, then there exists a switching $\mathbf{s}\in\{\pm1\}^n$ such that $G^{\mathbf{s}}$ has $d - O\!\left(\frac{d}{(\log d)^{1/4}}\right)$ main eigenvalues.

We prove power-loss results for some variants of Stein's Weighted $L^2$ Conjecture for Bochner-Riesz means. The variants concern multipliers by convex bodies with $C^k$ boundaries in every dimension $n$. The counterexamples are related to recently-constructed counterexamples to the Mizohata-Takeuchi Conjecture. Related results like logarithmic blowups for the endpoint case of the original conjecture are also discussed.

The slice rank method gives exponential bounds for sets with no three-term arithmetic progression in finite vector spaces of odd characteristic and for three-sunflower-free families of subsets of a fixed ground set. We show that for $k\ge4$, every tensor that is nonzero exactly on the $k$-term arithmetic progression relation or the $k$-sunflower relation has maximal slice rank over every coefficient field. When the support is prescribed only on pairwise distinct inputs, we obtain comparable lower bounds, which likewise rule out exponential savings.

We present a computational framework to verify a refinement of the McKay conjecture, proposed by E. Giannelli, for sporadic simple groups and their universal covers. This refinement asserts the existence of a bijection between the $p'$-characters of a finite group and those of the normalizer of its Sylow $p$-subgroup that does not increase character degrees. Our verification enforces stricter conditions, ensuring equivariance under the relevant outer automorphism group, preservation of the $p$-part of the character conductor, and compatibility with central characters. We establish theoretical reductions that limit the relevant outer automorphism action to an order of at most 2, allowing us to leverage Clifford theory and precomputed character tables to circumvent expensive explicit group constructions. Finally, we provide algorithms for classifying characters and constructing bijections that strictly satisfy these conditions.

Let $k$ be a field of characteristic $p\ge0$. We prove that $m$ points and $n$ lines in $k^2$ determine at most $3(mn)^{2/3}+m+n+2mn/p$ incidences, the last term being omitted in characteristic zero. Over the prime field $\mathbb{F}_p$ the coefficient of $mn/p$ can be replaced by $1$. The proof uses the polynomial method, and for $m=n$ the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which $-1$ is not a square we obtain the $L^2\to L^r$ extension estimate for the paraboloid in $\mathbb{F}_p^3$ for $r>10/3$. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than $1/3$. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.