“…A number of emerging trends are the growing use of generative models, dynamic (time-varying) and multilayer networks, as well as the application of algebraic topology. Overall, graph theory methods are centrally important to understanding the architecture, development, and evolution of brain networks.…”
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“…Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages: It leads more quickly to the essentials of the subject, An absence of signs and orientation considerations simplifies the theory, Computations and advanced applications can be presented at an earlier stage, Simple geometrical interpretations of (co)chains. …”
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“…He created new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to asymptotic expansions. …”
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“…Their use is well accepted in fundamental areas such as algebraic geometry, representation theory, algebraic topology, and combinatorics. Now, similar to having moved from groups to groupoids, it is becoming clear that generalizations of Hopf algebras must also be considered. …”
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“…These are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. …”
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“…A key tool for the new construction is the structure of a double complex, resembling the Čech-de Rham double complex of algebraic topology. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 541–551, 2015…”
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“…We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.…”
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“…The theory is new although it borrows from algebraic topology. A highlight is the proof that every generalized topological (co)homology theory has a counterpart in WSA(R) with in some sense "the same", or even better, properties as the topological theory. …”
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“…This book provides an advanced introduction to extended theories of quantum field theory and algebraic topology, including Hamiltonian quantization associated with some geometrical constraints, symplectic embedding and Hamilton-Jacobi quantization and Becci-Rouet-Stora-Tyutin (BRST) symmetry, as well as de Rham cohomology. …”
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“…The reader is assumed to have taken basic graduate courses in differential geometry and algebraic topology.…”
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“…This book will be particularly useful to graduate students and researchers who have basic literacy in symplectic geometry and algebraic topology, and would like to learn how to apply standard techniques from holomorphic curve theory without dwelling more than necessary on the analytical details. …”
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“…This study uses the methods of algebraic topology to characterize time series networks constructed from fMRI data measured for adult and children populations carrying out differentiated reading tasks. …”
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“…In the following, we propose a computational model for describing the emergence of the hippocampal cognitive map in a network of transient place cell assemblies and demonstrate, using methods of algebraic topology, how such a network can maintain spatial memory over time.…”
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“…Topological Data Analysis (TDA) is an approach to analyzing the shape of data using techniques from algebraic topology. The staple of TDA is Persistent Homology (PH). …”
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“…In this regard the functor framework makes better use of naturality, allowing it to reach calculations that remain beyond the grasp of classical algebraic topology. Lastly, Antoine Touzé’s introductory course on homological algebra makes the book accessible to graduate students new to the field. …”
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“…This monograph provides a unified, self-contained introduction to the theory and its applications, making it accessible to a student who has completed a first course in algebraic topology and manifold theory. The book can be used as a source for research projects for master's students, as a thorough introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the theory of bounded cohomology and links of the theory to other closely related areas. …”
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“…These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic. …”
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“…One of the key tools in this field for analyzing the shape or topological properties of a dataset is persistent homology, an algebraic topology-based method for estimating the topological features of a space of points that persists across several resolutions. …”
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“…We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. …”
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“…We address this question using a computational framework that allows evaluating the effect produced by the decaying connections between simulated hippocampal neurons on the properties of the cognitive map. Using novel Algebraic Topology techniques, we demonstrate that emergence of stable cognitive maps produced by networks with transient architectures is a generic phenomenon. …”
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