Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

We compare Stein fillings and Milnor fibers for rational surface singularities with reduced fundamental cycle. Deformation theory for this class of singularities was studied by de Jong-van Straten in [dJvS98]; they associated a germ of a singular plane curve to each singularity and described Milnor fibers via deformations of this singular curve. We consider links of surface singularities, equipped with their canonical contact structures, and develop a symplectic analog of de Jong-van Straten's construction. Using planar open books and Lefschetz fibrations, we describe all Stein fillings of the links via certain arrangements of symplectic disks, related by a homotopy to the plane curve germ of the singularity. As a consequence, we show that many rational singularities in this class admit Stein fillings that are not strongly diffeomorphic to any Milnor fibers. This contrasts with previously known cases, such as simple and quotient surface singularities, where Milnor fibers are known to give rise to all Stein fillings. On the other hand, we show that if for a singularity with reduced fundamental cycle, the self-intersection of each exceptional curve is at most -5 in the minimal resolution, then the link has a unique Stein filling (given by a Milnor fiber).Comment: 70 pages, 29 figures, additions to v2 are Theorem 1.3 and its corresponding discussion in 4.3, along with added references, v3 updated Remark 4.

Logic and intuition in architectural modelling: philosophy of mathematics for computational design

This dissertation investigates the relationship between the shift in the focus of architectural modelling from object to system and philosophical shifts in the history of mathematics that are relevant to that change. Particularly in the wake of the adoption of digital computation, design model spaces are more complex, multidimensional, arguably more logical, less intuitive spaces to navigate, less accessible to perception and visual comprehension. Such spatial issues were encountered much earlier in mathematics than in architectural modelling, with the growth of analytical geometry, a transition from Classical axiomatic proofs in geometry as the basis of mathematics, to analysis as the underpinning of geometry. Can the computational design modeller learn from the changing modern history, philosophy and psychology of mathematics about the construction and navigation of computational geometrical architectural system model space? The research is conducted through a review of recent architectural project examples and reference to three more detailed architectural modelling case studies. The spatial questions these examples and case studies raise are examined in the context of selected historical writing in the history, philosophy and psychology of mathematics and space. This leads to conclusions about changes in the relationship of architecture and mathematics, and reflections on the opportunities and limitations for architectural system models using computation geometry in the light of this historical survey. This line of questioning was motivated as a response to the experience of constructing digital associative geometry models and encountering the apparent limits of their flexibility as the graph of dependencies grew and the messiness of the digital modelling space increased. The questions were inspired particularly by working on the Narthex model for the Sagrada Família church, which extends to many tens of thousands of relationships and constraints, and which was modelled and repeatedly partially remodelled over a very long period. This experience led to the realisation that the limitations of the model were not necessarily the consequence of poor logical schema definition, but could be inevitable limitations of the geometry as defined, regardless of the means of defining it, the ‘shape’ of the multidimensional space being created. This led to more fundamental questions about the nature of Space, its relationship to geometry and the extent to which the latter can be considered simply as an operational and notational system. This dissertation offers a purely inductive journey, offering evidence through very selective examples in architecture, architectural modelling and in the philosophy of mathematics. The journey starts with some questions about the tendency of the model space to break out and exhibit unpredictable and not always desirable behaviour and the opportunities for geometrical construction to solve these questions is not conclusively answered. Many very productive questions about computational architectural modelling are raised in the process of looking for answers

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction

HyperCell: A Bio-inspired Design Framework for Real-time Interactive Architectures

This pioneering research focuses on Biomimetic Interactive Architecture using “Computation”, “Embodiment”, and “Biology” to generate an intimate embodied convergence to propose a novel rule-based design framework for creating organic architectures composed of swarm-based intelligent components. Furthermore, the research boldly claims that Interactive Architecture should emerge as the next truly Organic Architecture. As the world and society are dynamically changing, especially in this digital era, the research dares to challenge the Utilitas, Firmitas, and Venustas of the traditional architectural Weltanschauung, and rejects them by adopting the novel notion that architecture should be dynamic, fluid, and interactive. This project reflects a trajectory from the 1960’s with the advent of the avant-garde architectural design group, Archigram, and its numerous intriguing and pioneering visionary projects. Archigram’s non-standard, mobile, and interactive projects profoundly influenced a new generation of architects to explore the connection between technology and their architectural projects. This research continues this trend of exploring novel design thinking and the framework of Interactive Architecture by discovering the interrelationship amongst three major topics: “Computation”, “Embodiment”, and “Biology”. The project aims to elucidate pioneering research combining these three topics in one discourse: “Bio-inspired digital architectural design”. These three major topics will be introduced in this Summary.   “Computation”, is any type of calculation that includes both arithmetical and nonarithmetical steps and follows a well-defined model understood and described as, for example, an algorithm. But, in this research, refers to the use of data storage, parametric design application, and physical computing for developing informed architectural designs. “Form” has always been the most critical focus in architectural design, and this focus has also been a major driver behind the application computational design in Architecture. Nonetheless, this research will interpret the term “Form” in architecture as a continual “information processor” rather than the result of information processing. In other words, “Form” should not be perceived only as an expressive appearance based computational outcome but rather as a real-time process of information processing, akin to organic “Formation”. Architecture embodying kinetic ability for adjusting or changing its shape with the ability to process the surroundings and feedback in accordance with its free will with an inherent interactive intelligent movement of a living body. Additionally, it is also crucial to address the question of whether computational technologies are being properly harnessed, if they are only used for form-generating purposes in architecture design, or should this be replaced with real-time information communication and control systems to produce interactive architectures, with embodied computation abilities?   “Embodiment” in the context of this research is embedded in Umberto Eco’s vision on Semiotics, theories underlying media studies in Marshall McLuhan’s “Body Extension” (McLuhan, 1964), the contemporary philosophical thought of “Body Without Organs” (Gilles Deleuze and Félix Guattari, 1983), the computational Logic of ‘Swarm Behavior’ and the philosophical notion of “Monadology” proposed by Gottfried Leibniz (Leibniz, 1714). Embodied computation and design are predominant today within the wearable computing and smart living domains, which combine Virtual and Real worlds. Technical progress and prowess in VR development also contribute to advancing 3D smart architectural design and display solutions. The proposed ‘Organic body-like architectural spaces’ emphasize upon the realization of a body-like interactive space. Developing Interactive Architecture will imply eliciting the collective intelligence prevalent in nature and the virtual world of Big Data. Interactive Architecture shall thus embody integrated Information exchange protocols and decision-making systems in order to possess organic body-like qualities.   “Biology”, in this research explores biomimetic principles intended to create purposedriven kinetic and organic architecture. This involves a detailed study/critique of organic architecture, generating organic shapes, performance optimization based digital fabrication techniques and kinetic systems. A holistic bio-inspired architecture embodies multiple performance criteria akin to natural systems, which integrate structural, infrastructure performances throughout the growth of an organic body. Such a natural morphogenesis process of architectural design explores what Janine M. Benyus described as “learning the natural process”. Profoundly influenced by the processes behind morphogenesis, the research further explores Evolutionary Development Biology (Evo-Devo) explaining how embryological regulation strongly affect the resulting formations. Evo-Devo in interactive architecture implies the development of architecture based on three fundamental principles: “Simple to Complex”, “Geometric Information Distribution”, and “On/Off Switch and Trigger.” The research seeks to create a relatively intelligent architectural body, and the tactile interactive spatial environment by applying the extracted knowledge from the study of the aforementioned principles of Evo-Devo in the following fashion: A. Extract a Self-Similar Componential System based approach from the “Simple to Complex” principle of Evo-Devo B. Extract the idea of “Collective Intelligence” from “Geometric information Distribution” principle of Evo-Devo C. Extract the principle of “Assembly Regulation” from “On/Off switch and trigger” principle of Evo-Devo The “HyperCell” research, through an elaborate investigation on the three aforementioned topics, develops a design framework for developing real-time adaptive spatial systems. HyperCell does this, by developing a system of transformable cubic elements which can self-organize, adapt and interact in real-time. These Hypercells shall comprise an organic space which can adjust itself in relation to our human bodies. The furniture system is literally reified and embodied to develop an intra-active space that proactively provokes human movement. The space thus acquires an emotive dimension and can become your pet, partner, or even friend, and might also involve multiple usabilities of the same space. The research and its progression were also had actively connected with a 5-year collaborative European Culture project: “MetaBody”. The research thus involves exploration of Interactive Architecture from the following perspectives: architectural design, digital architectural history trajectory, computational technology, philosophical discourse related to the embodiment, media and digital culture, current VR and body-related technology, and Evolutionary Developmental Biology. “HyperCell” will encourage young architects to pursue interdisciplinary design initiatives via the fusion of computational design, embodiment, and biology for developing bio-inspired organic architectures

HyperCell: A Bio-inspired Design Framework for Real-time Interactive Architectures

This pioneering research focuses on Biomimetic Interactive Architecture using “Computationâ€, “Embodimentâ€, and “Biology†to generate an intimate embodied convergence to propose a novel rule-based design framework for creating organic architectures composed of swarm-based intelligent components. Furthermore, the research boldly claims that Interactive Architecture should emerge as the next truly Organic Architecture. As the world and society are dynamically changing, especially in this digital era, the research dares to challenge the Utilitas, Firmitas, and Venustas of the traditional architectural Weltanschauung, and rejects them by adopting the novel notion that architecture should be dynamic, fluid, and interactive. This project reflects a trajectory from the 1960’s with the advent of the avant-garde architectural design group, Archigram, and its numerous intriguing and pioneering visionary projects. Archigram’s non-standard, mobile, and interactive projects profoundly influenced a new generation of architects to explore the connection between technology and their architectural projects. This research continues this trend of exploring novel design thinking and the framework of Interactive Architecture by discovering the interrelationship amongst three major topics: “Computationâ€, “Embodimentâ€, and “Biologyâ€. The project aims to elucidate pioneering research combining these three topics in one discourse: “Bio-inspired digital architectural designâ€. These three major topics will be introduced in this Summary. “Computationâ€, is any type of calculation that includes both arithmetical and nonarithmetical steps and follows a well-defined model understood and described as, for example, an algorithm. But, in this research, refers to the use of data storage, parametric design application, and physical computing for developing informed architectural designs. “Form†has always been the most critical focus in architectural design, and this focus has also been a major driver behind the application computational design in Architecture. Nonetheless, this research will interpret the term “Form†in architecture as a continual “information processor†rather than the result of information processing. In other words, “Form†should not be perceived only as an expressive appearance based computational outcome but rather as a real-time process of information processing, akin to organic “Formationâ€. Architecture embodying kinetic ability for adjusting or changing its shape with the ability to process the surroundings and feedback in accordance with its free will with an inherent interactive intelligent movement of a living body. Additionally, it is also crucial to address the question of whether computational technologies are being properly harnessed, if they are only used for form-generating purposes in architecture design, or should this be replaced with real-time information communication and control systems to produce interactive architectures, with embodied computation abilities? “Embodiment†in the context of this research is embedded in Umberto Eco’s vision on Semiotics, theories underlying media studies in Marshall McLuhan’s “Body Extension†(McLuhan, 1964), the contemporary philosophical thought of “Body Without Organs†(Gilles Deleuze and Félix Guattari, 1983), the computational Logic of ‘Swarm Behavior’ and the philosophical notion of “Monadology†proposed by Gottfried Leibniz (Leibniz, 1714). Embodied computation and design are predominant today within the wearable computing and smart living domains, which combine Virtual and Real worlds. Technical progress and prowess in VR development also contribute to advancing 3D smart architectural design and display solutions. The proposed ‘Organic body-like architectural spaces’ emphasize upon the realization of a body-like interactive space. Developing Interactive Architecture will imply eliciting the collective intelligence prevalent in nature and the virtual world of Big Data. Interactive Architecture shall thus embody integrated Information exchange protocols and decision-making systems in order to possess organic body-like qualities. “Biologyâ€, in this research explores biomimetic principles intended to create purposedriven kinetic and organic architecture. This involves a detailed study/critique of organic architecture, generating organic shapes, performance optimization based digital fabrication techniques and kinetic systems. A holistic bio-inspired architecture embodies multiple performance criteria akin to natural systems, which integrate structural, infrastructure performances throughout the growth of an organic body. Such a natural morphogenesis process of architectural design explores what Janine M. Benyus described as “learning the natural processâ€. Profoundly influenced by the processes behind morphogenesis, the research further explores Evolutionary Development Biology (Evo-Devo) explaining how embryological regulation strongly affect the resulting formations. Evo-Devo in interactive architecture implies the development of architecture based on three fundamental principles: “Simple to Complexâ€, “Geometric Information Distributionâ€, and “On/Off Switch and Trigger.†The research seeks to create a relatively intelligent architectural body, and the tactile interactive spatial environment by applying the extracted knowledge from the study of the aforementioned principles of Evo-Devo in the following fashion: A. Extract a Self-Similar Componential Systembased approach from the “Simple to Complex†principle of Evo-Devo B. Extract the idea of “Collective Intelligence†from “Geometric information Distribution†principle of Evo-Devo C. Extract the principle of “Assembly Regulation†from “On/Off switch and trigger†principle of Evo-Devo The “HyperCell†research, through an elaborate investigation on the three aforementioned topics, develops a design framework for developing real-time adaptive spatial systems. HyperCell does this, by developing a system of transformable cubic elements which can self-organize, adapt and interact in real-time. These Hypercells shall comprise an organic space which can adjust itself in relation to our human bodies. The furniture system is literally reified and embodied to develop an intra-active space that proactively provokes human movement. The space thus acquires an emotive dimension and can become your pet, partner, or even friend, and might also involve multiple usabilities of the same space. The research and its progression were also had actively connected with a 5-year collaborative European Culture project: “MetaBodyâ€. The research thus involves exploration of Interactive Architecture from the following perspectives: architectural design, digital architectural history trajectory, computational technology, philosophical discourse related to the embodiment, media and digital culture, current VR and body-related technology, and Evolutionary Developmental Biology. “HyperCell†will encourage young architects to pursue interdisciplinary design initiatives via the fusion of computational design, embodiment, and biology for developing bio-inspired organic architectures

Play Among Books

How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

All At One Point: The New Physics of Italo Calvino and Jorge Luis Borges

This work of comparative literary criticism focuses on the presence of mathematical and scientific concepts and imagery in the works of Italo Calvino and Jorge Luis Borges, beginning with an historical overview of scientific philosophy and an introduction to the most significant scientific concepts of the last several centuries, before shifting to deep, scientifically-driven analyses of numerous individual fictions, and finally concluding with a meditation on the unexpectedly fictive aspects of science and mathematics. The close readings of these authors\u27 fictions are contextualized with thorough explanations of the potential literary implications of theories from physics, mathematics, neuroscience and chaos theory. While the mathematical studies highlight concepts such as Zeno\u27s Paradox, Cantorian set theory, and representations of numerical infinity, the discussions of physics isolate theoretical structures such as black holes, parallel universes and quantum-entangled particles for use in discussing the fictions of both authors. Underlying the main goals of this work is an equal focus on the existence of an ideal intellectual or, more broadly, an ideal liberal arts education that draws together concepts from diverse and seemingly-unrelated fields of knowledge with the intention of generating unexpected and novel ideas and connections. By demonstrating the numerous appearances of scientific and mathematical imagery in the works of Calvino and Borges, this work emphasizes the shared fictive basis of all human knowledge and strives to set science and fiction alongside each other as equals in the hope of preserving the richness and diversity of intellectual enterprises