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Mark Gross (mathematician)
Mark William Gross (born 30 November 1965)Anon (2017) is an American mathematician, specializing in differential geometry, algebraic geometry, and mirror symmetry. Education Gross studied from 1982 at Cornell University graduating with a bachelor's degree in 1984 and received in 1990 a PhD from the University of California, Berkeley for research supervised by Robin Hartshorne with a thesis on the ''Surfaces in the Four-Dimensional Grassmannian''. Career From 1990 to 1993 he was an assistant professor at the University of Michigan and spent the academic year 1992–1993 on leave as a postdoctoral researcher at the Mathematical Sciences Research Institute (MSRI) in Berkeley. He was at Cornell University in 1993–1997 an assistant professor and in 1997–2001 an associate professor and then at University of California, San Diego in 2001–2013 a full professor. He was a visiting professor at the University of Warwick in the academic year 2002–2003. Since 2013, he has been a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, education and public engagement and fostering international and global co-operation. Founded on 28 November 1660, it was granted a royal charter by King Charles II as The Royal Society and is the oldest continuously existing scientific academy in the world. The society is governed by its Council, which is chaired by the Society's President, according to a set of statutes and standing orders. The members of Council and the President are elected from and by its Fellows, the basic members of the society, who are themselves elected by existing Fellows. , there are about 1,700 fellows, allowed to use the postnominal title FRS ( Fellow of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Algebra
Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy various conditions. Definitions Suppose that ''F'' is an integral domain, such as the field Q(''x''1,...,''x''''n'') of rational functions in ''n'' variables over the rational numbers Q. A cluster of rank ''n'' consists of a set of ''n'' elements of ''F'', usually assumed to be an algebraically independent set of generators of a field extension ''F''. A seed consists of a cluster of ''F'', together with an exchange matrix ''B'' with integer entries ''b''''x'',''y'' indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be skew-symmetric, so that ''b''''x'',''y'' = –''b''''y'',''x'' for all ''x'' and ''y''. More generally the matrix might be skew-symmetrizable, meaning there are positive integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov–Witten Invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important. Definition Consider the following: *''X'': a closed symplectic manifold of dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Geometry
In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others. Motivation The idea is to study some algebraic variety (or scheme) ''U'' which is smooth but not necessarily proper by embedding it into ''X'', which is proper, and then looking at certain sheaves on ''X''. The problem is that the subsheaf of \mathcal_X consisting of functions whose restriction to ''U'' is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of \mathcal_X , multiplicatively. Remembering this additional structure on ''X'' corresponds to remembering the inclusion j \colon U \to X , which likens ''X'' with this extra structure to a varie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Archimedean Geometry
In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry. There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property (i.e. with respect to order or magnitude). Geometry over a non-Archimedean ordered field The first sense of the term is the geometry over a non-Archimedean ordered field, or a subset thereof. The aforementioned Dehn plane takes the self-product of the finite portion of a certain non-Archimedean ordered field based on the field of rational functions. In this geometry, there are significant differences from Euclidean geometry; in particular, there are infinitely many parallels to a straight line through a point—so the parallel postula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So for example, the classical polynomial x^3 + 2xy + y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi–Yau Manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after who first conjectured that such surfaces might exist, and who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. Definitions The motivational definition given by Shing-Tung Yau is of a compact K� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Zaslow
Eric Zaslow is an American mathematical physicist at Northwestern University. Biography Zaslow attended Harvard University, earning his Ph.D. in physics in 1995, with thesis "Kinks, twists, and folds : exploring the geometric musculature of quantum field theory" written under the direction of Cumrun Vafa. His research focuses on mathematical questions arising from duality symmetries in theoretical physics such as mirror symmetry. With Andrew Strominger and Shing-Tung Yau, he formulated the SYZ conjecture. He was named to the 2021 class of fellows of the American Mathematical Society "for contributions to mathematical physics and mirror symmetry". Zaslow is also known for being internationally ranked in ultimate Ultimate or Ultimates may refer to: Arts, entertainment, and media Music Albums * ''Ultimate'' (Jolin Tsai album) * ''Ultimate'' (Pet Shop Boys album) *''Ultimate!'', an album by The Yardbirds *''The Ultimate (Bryan Adams Album)'', a compilatio ..., with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
