Journal of Homotopy and Related Structures
Scope & Guideline
Charting New Territories in Mathematical Relationships
Introduction
Aims and Scopes
- Homotopy Theory and Algebraic Topology:
The journal focuses extensively on homotopy theory, examining its fundamental principles, structures, and applications in algebraic topology. This includes studies on homotopy types, homological algebra, and the relationships between different topological spaces. - Higher Category Theory:
A significant area of interest is higher category theory, where researchers explore (∞,1)-categories, their properties, and their implications for homotopy theory. This includes the study of simplicial sets, model categories, and enriched category theory. - Algebraic Structures and Computations:
The journal also covers research on various algebraic structures, such as L∞-algebras, associative algebras, and K-theory, with an emphasis on computational methods and theoretical advancements in these areas. - Geometric and Topological Applications:
Papers often explore geometric aspects of homotopy theory, including applications to diffeology, smooth manifolds, and foliations, highlighting the connections between abstract theory and geometric intuition. - Interdisciplinary Connections:
The journal encourages research that bridges homotopy theory with other mathematical fields, such as representation theory, algebraic geometry, and mathematical physics, fostering a rich dialogue between these disciplines.
Trending and Emerging
- Diffeology and Smooth Structures:
There is a growing interest in diffeological spaces and their applications in smooth geometry and homotopy theory. This theme reflects an increased focus on understanding smooth structures in a more flexible and generalized context. - Twisted Cohomology and Homotopy Theory:
Emerging research on twisted cohomology theories and their connections to homotopy theory suggests a trend towards exploring more nuanced algebraic structures that can capture complex phenomena in topology and algebra. - Infinity-Categories and Higher Algebra:
The exploration of (∞,1)-categories and higher algebra is gaining traction, indicating a shift towards more sophisticated categorical frameworks that provide deeper insights into homotopy theory and its applications. - Applications in Mathematical Physics:
There is an increasing trend of papers connecting homotopy theory with mathematical physics, particularly in the context of topological field theories and string theory, highlighting the interdisciplinary nature of current research. - Computational Homotopy Theory:
Advances in computational techniques for homotopy theory, including the use of spectral sequences and computational homological algebra, are becoming more prominent, reflecting a growing demand for practical applications of theoretical insights.
Declining or Waning
- Classical Algebraic Topology:
Research focused solely on classical algebraic topology, such as basic homology and cohomology theories, appears to be waning. This may be due to the increasing complexity and sophistication of newer theories that integrate algebraic topology with homotopy theory. - Basic Model Categories:
Papers discussing foundational model category theory without the integration of higher categorical perspectives are less frequent. The field seems to be moving towards more advanced frameworks that incorporate higher-dimensional categories. - Elementary Homotopy Concepts:
There is a noticeable decrease in publications centered on elementary homotopy concepts. Researchers are increasingly exploring more complex structures and their interrelations, leading to a decline in interest in basic homotopy properties. - Traditional Representation Theory:
Traditional approaches to representation theory, which do not leverage modern homotopical methods, have seen reduced attention. The integration of homotopy theory with representation theory is becoming more prominent, overshadowing classical studies.
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