Quantum gravity and effective topology

BackgroundAttention-deficit/hyperactivity disorder (ADHD) is currently diagnosed by a diagnostic interview, mainly based on subjective reports from parents or teachers. It is necessary to develop methods that rely on objectively measureable neurobiological data to assess brain-behavior relationship in patients with ADHD. We investigated the application of a topological data analysis tool, Mapper, to analyze the brain functional connectivity data from ADHD patients.MethodsTo quantify the disease severity using the neuroimaging data, the decomposition of individual functional networks into normal and disease components by the healthy state model (HSM) was performed, and the magnitude of the disease component (MDC) was computed. Topological data analysis using Mapper was performed to distinguish children with ADHD (n = 196) from typically developing controls (TDC) (n = 214).ResultsIn the topological data analysis, the partial clustering results of patients with ADHD and normal subjects were shown in a chain-like graph. In the correlation analysis, the MDC showed a significant increase with lower intelligence scores in TDC. We also found that the rates of comorbidity in ADHD significantly increased when the deviation of the functional connectivity from HSM was large. In addition, a significant correlation between ADHD symptom severity and MDC was found in part of the dataset.ConclusionsThe application of HSM and topological data analysis methods in assessing the brain functional connectivity seem to be promising tools to quantify ADHD symptom severity and to reveal the hidden relationship between clinical phenotypic variables and brain connectivity.

We explore an extended coupling constant space of 4d regularized Euclidean quantum gravity, dened via the formalism of dynamical triangulations. We add a measure term which can also serve as a generalized higher curvature term and determine the phase diagram and the geometries dominating in the various regions. A rst order phase tran- sition line is observed, but no second order transition point is located. As a consequence we cannot attribute any continuum physics interpretation to the so-called crinkled phase of 4d dynamical triangulations.

One can try to define the theory of quantum gravity as the sum over geometries. In two dimensions the sum over {\it Euclidean} geometries can be performed constructively by the method of {\it dynamical triangulations}. One can define a {\it proper-time} propagator. This propagator can be used to calculate generalized Hartle-Hawking amplitudes and it can be used to understand the the fractal structure of {\it quantum geometry}. In higher dimensions the philosophy of defining the quantum theory, starting from a sum over Euclidean geometries, regularized by a reparametrization invariant cut off which is taken to zero, seems not to lead to an interesting continuum theory. The reason for this is the dominance of singular Euclidean geometries. Lorentzian geometries with a global causal structure are less singular. Using the framework of dynamical triangulations it is possible to give a constructive definition of the sum over such geometries, In two dimensions the theory can be solved analytically. It differs from two-dimensional Euclidean quantum gravity, and the relation between the two theories can be understood. In three dimensions the theory avoids the pathologies of three-dimensional Euclidean quantum gravity. General properties of the four-dimensional discretized theory have been established, but a detailed study of the continuum limit in the spirit of the renormalization group and {\it asymptotic safety} is till awaiting.

Shaken, but not stirred—Potts model coupled to quantum gravity

Lorentzian simplicial quantum gravity is a non-perturbatively defined theory of quantum gravity which predicts a positive cosmological constant. Since the approach is based on a sum over space–time histories, it is perturbatively non-renormalizable even in three dimensions. By mapping the three-dimensional theory to a two-matrix model with ABAB interaction we show that both the cosmological and the (perturbatively) non-renormalizable gravitational coupling constant undergo additive renormalizations consistent with canonical quantization.

We formulate the string field theory in zero-dimensional target space corresponding to the two-dimensional quantum gravity theory defined through Causal Dynamical Triangulations. This third quantization of the quantum gravity theory allows us in principle to calculate the transition amplitudes of processes in which the topology of space changes in time, and to include non-trivial topologies of space-time. We formulate the corresponding Dyson-Schwinger equations and illustrate how they can be solved iteratively.

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounterbetween different approaches and different perspectives on the quantum-gravity problem. It contributedto a higher level of shared knowledge among the quantum-gravity communities pursuing each specific research program.There were plenary talks on many different approaches, including in particular string theory, loop quantum gravity, spacetime noncommutativity, causal dynamical triangulations, asymptotic safety and causal sets. Contributions from the perspective of philosophy of science were also welcomed.In addition several parallel sessions were organized. The present volume collects contributions from the Noncommutative Geometry and Quantum Gravity parallel session4, with additional invited contributions from specialists in the field.Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics:• from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory,• from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime,• from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions.This volume provides an overview of these various topics, interesting for the specialist as well as accessibleto the newcomer. 4partially funded by CNRS PEPS /PTI ‘‘Metric aspect of noncommutative geometry: from Monge to Higgs’’

Topological data analysis (TDA) is a mathematically well-founded set of methods to derive robust information about the structure and topology of data. It has been applied successfully in several biological contexts. Derived primarily from algebraic topology, TDA rigorously identifies persistent features in complex data, making it well-suited to better understand the key features of three-dimensional chromosome structure. Chromosome structure has a significant influence in many diverse genomic processes and has recently been shown to relate to cellular differentiation. While there exist many methods to study specific substructures of chromosomes, we are still missing a global view of all geometric features of chromosomes. By applying TDA to the study of chromosome structure through differentiation across three cell lines, we provide insight into principles of chromosome folding and looping. We identify persistent connected components and one-dimensional topological features of chromosomes and characterize them across cell types and stages of differentiation. Availability: Scripts to reproduce the results from this study can be found at https://github.com/Kingsford-Group/hictda

An Artificial Intelligence Ultrasound Platform for Screening and Staging of Thyroid Cancer

Asthma is a complex respiratory disease characterised by spatial ventilation heterogeneity (VH) in the lung. Multiple breath washout (MBW) and forced oscillation technique (FOT) testing has been shown to quantify VH at the mouth. Methods to provide fresh insight into the mechanisms driving VH are urgently needed to support treatment development. Computed tomography (CT) with parametric response map (PRM) analysis provides spatial and functional lung data. Integrated data sets of imaging features and VH markers present the problem of extracting meaningful information from high dimensional objects; topological data analysis (TDA) has demonstrated recent success in solving this problem. In this thesis topological methods are applied to extract information about lung anatomy and function from imaging data to study association with VH in innately high dimensional integrated data sets. It was sought to describe the spatial distributions of voxel based metrics, visualise the lungs using TDA methods, and combine PRM with simulated airway tree models to improve computational modelling of FOT. Data was obtained from a clinical trial involving 52 subjects, 41 asthmatic (11 healthy). An algorithm was developed to quantify directional Hounsfield unit change (ΔHU) gradients. TDA was applied to integrated data sets and voxel sets with measures of airway tree depth. FOT modelling was augmented with ΔHU about simulated acinar airways.Inferior-to-superior ΔHU gradient best differentiated VH markers. Visualisations of the lungs illustrated gradient reversal and increased heterogeneity in ΔHU in asthmatics. FOT modelling using ΔHU parameterisation showed increased correlation between clinically measured and simulated FOT small airway resistance (R5-R20). In this thesis additional insight is presented into mechanisms affecting VH markers derived from MBW and FOT. Topological methods to analyse imaging data were developed and implemented, including workflows to visualise the lungs and lung function. Linking simulated airway tree geometry with ΔHU showed improvement in FOT modelling.

We review some recent results from the causal dynamical triangulation (CDT) approach to quantum gravity. We review recent observations of dimensional reduction at a number of previously undetermined points in the parameter space of CDT, and discuss their possible relevance to the asymptotic safety scenario. We also present an updated phase diagram of CDT, discussing properties of a newly discovered phase and its possible relation to a signature change of the metric.

The paper will provide examples of computer vision tasks in which topological data analysis gave new effective solutions. Ideas underlying topological data analysis and its basic methods will be briefly described and illustrated with examples of computer vision problems. No prior knowledge in topological data analysis and computational geometry is assumed, a brief introduction to subject is given throughout the text.

Topological methods are very rarely used in structural health monitoring (SHM), or indeed in structural dynamics generally, especially when considering the structure and topology of observed data. Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly introduced here. Ideas from topological data analysis are applied to a Z24 Bridge case study, to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.

Single-cell technologies have revolutionized our ability to interrogate biological systems at unprecedented resolution, revealing complex cellular heterogeneity and dynamic processes that underlie development, disease, and immune responses. However, the high dimensionality and nonlinear structure of single-cell data present substantial analytical challenges. Topological data analysis offers a powerful mathematical framework for capturing the intrinsic shape of data, providing novel insights that complement and extend traditional statistical and machine learning methods. By leveraging tools such as persistent homology and the Mapper algorithm, TDA enables the detection of subtle, multiscale patterns – including rare cell populations, transitional states, and branching trajectories – that are often obscured by conventional approaches. In this review, we explore the theoretical foundations of topological data analysis and examine its emerging applications across single-cell transcriptomics, proteomics, and spatial biology. We highlight how this approach can unveil previously unrecognized biological phenomena, from alternative differentiation paths to complex tissue architectures, and discuss the growing ecosystem of computational tools that support its use. As single-cell datasets become increasingly large and multimodal, topological data analysis stands out as a uniquely robust and interpretable approach, with the potential to deepen our understanding of cellular identity and function in health and disease. TDA is specially suited for fields such as systems immunology since it can capture the complex, nonlinear structures inherent in high-dimensional immune data helping to identify distinct immune cell states, differentiation pathways, and dynamic responses to infection or therapy. This topological perspective complements traditional statistical approaches, providing a robust, scale-invariant framework for uncovering hidden organization within the immune system’s complexity.