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Thom Conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has genus given by the genus–degree formula :g = (d-1)(d-2)/2. The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology as C, then the genus g of \Sigma satisfies the inequality :g \geq (d-1)(d-2)/2. In particular, ''C'' is known as a ''genus minimizing representative'' of its homology class. It was first proved by Peter Kronheimer and Tomasz Mrowka in October 1994, using the then-new Seiberg–Witten invariants. Assuming that \Sigma has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Witten invariants. There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoltán Szabó (mathematician)
Zoltán Szabó (born November 24, 1965) is a professor of mathematics at Princeton University known for his work on Heegaard Floer homology. Education and career Szabó received his BA from Eötvös Loránd University in Budapest, Hungary in 1990, and he received his PhD from Rutgers University in 1994. Together with Peter Ozsváth, Szabó created Heegaard Floer homology, a homology theory for 3-manifolds. For this contribution to the field of topology, Ozsváth and Szabó were awarded the 2007 Oswald Veblen Prize in Geometry. In 2010, he was elected honorary member of the Hungarian Academy of Sciences The Hungarian Academy of Sciences ( , MTA) is Hungary’s foremost and most prestigious learned society. Its headquarters are located along the banks of the Danube in Budapest, between Széchenyi rakpart and Akadémia utca. The Academy's primar .... Selected publications *. *.Homology for Knots and Links'' American Mathematical Society, (2015) References External links ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surfaces
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, but, in the Italian school of algebraic geometry , and are up to 100 years old. Classification by the Kodaira dimension In the case of dimension one, varieties are classified by only the topological genus, but, in dimension two, one needs to distinguish the arithmetic genus p_a and the geometric genus p_g because one cannot distinguish birationally only the topological genus. Then, irregularity is introduced for the classification of varieties. A summary of the results (in detail, for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-manifolds
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-dimensional Geometry
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'', to describe the sizes or locations of objects in the everyday world. This concept of ordinary space is called Euclidean space because it corresponds to Euclid 's geometry, which was originally abstracted from the spatial experiences of everyday life. Single locations in Euclidean 4D space can be given as vectors or '' 4-tuples'', i.e., as ordered lists of numbers such as . For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled , , and ). It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D spaces emerge. A hint of that complexity can be seen in the accompanying 2D animation of one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Conjecture (knot Theory)
In knot theory, the Milnor conjecture says that the slice genus of the (p, q) torus knot is :(p-1)(q-1)/2. It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka. Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov. Overv ..., by means of the s-invariant.. References Geometric topology Knot theory 4-manifolds Conjectures that have been proved {{knottheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction. Adjunction for smooth varieties Formula for a smooth subvariety Let ''X'' be a smooth algebraic variety or smooth complex manifold and ''Y'' be a smooth subvariety of ''X''. Denote the inclusion map by ''i'' and the ideal sheaf of ''Y'' in ''X'' by \mathcal. The conormal exact sequence for ''i'' is :0 \to \mathcal/\mathcal^2 \to i^*\Omega_X \to \Omega_Y \to 0, where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism :\omega_Y = i^*\omega_X \otimes \operatorname(\mathcal/\mathcal^2)^\vee, where \vee denotes the dual of a line bundle. The particular case of a smooth divisor Suppose that ''D'' is a smooth div ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Ozsváth
Peter Steven Ozsváth (born October 20, 1967) is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds. Education Ozsváth received his PhD from Princeton in 1994 under the supervision of John Morgan; his dissertation was entitled ''On Blowup Formulas For SU(2) Donaldson Polynomials''. Awards In 2007, Ozsváth was one of the recipients of the Oswald Veblen Prize in Geometry. In 2008 he was named a Guggenheim Fellow. In July 2017, he was a plenary lecturer in the Mathematical Congress of the Americas. He was elected a member of the National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ... in 2018. Selected publications * *Homology for Knots and Links'' America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Hermann Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Taubes
Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taubes. Early career Taubes received his B.A. from Cornell University in 1975 and his Ph.D. in physics in 1980 from Harvard University under the direction of Arthur Jaffe, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex equations and the Bogomol'nyi monopole equations. Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space of solutions to the Yang-Mills equations was used by Simon Donaldson in his proof of Donaldson's theorem on diagonizability of intersection forms. He proved in that R4 has an uncountable number of smooth structures (see also exotic R4), and (with Raoul Bott in ) proved Witten's rigidity theorem on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
