호몰로지 - 편집 역사

2021년 2월 17일 (수) 10:51에 Pythagoras0님의 편집

2021-02-17T10:51:55Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T14:25:05Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2020-12-21T15:20:30Z

노트: 새 문단

2013년 6월 5일 (수) 08:31에 Pythagoras0님의 편집

2013-06-05T08:31:40Z

2013년 6월 3일 (월) 19:43에 Pythagoras0님의 편집

2013-06-03T19:43:14Z

Pythagoras0: /* 관련논문 */

2013-05-30T10:27:44Z

2013년 5월 27일 (월) 16:35에 Pythagoras0님의 편집

2013-05-27T16:35:58Z

2013년 5월 27일 (월) 07:56에 Pythagoras0님의 편집

2013-05-27T07:56:37Z

Pythagoras0: /* 관련된 항목들 */

2013-01-29T10:48:22Z

Pythagoras0: /* 역사 */

2013-01-29T10:47:54Z

역사

@@ 116번째 줄: / 116번째 줄: @@
   <references />
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q1144780 Q1144780]

← 이전 판		2020년 12월 28일 (월) 14:25 판
115번째 줄:		115번째 줄:
	===소스===		===소스===
	<references />		<references />
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1144780 Q1144780]

← 이전 판		2020년 12월 21일 (월) 15:20 판
47번째 줄:		47번째 줄:
	* Weibel, Charles A. 1999. [http://www.math.uiuc.edu/K-theory/0245/ History of Homological Algebra] History of Topology: 797–836.		* Weibel, Charles A. 1999. [http://www.math.uiuc.edu/K-theory/0245/ History of Homological Algebra] History of Topology: 797–836.
	* Hilton, Peter [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century] (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291		* Hilton, Peter [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century] (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291
		+
		+	== 노트 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1144780 Q1144780]
		+	===말뭉치===
		+	# Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf.<ref name="ref_ad36c7c4">[https://encyclopediaofmath.org/wiki/Homology_theory Encyclopedia of Mathematics]</ref>
		+	# Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory.<ref name="ref_ad36c7c4" />
		+	# There exists a cohomology theory dual to a homology theory (cf.<ref name="ref_ad36c7c4" />
		+	# The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms.<ref name="ref_ad36c7c4" />
		+	# By analysis of the lifting problem it introduces the funda mental group and explores its properties, including Van Kampen's Theorem and the relationship with the first homology group.<ref name="ref_2661a95e">[https://www.springer.com/gp/book/9780387941264 Homology Theory - An Introduction to Algebraic Topology]</ref>
		+	# Conceptually, however, it can be useful to understand homology as a special kind of homotopy.<ref name="ref_8773a6e8">[https://ncatlab.org/nlab/show/homology homology in nLab]</ref>
		+	# This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts.<ref name="ref_8773a6e8" />
		+	# One good way of understanding homology of CW complexes is with cellular homology.<ref name="ref_5f73e239">[https://math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups Intuition of the meaning of homology groups]</ref>
		+	# Homology groups were originally defined in algebraic topology .<ref name="ref_4f6d6ab8">[https://en.wikipedia.org/wiki/Homology_(mathematics) Homology (mathematics)]</ref>
		+	# The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes.<ref name="ref_4f6d6ab8" />
		+	# Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold .<ref name="ref_4f6d6ab8" />
		+	# A particular type of mathematical object, such as a topological space or a group , may have one or more associated homology theories.<ref name="ref_4f6d6ab8" />
		+	# Historically, the term "homology" was first used in a topological sense by Poincaré.<ref name="ref_53ee2d6c">[https://mathworld.wolfram.com/Homology.html Homology -- from Wolfram MathWorld]</ref>
		+	# To him, it meant pretty much what is now called a bordism, meaning that a homology was thought of as a relation between manifolds mapped into a manifold.<ref name="ref_53ee2d6c" />
		+	# To simplify the definition of homology, Poincaré simplified the spaces he dealt with.<ref name="ref_53ee2d6c" />
		+	# Eventually, Poincaré's version of homology was dispensed with and replaced by the more general singular homology.<ref name="ref_53ee2d6c" />
		+	# The starting point will be simplicial complexes and simplicial homology.<ref name="ref_c378c1ed">[https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/year3/ma3h6/ MA3H6 Algebraic Topology]</ref>
		+	# The simplicial homology depends on the way these simplices fit together to form the given space.<ref name="ref_c378c1ed" />
		+	# It is not that hard to prove that singular homology is a homotopy invariant but very hard to compute singular homology directly from the definition.<ref name="ref_c378c1ed" />
		+	# One of the main results in the module will be the proof that simplicial homology and singular homology agree for simplicial complexes.<ref name="ref_c378c1ed" />
		+	# We can think of homology as a perspective that allows one to distinguish shapes and forms disregarding changes made by stretching and bending.<ref name="ref_c6801dd4">[https://www.maths.ox.ac.uk/about-us/life-oxford-mathematics/oxford-mathematics-alphabet/h-homology Mathematical Institute]</ref>
		+	# For example, homology sees no difference between the beach ball and the jumping ball, which can be constructed from the ball shape by pulling out two fingers.<ref name="ref_c6801dd4" />
		+	# This is the reason why according to homology those objects are not the same.<ref name="ref_c6801dd4" />
		+	# Homology is a mathematical way of counting different types of loops and holes in topological spaces.<ref name="ref_c6801dd4" />
		+	# Through analysis of viral and simulated genomic datasets, we show how persistent homology captures fundamental evolutionary aspects not directly inferred from phylogeny.<ref name="ref_a24f2602">[https://www.pnas.org/content/110/46/18566 Topology of viral evolution]</ref>
		+	# We can define a topological invariant called the “homology group” H k as an algebraic structure that encompasses all holes in dimension k, and the “Betti number” b k is the count of these holes.<ref name="ref_a24f2602" />
		+	# C. We then perform persistent homology, which computes the homology groups of dimension k at all scales ε.<ref name="ref_a24f2602" />
		+	# Our aim, then, is to apply persistent homology to the study of evolution.<ref name="ref_a24f2602" />
		+	# Exercise Show that E(V) has zero homology.<ref name="ref_124a4c9e">[https://www.win.tue.nl/~aeb/at/algtop-6.html Algebraic Topology: Homology]</ref>
		+	# For each cover we obtain a homology group.<ref name="ref_124a4c9e" />
		+	# This leads to a homomorphism of homology groups.<ref name="ref_124a4c9e" />
		+	# Let us look at .Cech homology again.<ref name="ref_124a4c9e" />
		+	# In this section, we categorize the persistent homology enabled applications as single graph and multiple graph analysis.<ref name="ref_8fadc1af">[https://appliednetsci.springeropen.com/articles/10.1007/s41109-019-0179-3 Persistence homology of networks: methods and applications]</ref>
		+	# In some applications, persistent homology is used to detect global structural features of a single network such as complexity and distributions of strongly connected regions.<ref name="ref_8fadc1af" />
		+	# (2018) use persistent homology to detect clique communities and their evolution in weighted networks.<ref name="ref_8fadc1af" />
		+	# Persistent homology is also used to analyze the brain networks by computing distributions of cliques (brain regions) and cycles (strongly connected regions) in them.<ref name="ref_8fadc1af" />
		+	# If I is an ideal of R, he considers the homology of the kernel of F * → F * /I and shows that it is independent of the choice of resolution.<ref name="ref_ffe3293e">[https://www.sciencedirect.com/topics/mathematics/homology-group Homology Group - an overview]</ref>
		+	# At first, calculations of group homology were restricted to those groups π which were fundamental groups of familiar topological spaces, using the bar complex.<ref name="ref_ffe3293e" />
		+	# By measuring leaves from throughout the seed plants using persistent homology, a defined morphospace comparing all leaves is demarcated.<ref name="ref_5cf3f9f5">[https://www.frontiersin.org/articles/10.3389/fpls.2018.00553/full Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace]</ref>
		+	# Landmark analysis excels in its interpretability, because each landmark is an identifiable feature with biological meaning imparted by the shared homology between samples.<ref name="ref_5cf3f9f5" />
		+	# Here, we present a morphometric technique based on topology, using a persistent homology framework, to measure the outlines of leaves and classify them by plant family.<ref name="ref_5cf3f9f5" />
		+	# Using persistent homology, we then use a linear discriminant analysis (LDA) to classify leaves by plant family.<ref name="ref_5cf3f9f5" />
		+	# The theory has applications in many branches of mathematics, including spectral theory, the theory of de Rham homology in differential geometry, automatic continuity theory and K-theory.<ref name="ref_0dfe9e8f">[https://www.ncl.ac.uk/maths-physics/research/pure/topological-homology/ Mathematics, Statistics and Physics, School of]</ref>
		+	# Historically, the term ``homology'' was first used in a topological sense by Poincaré .<ref name="ref_985a0d8e">[https://archive.lib.msu.edu/crcmath/math/math/h/h342.htm Homology (Topology)]</ref>
		+	# To him, it meant pretty much what is now called a Cobordism, meaning that a homology was thought of as a relation between Manifolds mapped into a Manifold.<ref name="ref_985a0d8e" />
		+	# Eventually, Poincaré's version of homology was dispensed with and replaced by the more general Singular Homology.<ref name="ref_985a0d8e" />
		+	# In modern usage, however, the word homology is used to mean Homology Group.<ref name="ref_985a0d8e" />
		+	# This workshop will explore topics of current interest in the theory of Floer homology for 3-manifolds.<ref name="ref_3da8e5f0">[http://scgp.stonybrook.edu/archives/28596 Floer homology in low-dimensional topology: January 11-15, 2021]</ref>
		+	# Floer homology is a powerful tool for studying the topology of 3- and 4-dimensional manifolds, and the relations between them.<ref name="ref_3da8e5f0" />
		+	# There are a wide variety of ways to define the Floer homology of a 3-manifold.<ref name="ref_3da8e5f0" />
		+	# Many of these are now known to be equivalent, but their relation to the original instanton homology remains mysterious.<ref name="ref_3da8e5f0" />
		+	# In the Theory section, we have shown that the number of critical simplices determines the effective number of filtration weights to study the persistent homology of a clique complex (See Eq. 10).<ref name="ref_d8d23b01">[https://www.nature.com/articles/s41598-019-50202-3 Persistent homology of unweighted complex networks via discrete Morse theory]</ref>
		+	# Rather, our main goal is to develop a systematic filtration scheme to study persistent homology in unweighted and undirected networks.<ref name="ref_d8d23b01" />
		+	# A visual inspection of the barcode diagrams for the five model networks (Figs 3 and 5 and SI Figs S1–S4) suggests that the different models can be distinguished based on their persistent homology.<ref name="ref_d8d23b01" />
		+	# The last observation is a reflection of the differences in the persistent homology of the clique complexes corresponding to spherical and hyperbolic networks.<ref name="ref_d8d23b01" />
		+	# In this paper, we propose to use the persistent homology to systematically study the osmolytes’ molecular aggregation and their hydrogen-bonding network from a global topological perspective.<ref name="ref_b42e2c02">[https://pubs.rsc.org/en/content/articlelanding/2019/cp/c9cp03009c Persistent homology analysis of osmolyte molecular aggregation and their hydrogen-bonding networks]</ref>
		+	# The -th homology group of a simplicial complex , denoted , is the quotient vector space .<ref name="ref_0711f2e7">[https://jeremykun.com/2013/04/03/homology-theory-a-primer/ Homology Theory — A Primer]</ref>
		+	# Therefore, computing homology generalizes the graph-theoretic methods of computing connected components.<ref name="ref_0711f2e7" />
		+	# The quotient construction imposes that two vertices (two elements of the homology group) are considered equivalent if their difference is a boundary.<ref name="ref_0711f2e7" />
		+	# It is easy to see that (indeed, just by the first four generators of the image) all vertices are equivalent to 0, so there is a unique generator of homology, and the vector space is isomorphic to .<ref name="ref_0711f2e7" />
		+	===소스===
		+	<references />

@@ 2번째 줄: / 2번째 줄: @@
 * [[단체 호몰로지 (simplicial homology)]]
 * 특이 호몰로지 (singular homology)
-* [[드람 코호몰로지]]
+* <math>\langle S,d\omega\rangle=\langle \partial S,\omega \rangle</math>
-* Eilenberg-Steenrod 공리
@@ 43번째 줄: / 45번째 줄: @@
 ==관련논문==
 * Hilton, Peter [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century] (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
-* simplicial 호몰로지(코호몰로지)
+* [[단체 호몰로지 (simplicial homology)]]
-* singular 호몰로지
+* 특이 호몰로지 (singular homology)
 * [[드람 코호몰로지]]
 * Eilenberg-Steenrod 공리
@@ 39번째 줄: / 39번째 줄: @@
 ==사전 형태의 자료==
-* http://en.wikipedia.org/wiki/Simplicial_homology
+* http://en.wikipedia.org/wiki/Homology_theory
 ==관련논문==
+* Hilton, Peter [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century] (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291

← 이전 판		2013년 5월 30일 (목) 10:27 판
47번째 줄:		47번째 줄:
	* [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century]<br>		* [http://www.jstor.org/stable/2689545 A Brief, Subjective History of Homology and Homotopy Theory in This Century]<br>
	** Hilton, Peter (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291		** Hilton, Peter (1988), Mathematics Magazine (Mathematical Association of America) 60 (5): 282-291
−	* http://www.jstor.org/action/doBasicSearch?Query=
−	* http://www.ams.org/mathscinet
−	* http://dx.doi.org/

← 이전 판		2013년 5월 27일 (월) 16:35 판
33번째 줄:		33번째 줄:
	* [[스미스 표준형 (Smith normal form)]]		* [[스미스 표준형 (Smith normal form)]]

		+
		+	==계산 리소스==
		+	* [http://page.math.tu-berlin.de/~lutz/stellar/ The Manifold Page]

@@ 18번째 줄: / 18번째 줄: @@
 * 1920년대 Veblen, Alexander, Lefschetz
 * [[수학사 연표]]
 ==메모==
@@ 30번째 줄: / 31번째 줄: @@
 * [[드람 코호몰로지]]
 * [[스토크스 정리]]
 ==사전 형태의 자료==

@@ 25번째 줄: / 25번째 줄: @@
 ==관련된 항목들==
 * [[대수적위상수학]]
 * [[다면체에 대한 오일러의 정리 V-E+F=2]]

@@ 9번째 줄: / 9번째 줄: @@
 ==역사==
 * 푸앵카레
 * 브라우어
 * 1920년대 Veblen, Alexander, Lefschetz
 * [[수학사 연표]]
 ==메모==