arXiv subjects

Steven Rayan

Publications and source records attributed to Steven Rayan.

At least 37 records · Page 2Linked to original sources

arXiv2025

A generalized spectral correspondence

We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of cyclic pairs and co-Higgs bundles over $\mathbb{P}^1$. By appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$ through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt.

math.AG↗

arXiv2025

Identifying Protein Co-regulatory Network Logic by Solving B-SAT Problems through Gate-based Quantum Computing

There is growing awareness that the success of pharmacologic interventions on living organisms is significantly impacted by context and timing of exposure. In turn, this complexity has led to an increased focus on regulatory network dynamics in biology and our ability to represent them in a high-fidelity way, in silico. Logic network models show great promise here and their parameter estimation can be formulated as a constraint satisfaction problem (CSP) that is well-suited to the often sparse, incomplete data in biology. Unfortunately, even in the case of Boolean logic, the combinatorial complexity of these problems grows rapidly, challenging the creation of models at physiologically-relevant scales. That said, quantum computing, while still nascent, facilitates novel information-processing paradigms with the potential for transformative impact in problems such as this one. In this work, we take a first step at actualizing this potential by identifying the structure and Boolean decisional logic of a well-studied network linking 5 proteins involved in the neural development of the mammalian cortical area of the brain. We identify the protein-protein connectivity and binary decisional logic governing this network by formulating it as a Boolean Satisfiability (B-SAT) problem. We employ Grover's algorithm to solve the NP-hard problem faster than the exponential time complexity required by deterministic classical algorithms. Using approaches deployed on both quantum simulators and actual noisy intermediate scale quantum (NISQ) hardware, we accurately recover several high-likelihood models from very sparse protein expression data. The results highlight the differential roles of data types in supporting accurate models; the impact of quantum algorithm design as it pertains to the mutability of quantum hardware; and the opportunities for accelerated discovery enabled by this approach.

quant-ph↗

arXiv2025

Reconstructing Biological Pathways by Applying Selective Incremental Learning to (Very) Small Language Models

The use of generative artificial intelligence (AI) models is becoming ubiquitous in many fields. Though progress continues to be made, general purpose large language AI models (LLM) show a tendency to deliver creative answers, often called "hallucinations", which have slowed their application in the medical and biomedical fields where accuracy is paramount. We propose that the design and use of much smaller, domain and even task-specific LM may be a more rational and appropriate use of this technology in biomedical research. In this work we apply a very small LM by today's standards to the specialized task of predicting regulatory interactions between molecular components to fill gaps in our current understanding of intracellular pathways. Toward this we attempt to correctly posit known pathway-informed interactions recovered from manually curated pathway databases by selecting and using only the most informative examples as part of an active learning scheme. With this example we show that a small (~110 million parameters) LM based on a Bidirectional Encoder Representations from Transformers (BERT) architecture can propose molecular interactions relevant to tuberculosis persistence and transmission with over 80% accuracy using less than 25% of the ~520 regulatory relationships in question. Using information entropy as a metric for the iterative selection of new tuning examples, we also find that increased accuracy is driven by favoring the use of the incorrectly assigned statements with the highest certainty (lowest entropy). In contrast, the concurrent use of correct but least certain examples contributed little and may have even been detrimental to the learning rate.

q-bio.MN↗

arXiv2025

Spectral coverings without embeddings

In this article, we investigate a weakened version of the spectral correspondence for twisted Higgs bundles. Namely, we construct twisted Higgs bundles from a finite covering map and a vector bundle on that covering but without requiring that they match the eigen-data for some fixed twisted Higgs bundle. We investigate stability for twisted Higgs bundles constructed in this way, and compare our covering data to that of the traditional spectral cover.

math.AG↗

arXiv2025

A Graph-Based Classical and Quantum Approach to Deterministic L-System Inference

L-systems can be made to model and create simulations of many biological processes, such as plant development. Finding an L-system for a given process is typically solved by hand, by experts, in a massively time-consuming process. It would be significant if this could be done automatically from data, such as from sequences of images. In this paper, we are interested in inferring a particular type of L-system, deterministic context-free L-system (D0L-system) from a sequence of strings. We introduce the characteristic graph of a sequence of strings, which we then utilize to translate our problem (inferring D0L-systems) in polynomial time into the maximum independent set problem (MIS) and the SAT problem. After that, we offer a classical exact algorithm and an approximate quantum algorithm for the problem.

quant-ph↗

arXiv2025

GPU-accelerated Modeling of Biological Regulatory Networks

The complex regulatory dynamics of a biological network can be succinctly captured using discrete logic models. Given even sparse time-course data from the system of interest, previous work has shown that global optimization schemes are suitable for proposing logic models that explain the data and make predictions about how the system will behave under varying conditions. Considering the large scale of the parameter search spaces associated with these regulatory systems, performance optimizations on the level of both hardware and software are necessary for making this a practical tool for in silico pharmaceutical research. We show here how the implementation of these global optimization algorithms in a GPU-computing environment can accelerate the solution of these parameter search problems considerably. We carry out parameter searches on two model biological regulatory systems that represent almost an order of magnitude scale-up in complexity, and we find the gains in efficiency from GPU to be a 33%-43% improvement compared to multi-thread CPU implementations and a 33%-1866% increase compared to CPU in serial. These improvements make global optimization of logic model identification a far more attractive and feasible method for in silico hypothesis generation and design of experiments.

q-bio.MN↗

arXiv2025

Moduli stacks of quiver bundles with applications to Higgs bundles

We provide a general method for constructing moduli stacks whose points are diagrams of vector bundles over a fixed base, indexed by a fixed simplicial set -- that is, quiver bundles of a fixed shape. We discuss some constraints on the base for these moduli stacks to be Artin and observe that a large class of interesting schemes satisfy these constraints. Using this construction, we recover Nakajima quiver varieties and provide an alternate construction for moduli stacks of Higgs bundles along with a proof of algebraicity following readily from the algebraicity of moduli stacks of quiver bundles. One feature of our approach is that, for each of the moduli stacks we discuss, there are moduli stacks that are Artin, parametrizing morphisms of the objects being classified. We discuss some potential applications of this in categorifying non-abelian Hodge theory in a sense we will make precise. We also discuss potential applications of our methods and perspectives to the subjects of quiver varieties, abstract moduli theory, and homotopy theory.

math.AG↗

arXiv2024

Asynchronous Telegate and Teledata Protocols for Distributed Quantum Computing

The cost of distributed quantum operations such as the telegate and teledata protocols is high due to latencies from distributing entangled photons and classical information. This paper proposes an extension to the telegate and teledata protocols to allow for asynchronous classical communication which hides the cost of distributed quantum operations. We then discuss the benefits and limitations of these asynchronous protocols and propose a potential way to improve these asynchronous protocols using nonunitary operators. Finally, a quantum network card is described as an example of how asynchronous quantum operations might be used.

quant-ph↗

arXiv2024

Nakajima quiver bundles

We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold $X$, such a representation involves an assignment of a complex vector bundle on $X$ to each node of the doubled quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the ADHM equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperkähler. Finally, we produce concrete examples that recover known and new moduli spaces.

math.AG↗

arXiv2024

Very stable and wobbly loci for elliptic curves

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus $1$ curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

math.AG↗

arXiv2024

Topological recursion and variations of spectral curves for twisted Higgs bundles

Prior works relating meromorphic Higgs bundles to topological recursion, in particular those of Dumitrescu-Mulase, have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. We start from an $\mathcal{L}$-twisted Higgs bundle for some fixed holomorphic line bundle $\mathcal{L}$ on the surface. We decorate the Higgs bundle with the choice of a section $s$ of $K^*\otimes\mathcal{L}$, where $K$ is the canonical line bundle, and then encode this data as a $b$-structure on the base Riemann surface which lifts to the associated Hitchin spectral curve. We then propose a so-called twisted topological recursion on the spectral curve, after which the corresponding Eynard-Orantin differentials live in a twisted cotangent bundle. This formulation retains, and interacts explicitly with, the singular structure of the original meromorphic setting -- equivalently, the zero divisor of $s$ -- while performing the recursion. Finally, we show that the $g=0$ twisted Eynard-Orantin differentials compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of Baraglia-Huang for ordinary Higgs bundles and topological recursion. Starting from the spectral curve as a polynomial form in an affine coordinate rather than a Higgs bundle, our result implies that, under certain conditions on $s$, the expansion is independent of the ambient space $\mbox{Tot}(\mathcal{L})$ in which the curve is interpreted to reside.

math.AG↗

arXiv2023

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of the quotient singularity C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n - 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n >= 6.

math.AG↗

arXiv2023

Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized hyperkähler reduction

Let $G$ be a compact Lie group. We study a class of Hamiltonian $(G \times S^{1})$-manifolds decorated with a function $s$ with certain equivariance properties, under conditions on the $G$-action which we call of (semi-)linear type. In this context, a close analogue of hyperkähler reduction is defined, and our main result establishes surjectivity of an appropriate analogue of Kirwan's map. As a particular case, our setting includes a class of hyperkähler manifolds with trihamiltonian torus actions, to which our surjectivity result applies.

math.SG↗

arXiv2022

The Kapustin-Witten equations and nonabelian Hodge theory

Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $t\in\mathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kähler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $λ$-connection in the nonabelian Hodge theory of Donaldson-Corlette-Hitchin-Simpson in which $λ$ is also valued in $\mathbb{P}^1$. Varying $λ$ interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at $λ=0$) and the moduli space of semisimple local systems on the same variety (at $λ=1$) in the twistor space. In this article, we utilise the correspondence furnished by nonabelian Hodge theory to describe a relation between the moduli spaces of solutions to the equations by Kapustin and Witten at $t=0$ and $t \in \mathbb{R} \setminus \{ 0 \}$ on a smooth, compact Kähler surface. We then provide supporting evidence for a more general form of this relation on a smooth, closed four-manifold by computing its expected dimension of the moduli space for each of $t=0$ and $t \in \mathbb{R} \setminus \{ 0 \}$.

math.DG↗

arXiv2022

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of $\textbf{Vect}_{\mathbb C}$ generated by $\mathbb C^2$ -- in particular, quantum circuits are diagrams in $\textbf{Vect}_{\mathbb C}$ -- while topological quantum field theories, in the sense of Atiyah, are diagrams in $\textbf{Vect}_{\mathbb C}$ indexed by cobordisms. We initiate a program that formalizes this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a double category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable double categorical structure which we call $\mathbb F\textbf{Vect}_{\mathbb C}$. We realize quantum circuits as images of cobordisms under monoidal double functors from these modified cobordisms to $\mathbb F\textbf{Vect}_{\mathbb C}$, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain double category.

quant-ph↗

arXiv2022

On the hyperbolic Bloch transform

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" already in the simplest case, that is when the Hilbert space is the regular representation of the Fuchsian group, $Γ$. Second, when $Γ\subset \mathrm{PSU} (1, 1)$ acts isometrically on the hyperbolic plane, $\mathbb{H}$, and the Hilbert space is $L^2 \left( \mathbb{H} \right)$, then we define a modified, geometric Bloch transform, that sends wave functions to sections of stable, flat bundles over $Σ= \mathbb{H} / Γ$ and transforms the hyperbolic Laplacian into the covariant Laplacian.

math-ph↗

arXiv2022

Hyperbolic band theory through Higgs bundles

Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's theorem have revealed an intrinsic role for algebro-geometric moduli spaces, notably those of stable bundles on a curve. We expand this picture to include Higgs bundles, which enjoy natural interpretations in the context of band theory. First, their spectral data encodes a crystal lattice and momentum, providing a framework for symmetric hyperbolic crystals. Second, they act as a complex analogue of crystal momentum. As an application, we elicit a new perspective on Euclidean band theory. Finally, we speculate on potential interactions of hyperbolic band theory, facilitated by Higgs bundles, with other themes in mathematics and physics.

math-ph↗

arXiv2022

Automorphic Bloch theorems for hyperbolic lattices

Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of 2D hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; whether a formal Bloch theorem can be rigorously established; and whether hyperbolic band theory can describe finite lattices realized in experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved, but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps corrrespond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only 1D irreps appear, and the hyperbolic band theory previously developed becomes exact.

cond-mat.mes-hall↗