5. Flavours of topology

2019 ◽  
pp. 90-114
Author(s):  
Richard Earl
Keyword(s):  
19Th Century ◽  
Set Theory ◽  
Algebraic Topology ◽  
General Topology ◽  
Major Theme ◽  
Henri Poincaré ◽  
Combinatorial Topology ◽  
Differential Topology ◽  
French Mathematician ◽  
Modern Mathematics

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

scholarly journals Poincaré conjecture: A problem solved after a century of new ideas and continued work

2018 ◽  
pp. 59
Author(s):  
María Teresa Lozano Imízcoz
Keyword(s):  
Fundamental Group ◽  
Algebraic Topology ◽  
Three Dimensional ◽  
Henri Poincaré ◽  
French Mathematician ◽  
New Ideas ◽  
Poincare Conjecture

The Poincaré conjecture is a topological problem established in 1904 by the French mathematician Henri Poincaré. It characterises three-dimensional spheres in a very simple way. It uses only the first invariant of algebraic topology – the fundamental group – which was also defined and studied by Poincaré. The conjecture implies that if a space does not have essential holes, then it is a sphere. This problem was directly solved between 2002 and 2003 by Grigori Perelman, and as a consequence of his demonstration of the Thurston geometrisation conjecture, which culminated in the path proposed by Richard Hamilton.

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

scholarly journals Mother Behavior to Their Daughters As Seen in ''Pride and Prejudice" and "Little Women"

2019 ◽  
Vol 2 (4) ◽  
pp. 620-625
Author(s):  
Fitri Arniati ◽  
Muhammad Darwis ◽  
Nurhayati Rahman ◽  
Fathu Rahman
Keyword(s):  
19Th Century ◽  
Major Theme ◽  
The Novel ◽  
Pride And Prejudice ◽  
Late 19Th Century ◽  
Early 19Th Century ◽  
Little Women ◽  
Women's Role ◽  
The Right

This research is to study about the mother behavior to their daughters as seen in "Pride and Prejudice" and "Little Women". The mother behavior to their daughters show the different way of women as a mother in bringing up their children according to their social and condition at the time. The data were taken from two novels entitled "pride and prejudice" and "little women" is the topic of the study. The  women held in the early 19th century and the late 19th century was described as one that belonged in the home as a wife and mother, and that should marry a man who can support their family. Also throughout the novel women's role in society was described as one that is to be accomplished in household  chore and those of entertainment, such as singing  and playing music. The role of women in society was a major theme throughout the novel "Pride and Prejudice" and "Little Women" The method used in this research  is a study of comparative literature to analyze mother behavior especially for Mrs. Bennet, Lady Catherine de Bourgh, These women have similarities and different behavior in find the right mate for their daughters. This study shows that every woman has characteristics in caring for their children and paying attention to the survival of their children.

Optimization of Utility Functions in an Admissible Space of Higher Dimension

2016 ◽  
pp. 102-113
Author(s):  
German Almanza ◽  
Victor M. Carrillo ◽  
Cely C. Ronquillo
Keyword(s):  
Graduate Students ◽  
Linear Algebra ◽  
Algebraic Topology ◽  
Morse Theory ◽  
Utility Functions ◽  
Higher Dimension ◽  
Differential Topology ◽  
Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

Review Lecture Periods of integrals and topology of algebraic varieties

1984 ◽  
Vol 391 (1801) ◽  
pp. 231-254
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Topology ◽  
Major Theme ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Algebraic Functions ◽  
Several Variables ◽  
And Topology ◽  
Satisfactory Theory

A major theme of nineteenth century mathematics was the study of integrals of algebraic functions of one variable. This culminated in Riemann’s introduction of the surfaces that bear his name and analysis of periods of integrals on cycles on the surface. The creation of a correspondingly satisfactory theory for functions of several variables had to wait on the development of algebraic topology and its application by Lefschetz to algebraic varieties. These results were refined by Hodge’s theory of harmonic integrals. A closer analysis of Hodge structures by P. A. Griffiths and P. Deligne in recent years has led to unexpectedly strong restrictions on the topology of the variety and to a diversity of other applications. This advance is closely linked to the study of variation of integrals under deformations, particularly in the neighbourhood of a singular point.

The Empty Set, The Singleton, and the Ordered Pair

2003 ◽  
Vol 9 (3) ◽  
pp. 273-298 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Mathematical Logic ◽  
19Th Century ◽  
Set Theory ◽  
Formal Semantics ◽  
Recursive Definition ◽  
Iterative Conception ◽  
Power Set ◽  
Definition Of ◽  
Ernst Zermelo ◽  
Considerable Concern

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

scholarly journals Particulate Exotica

2021 ◽  
Author(s):  
Fan Zhang
Keyword(s):  
Algebraic Topology ◽  
Complete Information ◽  
Differential Topology ◽  
Four Dimensions ◽  
Recent Advances ◽  
Obvious Reason ◽  
Microscopic World

Recent advances in differential topology single out four-dimensions as being special, allowing for vast varieties of exotic smoothness (differential) structures, distinguished by their handlebody decompositions, even as the coarser algebraic topology is fixed. Should the spacetime we reside in takes up one of the more exotic choices, and there is no obvious reason why it shouldn't, apparent pathologies would inevitably plague calculus-based physical theories assuming the standard vanilla structure, due to the non-existence of a diffeomorphism and the consequent lack of a suitable portal through which to transfer the complete information regarding the exotic physical dynamics into the vanilla theories. An obvious plausible consequence of this deficiency would be the uncertainty permeating our attempted description of the microscopic world. We tentatively argue here, that a re-inspection of the key ingredients of the phenomenological particle models, from the perspective of exotica, could possibly yield interesting insights. Our short and rudimentary discussion is qualitative and speculative, because the necessary mathematical tools have only just began to be developed.

scholarly journals Homotopy Type Theory: A Synthetic Approach to Higher Equalities

2018 ◽  
Author(s):  
Michael Shulman
Keyword(s):  
Set Theory ◽  
Homotopy Type ◽  
Type Theory ◽  
Synthetic Approach ◽  
Homotopy Type Theory ◽  
Inductive Types ◽  
Basic Ideas ◽  
Univalent Foundations ◽  
Modern Mathematics

Homotopy type theory and univalent foundations (HoTT/UF) is a new foundation of mathematics, based not on set theory but on “infinity-groupoids”, which consist of collections of objects, ways in which two objects can be equal, ways in which those ways-to-be-equal can be equal, ad infinitum. Though apparently complicated, such structures are increasingly important in mathematics. Philosophically, they are an inevitable result of the notion that whenever we form a collection of things, we must simultaneously consider when two of those things are the same. The “synthetic” nature of HoTT/UF enables a much simpler description of infinity groupoids than is available in set theory, thereby aligning with modern mathematics while placing “equality” back in the foundations of logic. This chapter will introduce the basic ideas of HoTT/UF for a philosophical audience, including Voevodsky’s univalence axiom and higher inductive types.

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