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Rizwan Jahangir

Publications and source records attributed to Rizwan Jahangir.

8 recordsLinked to original sources
arXiv2026

Canonical Lattices of Integer Relations Associated to Rational Fans: Wall Generation and a Two-Step Support Filtration

We study the lattice $L_{\mathrm{rel}}(Σ)=\ker\big(\mathbb{Z}^{Σ(1)}\to N\big)$ of integer relations among the primitive ray generators of a rational fan $Σ$, from an intrinsic, coordinate-free point of view. For each cone $τ\inΣ$ we introduce the \emph{star-supported} sublattice $L_{\mathrm{rel}}(\operatorname{Star}(τ))$ of relations whose support lies in the star of $τ$, and we organize these by codimension into a support filtration $F_\bullet L_{\mathrm{rel}}(Σ)$. Our main result is a sharp local generation theorem: for a complete fan the relation lattice is generated \emph{integrally} by the relations supported on the stars of walls (codimension-one cones). Equivalently, the support filtration collapses after a single step, $F_1 L_{\mathrm{rel}}(Σ)=L_{\mathrm{rel}}(Σ)$. This is an intrinsic repackaging of the classical wall (wall-crossing) relations that generate the group of numerically trivial classes on a complete toric variety. We make the resulting two-step structure precise: for simplicial fans one has $0=F_0\subsetneq F_1=L_{\mathrm{rel}}(Σ)$, while for general fans $F_0$ records the intrinsic relations of non-simplicial maximal cones and $F_1$ adds exactly the wall relations. We prove functoriality of $L_{\mathrm{rays}}$ and $L_{\mathrm{rel}}$ under fan isomorphisms and ray-preserving subdivisions, deduce that every primitive collection of size $m$ is wall-generated, and illustrate the theory on $\mathbb{P}^2\times\mathbb{P}^1$, products of projective lines, weighted projective spaces, and the (non-simplicial) fan over a cube. We are careful throughout to distinguish what the filtration does and does not detect, correcting a natural but false expectation that support-codimension yields a strictly increasing multi-step invariant.

math.CO↗
arXiv2026

Combinatorial Cycle Classes in the Intersection Cohomology of Projective Toric Varieties

We investigate cycle-class realizations inside the combinatorial intersection cohomology for fans developed by Barthel, Brasselet, Fieseler, and Kaup (BBFK). For projective toric varieties, the intersection cohomology is Hodge-Tate, and thus the space of rational Hodge classes coincides with the full rational even-degree intersection cohomology. We formulate a compatibility statement between combinatorial and geometric cycle classes and explore it in the torus-invariant setting under standard functoriality assumptions. The central question we address is whether these invariant combinatorial cycle classes span the even-degree combinatorial intersection cohomology $IH^{2k}_{\mathrm{comb}}(Σ, \mathbb{Q})$. Assuming the stated BBFK--BL compatibility, we verify this linear-generation statement for projective toric varieties of dimension at most $3$; the simplicial case follows unconditionally from standard rational cohomology descriptions. We illustrate the framework with a non-simplicial example in dimension $3$ for which the Betti numbers and spanning property are derived directly from Stanley's toric $h$-vector formula and Fieseler's surjectivity theorem.

math.AG↗
arXiv2026

The Accessibility Capability Boundary: Operational Limits and Expansion Potential of AI-Generated Browser-Native Accessibility Systems

As large language models (LLMs) demonstrate increasing competence in synthesizing functional user interfaces, a fundamental question emerges in accessibility computing: \textit{how far can AI-driven accessibility systems go?} This paper introduces the \textit{Accessibility Capability Boundary} (ACB), a formal framework for reasoning about the operational limits and expansion potential of autonomous accessibility systems, and grounds this theory in a real-world systems artifact. We model accessibility not as a binary compliance property but as a dynamic, multidimensional capability space constrained by measurable variables including deployment latency, cognitive load, infrastructure dependency, offline persistence, interaction complexity, and adaptability. We argue that AI-generated, browser-native systems constructed as single-file HTML artifacts leveraging standard browser APIs may dramatically shift the ACB outward by reducing deployment friction to near-zero and enabling rapid, context-specific interface adaptation. We ground our theoretical framework in the analysis of two real-world exploratory prototypes. The first is an AI-generated browser-native accessibility interface deployed for a blind user in Nepal. The second is a fully functional, open-source webcam alignment assistant for visually impaired users, serving as a concrete systems artifact. Through formal definitions, propositions, and a comparative evaluation matrix, we characterize the regions of the accessibility capability space that such systems can and cannot reach. We further identify remaining computational, infrastructural, and verification constraints that constitute the hard boundaries of this paradigm. This work contributes a theoretical foundation for understanding the scalable limits of autonomous accessibility computing and proposes a research agenda for future work in accessibility-aware AI systems.

cs.HC↗
arXiv2026

Explainable PQC: A Layered Interpretive Framework for Post-Quantum Cryptographic Security Assumptions

This paper studies how post-quantum cryptographic (PQC) security assumptions can be represented and communicated through a structured, layered framework that is useful for technical interpretation but does not replace formal cryptographic proofs. We propose ``Explainable PQC,'' an interdisciplinary framework connecting three layers: (1) a complexity-based interpretive model that distinguishes classical security, quantum security, and reduction-backed hardness, drawing on computational complexity classes as supporting language; (2) an exploratory mathematical investigation applying combinatorial Hodge theory and polyhedral geometry to study structural aspects of lattice hardness; and (3)~an empirical experimentation platform, implemented in Julia, for measuring the behavior of lattice basis reduction algorithms (LLL, BKZ) in low-dimensional settings. The motivating case study throughout the paper is lattice-based PQC, including ML-KEM (FIPS 203) and ML-DSA (FIPS 204). The contribution of this paper is conceptual and organizational: it defines a layered interpretive framework, clarifies its scope relative to formal cryptographic proofs and reduction-based security arguments, and identifies mathematical and implementation-level directions through which PQC security claims may be more transparently communicated. This paper does not claim new cryptographic hardness results, new attacks, or concrete security parameter estimates.

cs.CR↗
arXiv2025

On Cohen-Macaulay non-prime collections of cells

In this paper we investigate Cohen-Macaulayness, Gorensteinness and the Hilbert-Poincaré series for some classes of non-prime collections of cells. In particular, we show that all closed path polyominoes are Cohen-Macaulay and we characterize those that are Gorenstein.

math.AC↗
arXiv2023

Shellable simplicial complex and switching rook polynomial of frame polyominoes

Let $\mathcal{P}$ be a frame polyomino, a new kind of non-simple polyomino. In this paper we study the $h$-polynomial of $K[\mathcal{P}]$ in terms of the switching rook polynomial of $\mathcal{P}$ using the shellable simplicial complex $Δ(\mathcal{P})$ attached to $\mathcal{P}$. We provide a suitable shelling order for $Δ(\mathcal{P})$ and we define a bijection between the set of the canonical configurations of $j$ rooks in $\mathcal{P}$ and the facets of $Δ(\mathcal{P})$ with $j$ steps. Finally we use a well-known combinatorial result, due to McMullen and Walkup, about the $h$-vector of a shellable simplicial complex to interpret the $h$-polynomial of $K[\mathcal{P}]$ as the switching rook polynomial of $\mathcal{P}$.

math.CO↗
arXiv2023

On Cohen-Macaulay posets of dimension two and permutation graphs

We characterize Cohen-Macaulay posets of dimension two; they are precisely the shellable and strongly connected posets of dimension two. We also give a combinatorial description of these posets. Using the fact that co-comparability graph of a 2-dimensional poset is a permutation graph, we characterize Cohen-Macaulay permutation graphs.

math.CO↗