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...of [[topology|topological]] and [[algebraic topology|algebro-topological]] methods to solving problems in [[combinatorics]]. ...in topology and in the early 20th century this turned into the field of [[algebraic topology]]. ...

...Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry | subject = [[Topological combinatorics]] ...

...structure. They have application in the [[cobordism|cobordism ring]] in [[algebraic topology]]. is an [[endomorphism]] of the multiplicative [[monoid]] <math>(A[x_1, x_2,\cdots][[z]],\cdot)</math>, ...

...ing values on ''F''. The theorem states that, if the first [[Chern class]] of ''F'' is negative,<math display="block">H^q(M; \Omega^p(F)) = 0 \text{ when Alternatively, if the first Chern class of ''F'' is positive,<math display="block">H^q(M; \Omega^p(F)) = 0 \text{ when ...

...e natural objects to study in [[algebraic geometry]]. The following notion of ''finite vector bundle'' is due to [[André Weil]] and will be needed to def ...bundles. <ref name=":BV">N. Borne, A. Vistoli ''The Nori fundamental gerbe of a fibered category'', J. Algebr. Geom. 24, No. 2, 311-353 (2015)</ref> ...

{{Short description|Type of vector bundle}} ...ble vector bundles contains the [[Tannakian formalism|Tannakian category]] of [[essentially finite vector bundle]]s, whose naturally associated group sch ...

...le=Erratum for "Heights of vector bundles and the fundamental group scheme of a curve" |year=2020 |last1=Antei |first1=Marco |last2=Emsalem |first2=Miche ...ion of the existence of <math>\pi_1(X,x)</math> defined on regular schemes of dimension 1 had to wait about forty more years. There are various generaliz ...

...specifically in [[differential geometry]] and [[topology]], various types of [[Function (mathematics)|functions]] between [[manifold]]s are studied, bot == Types of maps == ...

...the geometric properties of formal duals of [[non-commutative]] [[algebra|algebraic object]]s such as [[ring (mathematics)|rings]] as well as geometric objects ...is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric in ...

{{Short description|Conjecture in algebraic geometry}} ...ure about [[intersection number]]s of stable classes on the [[moduli space of curves]], introduced by [[Edward Witten]] in the paper {{harvs|txt|last=Wit ...

...multiple of four.<ref>{{Harvnb|Atiyah|Bott|1967}}</ref> It is an instance of a Dirac-type operator. Let <math>M</math> be a compact [[Riemannian manifold]] of even dimension <math>2l</math>. Let ...

...ve structure on the [[tangent space]]. The concept generalizes the notion of [[Frobenius algebra]] to tangent bundles. ...ically [[quantum cohomology]]. The broadest definition is in the category of Riemannian [[supermanifold]]s. We will limit the discussion here to smooth ...

{{short description|List of concrete topologies and topological spaces}} {{Main|List of topology topics}} ...

...logue/catalogue.asp?isbn=9780521760362 Directed Algebraic Topology: Models of Non-Reversible Worlds], Marco Grandis, Cambridge University Press, {{ISBN|9 ...t the execution traces of the program. From the [[computer science]] point of view, however, the resulting pospaces have a severe drawback. Because parti ...

{{Short description|Concept in algebraic geometry}} ...ng why they do not exist. It also provides invariants to distinguish one [[algebraic variety]] from another. ...

...uclidean space]] given by the set of points with [[Cartesian coordinate]]s of the form ...three-in-line problem]], and a geometric proof of the [[chromatic number]] of [[Kneser graph]]s. ...

{{Short description|Moduli spaces of ramified covers}} ...is a [[symmetric group]] and the monodromy classes are the conjugacy class of [[Transposition (mathematics)|transpositions]]). ...

{{Short description|Generalization of a complex manifold that allows the use of singularities}} In [[mathematics]], particularly [[differential geometry]] and [[complex geometry]], a '''complex analytic variety'''<ref group="note">Complex analytic varie ...

{{Short description|Multiscale analog of homology group}} ...st1=Peter Tsung-Wen |last2=Cheong |first2=Siew Ann |date=2021 |title=Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in ...

...2=Wright |first2=Matthew |date=2015-12-01 |title=Interactive Visualization of 2-D Persistence Modules |class=math.AT |eprint=1512.00180 }}</ref> ...y to do this is by considering the sequence of [[Vietoris–Rips complex]]es of a finite set in a [[metric space]] indexed over all scale parameters. ...