New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
History of Information: Classical, Medieval, Modern theory
Open problem of Information: The unification of various theories of information; What is useful/meaningful information?What is an adequate logic of information? Continuous versus discrete models of nature; Computation versus thermodynamics; Classical information versus quantum information; Information and the theory of everything; The Church-Turing Hypothesis; P versus NP?
It from Bit: Why the Quantum? It from Bit? A Participatory Universe?: Three Far-reaching, Visionary Questions from John Archibald Wheeler
Physic, Math, Information: String Theory, Quantum, Sporadic finite Groups, Leech Latice, Gravity as emergent,
Universe digital copy conjecture: representation of universal information
Emergent Transformation Conjecture: the math of emergent
Potential applications: Deep Learning; Capability Transformation using Enterprise Architect
What’s in it for us: Information science, getting ready for Industry 4.0
Topological Data Analysis: visual presentation of multidimensional data setsDataRefiner
Topology data analysis (TDA) is an unsupervised approach which may revolutionise the way data can be mined and eventually drive the new generation of analytical tools. The idea behind TDA is an attempt to "measure" shape of data and find compressed combinatorial representation of the shape. In ordinary topology, the combinatorial representations serve the purpose of providing the compressed representation of high dimensional data sets which retains information about the geometric relationships between data points. TDA can also be used as a very powerful clustering technique. Edward will present the comparison between TDA and other dimension reduction algorithms like PCA, LLE, Isomap, MDS, and Spectral Embedding.
Quantum Information Science and Quantum Neuroscience.pptMelanie Swan
Mathematical advance in quantum information science is proceeding quickly and applies to many fields, particularly the complexities of neuroscience (here focusing on image-readable physical behaviors such as neural signaling, as opposed to higher-order operations of cognition, memory, and attention). Quantum mathematical models are extensible to neuroscience problem classes treating dynamical time series, diffusion, and renormalization in multiscalar systems. Approaches first reconstruct wavefunctions observed in EEG and fMRI scans. Second, single-neuron models (Hodgkin-Huxley, integrate-and-fire, theta neurons) and collective neuron models (neural field theories, Kuramoto oscillators) are employed to model empirical data. Third, genome physics is used to study time series sequence prediction in DNA, RNA, and proteins based on 3d+ complex geometry involving fields, curvature, knotting, and information compaction. Finally, quantum neuroscience physics is applied in AdS/Brain modeling, Chern-Simons biology (topological invariance), neuronal gauge theories, network neuroscience, and the chaotic dynamics of bifurcation and bistability (to explain epileptic and resting states). The potential benefit of this work is an improved understanding of disease and pathology resolution in humans.
History of Information: Classical, Medieval, Modern theory
Open problem of Information: The unification of various theories of information; What is useful/meaningful information?What is an adequate logic of information? Continuous versus discrete models of nature; Computation versus thermodynamics; Classical information versus quantum information; Information and the theory of everything; The Church-Turing Hypothesis; P versus NP?
It from Bit: Why the Quantum? It from Bit? A Participatory Universe?: Three Far-reaching, Visionary Questions from John Archibald Wheeler
Physic, Math, Information: String Theory, Quantum, Sporadic finite Groups, Leech Latice, Gravity as emergent,
Universe digital copy conjecture: representation of universal information
Emergent Transformation Conjecture: the math of emergent
Potential applications: Deep Learning; Capability Transformation using Enterprise Architect
What’s in it for us: Information science, getting ready for Industry 4.0
Topological Data Analysis: visual presentation of multidimensional data setsDataRefiner
Topology data analysis (TDA) is an unsupervised approach which may revolutionise the way data can be mined and eventually drive the new generation of analytical tools. The idea behind TDA is an attempt to "measure" shape of data and find compressed combinatorial representation of the shape. In ordinary topology, the combinatorial representations serve the purpose of providing the compressed representation of high dimensional data sets which retains information about the geometric relationships between data points. TDA can also be used as a very powerful clustering technique. Edward will present the comparison between TDA and other dimension reduction algorithms like PCA, LLE, Isomap, MDS, and Spectral Embedding.
Quantum Information Science and Quantum Neuroscience.pptMelanie Swan
Mathematical advance in quantum information science is proceeding quickly and applies to many fields, particularly the complexities of neuroscience (here focusing on image-readable physical behaviors such as neural signaling, as opposed to higher-order operations of cognition, memory, and attention). Quantum mathematical models are extensible to neuroscience problem classes treating dynamical time series, diffusion, and renormalization in multiscalar systems. Approaches first reconstruct wavefunctions observed in EEG and fMRI scans. Second, single-neuron models (Hodgkin-Huxley, integrate-and-fire, theta neurons) and collective neuron models (neural field theories, Kuramoto oscillators) are employed to model empirical data. Third, genome physics is used to study time series sequence prediction in DNA, RNA, and proteins based on 3d+ complex geometry involving fields, curvature, knotting, and information compaction. Finally, quantum neuroscience physics is applied in AdS/Brain modeling, Chern-Simons biology (topological invariance), neuronal gauge theories, network neuroscience, and the chaotic dynamics of bifurcation and bistability (to explain epileptic and resting states). The potential benefit of this work is an improved understanding of disease and pathology resolution in humans.
WiDS Alexandria, Egypt workshop in topological data analysis (Python and R code available on request), covering persistent homology, the Mapper algorithm, and discrete Ricci curvature. Examples include text data and social network data.
AdS Biology and Quantum Information ScienceMelanie Swan
Quantum Information Science is a fast-growing discipline advancing many areas of science such as cryptography, chemistry, finance, space science, and biology. In particular AdS/Biology, an interpretation of the AdS/CFT correspondence in biological systems, is showing promise in new biophysical mathematical models of topology (Chern-Simons (solvable QFT), knotting, and compaction). For example, one model of neurodegenerative disease takes a topological view of protein buildup (AB plaques and tau tangles in Alzheimer’s disease, alpha-synuclein in Parkinson’s disease, TDP-43 in ALS). AdS/Neuroscience methods are implicated in integrating multiscalar systems with different bulk-boundary space-time regimes (e.g. oncology tumors, fMRI + EEG imaging), entanglement (correlation) renormalization across scales (MERA, random tensor networks, melonic diagrams), entropy (possible system states), entanglement entropy (interrelated fluctuations and correlations across system tiers), and non-ergodicity (implied efficiency mechanisms since biology does not cycle through all possible configurations per temperature (thermotaxis), chemotaxis, and energy cues); Maxwell’s demon of biology (partition functions), conservation across system scales (biophysical gauge symmetry (system-wide conserved quantity)), and the presence of codes (DNA, codons, neural codes). A multiscalar AdS/CFT correspondence is mobilized in 4-tier ecosystem models (light-plankton-krill-whale and ion-synapse-neuron-network (AdS/Brain)).
CCS2019-opological time-series analysis with delay-variant embeddingHa Phuong
Q. H. Tran and Y. Hasegawa, Topological time-series analysis with delay-variant embedding, Oral Presentation at Conference on Complex Systems, Singapore, Singapore, Oct. 2019.
Predictability of the Dynamic Mode Decomposition in Coastal ProcessesRuo-Qian (Roger) Wang
Dynamic Mode Decomposition (DMD) is a model order reduction technique that helps reduce the complexity and high dimensionality of computational models. Since each decomposed mode has only a single frequency, the result of DMD is frequently easier to interpret physically than the conventional Proper Orthogonal Decomposition (POD). The DMD can also produce the eigenvalues of each mode to show the trend of the mode, establishing the rate of growth or decay, but the original DMD cannot produce the contributing weights of the modes. As a result, a challenge with the technique is selecting the important modes to build a reduced order model.
DMD variants have been developed to estimate the weights of each mode in the DMD. One of the popular methods is called Optimal Mode Decomposition (OMD). This method decomposes the data matrix into a product of the DMD modes, the diagonal weight matrix, and the Vandermonde matrix by minimizing the error between this product and the original data. The weight matrix can be used to rank the importance of the mode contributions and ultimately leads to the reduced order model for prediction and controlling purpose.
We are currently applying DMD and OMD to a large-scale numerical simulation of the San Francisco Bay, which features complicated coastal geometry, multiple frequency components, and high periodicity. Since DMD defines modes with specific frequencies, we expect DMD would produce a good approximation of the original model, but the preliminary results show that the predictability of the DMD is poor if unimportant modes are dropped according to the OMD. We are currently testing other DMD variants and will report our findings in the presentation.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
SIAM-AG21-Topological Persistence Machine of Phase TransitionHa Phuong
Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021.
Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.
In this tutorial, we study various statistical problems such as community detection on graphs, Principal Component Analysis (PCA), sparse PCA, and Gaussian mixture clustering in a Bayesian framework. Using a statistical physics point of view, we show that there exists a critical noise level above which it is impossible to estimate better than random guessing. Below this threshold, we compare the performance of existing polynomial-time algorithms to the optimal one and observe a gap in many situations: even if non-trivial estimation is theoretically possible, computationally efficient methods do not manage to achieve optimality. This tutorial will present how we adapted the tools and techniques from the mathematical study of spin glasses to study high-dimensional statistics and Approximate Message Passing (AMP) algorithm.
This tutorial was presented by Marc Lelarge at the 21st INFORMS Applied Probability Society Conference (2023)
https://informs-aps2023.event.univ-lorraine.fr/
UMAP is a technique for dimensionality reduction that was proposed 2 years ago that quickly gained widespread usage for dimensionality reduction.
In this presentation I will try to demistyfy UMAP by comparing it to tSNE. I also sketch its theoretical background in topology and fuzzy sets.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
This presentation on Lattice-based Digital Signatures from April 2018 was given to the Chinese academy of science from OnBoard Security's Zhenfei Zhang.
Cocentroidal and Isogonal Structures and Their Matricinal Forms, Procedures a...inventionjournals
The vector space of all matrices of the same order on a set of real numbers with a non-negative metric defined on it satisfying certain axioms, we call it a JS metric space. This space can be regarded as the infinite union of matrices possessing a special property that there exist infinite structures having the same (virtual) point-centroid. In this paper we introduce the notion of cocentroidal matrices to a given matrix (Root Matrix) in JS metric space wherein the metric is the Euclidean metric measuring the distance between two matrices to be the distance between their centroids. We describe mathematical routines that we envisaged and then theoretically verified for its applicability in a system that is under rotational motion about a point-centroid, and various cases in physics. We have sounded the same concept by considering necessary graphs drawn on the basis of mathematical equations as an additional feature of this article. Isogonality of different cocentroidal structures is the conceptual origin of one of the main concepts, necessitating the notion of convergence in terms of determinant values of a system of cocentroidal matrices.
WiDS Alexandria, Egypt workshop in topological data analysis (Python and R code available on request), covering persistent homology, the Mapper algorithm, and discrete Ricci curvature. Examples include text data and social network data.
AdS Biology and Quantum Information ScienceMelanie Swan
Quantum Information Science is a fast-growing discipline advancing many areas of science such as cryptography, chemistry, finance, space science, and biology. In particular AdS/Biology, an interpretation of the AdS/CFT correspondence in biological systems, is showing promise in new biophysical mathematical models of topology (Chern-Simons (solvable QFT), knotting, and compaction). For example, one model of neurodegenerative disease takes a topological view of protein buildup (AB plaques and tau tangles in Alzheimer’s disease, alpha-synuclein in Parkinson’s disease, TDP-43 in ALS). AdS/Neuroscience methods are implicated in integrating multiscalar systems with different bulk-boundary space-time regimes (e.g. oncology tumors, fMRI + EEG imaging), entanglement (correlation) renormalization across scales (MERA, random tensor networks, melonic diagrams), entropy (possible system states), entanglement entropy (interrelated fluctuations and correlations across system tiers), and non-ergodicity (implied efficiency mechanisms since biology does not cycle through all possible configurations per temperature (thermotaxis), chemotaxis, and energy cues); Maxwell’s demon of biology (partition functions), conservation across system scales (biophysical gauge symmetry (system-wide conserved quantity)), and the presence of codes (DNA, codons, neural codes). A multiscalar AdS/CFT correspondence is mobilized in 4-tier ecosystem models (light-plankton-krill-whale and ion-synapse-neuron-network (AdS/Brain)).
CCS2019-opological time-series analysis with delay-variant embeddingHa Phuong
Q. H. Tran and Y. Hasegawa, Topological time-series analysis with delay-variant embedding, Oral Presentation at Conference on Complex Systems, Singapore, Singapore, Oct. 2019.
Predictability of the Dynamic Mode Decomposition in Coastal ProcessesRuo-Qian (Roger) Wang
Dynamic Mode Decomposition (DMD) is a model order reduction technique that helps reduce the complexity and high dimensionality of computational models. Since each decomposed mode has only a single frequency, the result of DMD is frequently easier to interpret physically than the conventional Proper Orthogonal Decomposition (POD). The DMD can also produce the eigenvalues of each mode to show the trend of the mode, establishing the rate of growth or decay, but the original DMD cannot produce the contributing weights of the modes. As a result, a challenge with the technique is selecting the important modes to build a reduced order model.
DMD variants have been developed to estimate the weights of each mode in the DMD. One of the popular methods is called Optimal Mode Decomposition (OMD). This method decomposes the data matrix into a product of the DMD modes, the diagonal weight matrix, and the Vandermonde matrix by minimizing the error between this product and the original data. The weight matrix can be used to rank the importance of the mode contributions and ultimately leads to the reduced order model for prediction and controlling purpose.
We are currently applying DMD and OMD to a large-scale numerical simulation of the San Francisco Bay, which features complicated coastal geometry, multiple frequency components, and high periodicity. Since DMD defines modes with specific frequencies, we expect DMD would produce a good approximation of the original model, but the preliminary results show that the predictability of the DMD is poor if unimportant modes are dropped according to the OMD. We are currently testing other DMD variants and will report our findings in the presentation.
Introduction to Topological Data AnalysisMason Porter
Here are slides for my 3/14/21 talk on an introduction to topological data analysis.
This is the first talk in our Short Course on topological data analysis at the 2021 American Physical Society (APS) March Meeting: https://march.aps.org/program/dsoft/gsnp-short-course-introduction-to-topological-data-analysis/
SIAM-AG21-Topological Persistence Machine of Phase TransitionHa Phuong
Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021.
Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.
In this tutorial, we study various statistical problems such as community detection on graphs, Principal Component Analysis (PCA), sparse PCA, and Gaussian mixture clustering in a Bayesian framework. Using a statistical physics point of view, we show that there exists a critical noise level above which it is impossible to estimate better than random guessing. Below this threshold, we compare the performance of existing polynomial-time algorithms to the optimal one and observe a gap in many situations: even if non-trivial estimation is theoretically possible, computationally efficient methods do not manage to achieve optimality. This tutorial will present how we adapted the tools and techniques from the mathematical study of spin glasses to study high-dimensional statistics and Approximate Message Passing (AMP) algorithm.
This tutorial was presented by Marc Lelarge at the 21st INFORMS Applied Probability Society Conference (2023)
https://informs-aps2023.event.univ-lorraine.fr/
UMAP is a technique for dimensionality reduction that was proposed 2 years ago that quickly gained widespread usage for dimensionality reduction.
In this presentation I will try to demistyfy UMAP by comparing it to tSNE. I also sketch its theoretical background in topology and fuzzy sets.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
This presentation on Lattice-based Digital Signatures from April 2018 was given to the Chinese academy of science from OnBoard Security's Zhenfei Zhang.
Cocentroidal and Isogonal Structures and Their Matricinal Forms, Procedures a...inventionjournals
The vector space of all matrices of the same order on a set of real numbers with a non-negative metric defined on it satisfying certain axioms, we call it a JS metric space. This space can be regarded as the infinite union of matrices possessing a special property that there exist infinite structures having the same (virtual) point-centroid. In this paper we introduce the notion of cocentroidal matrices to a given matrix (Root Matrix) in JS metric space wherein the metric is the Euclidean metric measuring the distance between two matrices to be the distance between their centroids. We describe mathematical routines that we envisaged and then theoretically verified for its applicability in a system that is under rotational motion about a point-centroid, and various cases in physics. We have sounded the same concept by considering necessary graphs drawn on the basis of mathematical equations as an additional feature of this article. Isogonality of different cocentroidal structures is the conceptual origin of one of the main concepts, necessitating the notion of convergence in terms of determinant values of a system of cocentroidal matrices.
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The concepts of connectedness and countability in digital image processing are used for establishing boundaries of objects and components of regions in an image. The purpose of this paper is to investigate some notions of connectedness and countability of Khalimsky line topology.
https://utilitasmathematica.com/index.php/Index
Our Journal has steadfast in its commitment to promoting justice, equity, diversity, and inclusion within the realm of statistics. Through collaborative efforts and a collective dedication to these principles, we believe in building a statistical community that not only advances the profession. Paper publication
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Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
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💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
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LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
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Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
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The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
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Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
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Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
2. A Puzzle and an Adage
Eugene Wigner famously commented on
“the unreasonable effectiveness of
mathematics in the natural sciences”.
In one form, this is a puzzle about why
mathematical constructions are so apt as a
medium of representation of the physical
world.
3. Numbers
The puzzle seems particularly acute with respect to
certain numerical representations, and especially
those using complex numbers.
The puzzle arises from the fact that the physical world
in not a numerical entity and does not contain
numbers. So one must reflect on exactly which
structures of numerical constructions could be
isomorphic to, and hence faithfully represent, physical
structures.
4. The Adage
If we think of certain mathematical constructions
as abstract tools by means of which we represent
the physical structure of the world, then we must
bear in mind a profound saying…
If the only tool you have is a hammer, everything
looks like a nail.
That is, we should strive to construct and evaluate
many mathematical tools, and be sensitive to
when one seems to work better than another.
Perhaps that is because it is better suited to the
problem at hand.
5. Geometry
The Greeks divided mathematics into geometry (the
theory of magnitudes) and arithmetic (the theory of
numbers).
Much of the power of modern science derives from
the use of numerical or algebraic representations of
geometrical structure. This was first made possible via
the introduction of coordinate systems, which
associate points in a geometrical space with sets of
real numbers.
Nonetheless, the most basic geometrical structure is
still described without numbers.
6. Topology
In order to organize a set of points into a space,
some additional structure must be imposed on them.
The most fundamental such structure determines
facts about continuity in the space, including the
continuity of functions from one space to another.
This is the topological structure.
This level of structure is defined without regard to
either distance (metrical structure) or straightness
(affine structure): hence the rubric rubber sheet
geometry.
8. Standard Topology
The basic notion in the usual formulations
of topology is the open set.
Every other notion–closed set, connected
space, continuous function, boundary,
compactness, Hausdorff, etc.– is
ultimately defined in terms of the open
set structure.
9. The Architecture of Topology
Open set ⇔ Closed set ⇔ Neighborhoods
Connected
Space
Continuous
Function
Boundary of
a Set
Curve (continuous function
from real line into space)
Path
(image of a curve)
10. Informal Explication
“an open set is one in which every point has
some breathing space” M. Crossley, Essential
Topology
“In topology and related fields of mathematics, a
set U is called open if, intuitively speaking, you
can ‘wiggle’ or ‘change’ any point x in U by a
small amount in any direction and still be inside U.
In other words, x is surrounded only by elements
of U; it can’t be on the edge of U.”–Wikipedia
12. The Axioms
Definition: A topological space is a set, X,
together with a collection of subsets of X,
called “open” sets, which satisfy the following
rules:
T1. The set X itself is “open”.
T2. The empty set is “open”.
T3. Arbitrary unions of “open” sets are “open”.
T4. Finite intersections of “open” sets are
“open”.
13. For Example
Consider a space with only 2 points, p
and q. There are four standard
topologies:
The discrete topology: {p, q}, {p}, {q}, ∅.
The indiscrete topology: {p, q}, ∅.
Two Sierpinski spaces: {p, q} , {p}, ∅ and
{p, q}, {q}, ∅.
14. Why Should This Work?
If this particular mathematical tool—the analysis of the
continuity properties of a space in terms of its open set
structure—is a direct way to describe physical space or
space-time, then there should be some physical feature of
the world that determines which sets of events are open
sets.
It is not obvious what such a physical feature would be.
We could, of course, postulate it as a primitive fact about
sets of events—that some, but not others constitute open
sets—but that should be a last resort.
15. Alternative Geometrical
Primitive: the Line
Rather than the open set, there is a better
fundamental notion upon which a theory of sub-
metrical geometry can be built: the line.
More exactly, the “open” line, in the sense that
both open and closed line segments are “open”
and a circle is “closed”: from any point on the
line one can move continuously to any other
given point, but only by moving in one direction.
An open line in this sense has a structure
represented by a linear order among the points.
17. Theory of Linear Structures
lines
neighborhoods
≠ neighborhoods
open sets
initial-part
open sets
initial-part
closed sets
(=)continuous
functions
≠ continuous
functions
connected space
≠ connected space
18. Linear Orders
A linear order on a set S is a relation, which we
will symbolize by “≥”, that satisfies three
conditions:
For all p, q, r ∈ S
1) If p ≥ q and q ≥ p, then p = q (Antisymmetry)
2) If p ≥ q and q ≥ r, then p ≥ r (Transitivity)
3) p ≥ q or q ≥ p (Totality)
19. Intervals
An interval in set with a linear order is a
subset of at least two points such that
for any p, q in the subset, all points
between p and q in the order are in the
subset. (Dedekind)
20. Linear Structures (1st
type)
A Linear Structure is a set S together with Λ a
set of subsets of S called the lines of S that
satisfy:
LS1 (Minimality Axiom): Each line contains
at least two points.
LS2 (Segment Axiom): Every line λ admits of
a linear order among its points such that a
subset of λ is itself a line if and only if it is an
interval of that linear order.
21. Linear Structures con’t
LS3 (Point-Splicing Axiom): If λ and µ are lines
that have in common only a single point p that is
an endpoint of both, then λ ∪ µ is a line provided
that no lines in the set (λ ∪ µ) – p have a point in
λ and a point in µ.
LS4 (Completion Axiom): Any linearly ordered
set σ such that all and only the closed intervals in
the order are closed lines is a line.
22. Non-Uniqueness of Order
According to this first set of axioms, every
line can be represented by a linear order
among its points. But evidently there are
two such linear orders that will do the job,
one the inverse of the other. Each will
imply the same intervals, and so the same
structure of segments. (A segment of a line λ is
a subset of λ that is a line.)
26. Open Sets
A set Σ in a Linear Structure is an open set iff it
is a neighborhood of all of its members.
(NB: this definition looks identical to a
definition that appears in standard topology, with
neighborhood replaced by neighborhood. But in
standard topology, a neighborhood of a point is a
set containing an open set containing the point.)
27. Theorem
The collection of open sets in a
Linear Structure satisfy the axioms
of standard topology, i.e., the open
sets are open sets.
29. This suggests….
Evidently many (in an obvious
sense most) standard topologies
on a finite point set cannot be
generated from a Linear Structure
on that set.
I call such topologies
geometrically uninterpretable.
31. But…
A little further thought shows this to be incorrect! We
can understand all finite point topologies in terms of
“wiggles”.
In the sort of Linear Structure we have constructed
so far, we have treated the lines as two-way streets:
if a small “wiggle” along a line can take you from p
to q, then a small wiggle along the same line can
take you from q to p.
32. However…
Suppose we treat the lines as one-way streets: to
specify a line one has to specify both a set of
points that constitute it and a direction, i.e., only
one linear order represents a line, not two.
The intuitive notion of a “small wiggle” is a
continuous motion long a line in the direction of
the line.
33. Directed Linear Structures
This gives rise to the notion of a Directed Linear
Structure. The axioms are modified in the obvious
way: all and only the directed intervals in a linear
order are segments of a line, etc.
The Splicing Axiom now requires that to splice two
lines, the point p must be the final point of one
and the initial point of the other.
34. Outward Neighborhoods,
Outward Open Sets
A set Σ is an outward neighborhood of a
point p iff every line with p as an initial
endpoint has a segment with p as an initial
endpoint in Σ.
A set Σ in a Linear Structure is an outward
open set iff it is an outward neighborhood of
all of its members.
35. Directed LS for Two-Point Space
p q
p q
p q
p q
Outward open sets
{p,q}, {p}, {q}, Ø
{p,q}, {q}, Ø
{p,q}, {p}, Ø
{p,q}, Ø
36. More Numbers
# of points topologies Directed LS Topologies
from DLS
1 1 1 1
2 4 4 4
3 29 64 29
4 355 4,096 355
5 6,942 1,048,576 6,942
37. Theorem
Every finite-point topology is generated
by some finite-point Directed Linear
Structure. Typically, many distinct Linear
Structures give rise to the same
topology, so one loses geometrical
information if one only knows the
topology.
40. The Geometry of a Part of a
Space
In the Theory of Linear Structures, unlike
standard topology, the geometry of a
part of a space is defined in the natural
way: by simple restriction. That is, the
Linear Structure of a part of a space is
given by the lines that are contained in
that part.
41. Space-Time: Why a 4-d
Manifold is Unmotivated
A topological 4-dimensional manifold has an
open set structure that everywhere is locally
isomorphic to a 4-d Euclidean space. From our
point of view, the obvious reason to expect this
would be because the Linear Structure of space-
time is locally isomorphic to the Linear Structure
of 4-d Euclidean space.
45. Wald on “Mixed” Lines
”The length of curves which change from timelike to
spacelike is not defined” (General Relativity, p. 44).
So let’s eliminate those curves: the Linear Structure
of a Relativistic space-time is not the same as that of
any Euclidean space.
46. Physics
If the fundamental sub-metrical geometrical
structure is the line, then when we turn to
physics, we should ask: what physical feature
of the universe could generate physical lines?
More generally, what physical feature of the
universe naturally generates a linear order
among the points of space-time?
47. Time
Intuitively, time provides a directed linear
ordering of events. It is the natural place to
look for a source of physical lines.
48. Newtonian and Neo-
Newtonian Space-Time
In Newtonian or Neo-Newtonian space-time, if
one asks for a maximal set of events which is
linearly ordered in time, one gets a set of
points, one at each instant of time. This set of
points will not typically look like any sort of line:
50. Relativistic Space-Time
(Globally Hyperbolic)
In a Relativistic space-time (Lorentzian
pseudo-metric) with no closed time-like
curves (no “time travel”), if one asks for a
maximal set of events which is linearly
ordered in time, what one gets is a
continuous time-like or null curve.
The light-cone structure forces such a set
to intuitively form a line.
53. Only Trivial Geometry on a
Spacelike Hypersurface
If we only admit timelike-or-null lines, then
when we restrict the geometry to a space-like
hypersurface we get no lines at all: there is no
intrinsic spatial geometry.
Other slices, though, have the expected
Relativistic structure.
55. Relativistic Structure is Built
Into the Linear Structure
If we follow this recipe, the light-cone
structure of a space-time is already definable
from its Linear Structure, without use of any
metrical notions.
In particular, a closed line with endpoints p and
q is a straight lightlike line just in case it is the
only closed line with these endpoints. So one
can recover the light-cone structure directly
from the Directed Linear Structure.
56. Recovering the Whole
Relativistic Metric
To get the full Relativistic
(pseudo-)metric, one needs to attribute
a “length” to these lines, i.e. the proper
time along them. This is enough to
determine all the spatio-temporal
structure postulated by Relativity.
57. How the Mathematics
Describes Physics
If we use the Theory of Linear Structures to characterize
the geometry of a space, then the topology is
determined by the directed lines—linearly ordered sets of
points—in the space.
If we accept that time linearly orders events, then the
maximal sets of temporally ordered events form a
natural physical directed linear structure in space-time.
In Relativity—but not classical physics—this turns out to
be just the geometrical structure the physics need. Time
invests space-time with this geometrical structure.