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Marius Crainic
Marius Nicolae Crainic (February 3, 1973, Aiud) is a Romanian mathematician working in the Netherlands. Education and career Born in Aiud, Romania, Crainic obtained a bachelor's degree at Babeș-Bolyai University (Cluj-Napoca) in 1995. He then moved to the Netherlands and obtained a master's degree in 1996 at Nijmegen University. He received his Ph.D. in 2000 from Utrecht University under the supervision of Ieke Moerdijk. His Ph.D. thesis is titled "''Cyclic cohomology and characteristic classes for foliations''". He was a Miller Research Fellow at the University of California, Berkeley from 2001 to 2002. He then returned to Utrecht University as a Fellow of the Royal Netherlands Academy of Arts and Sciences (KNAW). In 2007 he became an associate professor at Utrecht University, and since 2012 he is a full professor. In 2016 he was elected member of KNAW. In 2008 Crainic was awarded the André Lichnerowicz Prize in Poisson Geometry and in 2016 he received the De Bruijn Priz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aiud
Aiud (; la, Brucla, hu, Nagyenyed, Hungarian pronunciation: ; german: Straßburg am Mieresch) is a city located in Alba County, Transylvania, Romania. The city's population is 22,876. It has the status of municipality and is the 2nd-largest city in the county, after county seat Alba Iulia. The city derives its name ultimately from Saint Giles (Aegidius), to whom the first church in the settlement was dedicated when built. Administration The municipality of Aiud is made up of the city proper and of ten villages. These are divided into four urban villages and six villages which are located outside the city proper but belong to the municipality. The four urban villages are: Aiudul de Sus, Gâmbaș, Măgina and Păgida. The rural villages are: Ciumbrud (0.81 km2), Sâncrai (0.65 km2), Gârbova de Jos (1.04 km2), Țifra (0.06 km2), Gârbova de Sus (0.52 km2) and Gârbovița (0.28 km2). Demographics , the total population is 26,296 (by gender: 12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
COVID-19 Pandemic
The COVID-19 pandemic, also known as the coronavirus pandemic, is an ongoing global pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The novel virus was first identified in an outbreak in the Chinese city of Wuhan in December 2019. Attempts to contain it there failed, allowing the virus to spread to other areas of Asia and later worldwide. The World Health Organization (WHO) declared the outbreak a public health emergency of international concern on 30 January 2020, and a pandemic on 11 March 2020. As of , the pandemic had caused more than cases and confirmed deaths, making it one of the deadliest in history. COVID-19 symptoms range from undetectable to deadly, but most commonly include fever, dry cough, and fatigue. Severe illness is more likely in elderly patients and those with certain underlying medical conditions. COVID-19 transmits when people breathe in air contaminated by droplets and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1973 Births
Events January * January 1 - The United Kingdom, the Republic of Ireland and Denmark 1973 enlargement of the European Communities, enter the European Economic Community, which later becomes the European Union. * January 15 – Vietnam War: Citing progress in peace negotiations, U.S. President Richard Nixon announces the suspension of offensive action in North Vietnam. * January 17 – Ferdinand Marcos becomes President for Life of the Philippines. * January 20 – Richard Nixon is Second inauguration of Richard Nixon, sworn in for a second term as President of the United States. Nixon is the only person to have been sworn in twice as President (First inauguration of Richard Nixon, 1969, Second inauguration of Richard Nixon, 1973) and Vice President of the United States (First inauguration of Dwight D. Eisenhower, 1953, Second inauguration of Dwight D. Eisenhower, 1957). * January 22 ** George Foreman defeats Joe Frazier to win the heavyweight world boxing championship. ** A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Up To Homotopy
A representation up to homotopy has several meanings. One of the earliest appeared in physics, in constrained Hamiltonian systems. The essential idea is lifting a non-representation on a quotient to a representation up to strong homotopy on a resolution of the quotient. As a concept in differential geometry, it generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. As such, it was introduced by Abad and Crainic. As a motivation consider a regular Lie algebroid (''A'',''ρ'', ,. (regular meaning that the anchor ''ρ'' has constant rank) where we have two natural ''A''- connections on ''g''(''A'') = ker ''ρ'' and ''ν''(''A'')= ''TM''/im ''ρ'' respectively: :\nabla\colon \Gamma(A)\times\Gamma(\mathfrak(A))\to\Gamma(\mathfrak(A)): \nabla_\psi:= phi,\psi :\nabla\colon \Gamma(A)\times\Gamma(\nu(A))\to\Gamma(\nu(A)): \nabla_\overline:=\overline. In the deformation theory of the Lie algebroid ''A'' there is a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rui Loja Fernandes
Rui António Loja Fernandes (July 20, 1965, Coimbra) is a Portuguese mathematician working in the USA. Education and career Fernandes obtained a bachelor's degree in Physics Engineering at Instituto Superior Técnico (Lisbon, Portugal) in 1988. He then moved to the USA and earned a master's degree in Mathematics in 1991 and a PhD in Mathematics in 1994 from the University of Minnesota. His PhD thesis was entitled "''Completely Integrable bi-Hamiltonian Systems"'' and has been written under the supervision of Peter J. Olver. In 1994 he returned to Instituto Superior Técnico, where he worked first as Assistant Professor (1994-2002), and then as Associated Professor (2003-2007) and Full Professor (2007-2012). In 2012 he moved back to the USA and since then he is the Lois M. Lackner Professor of Mathematics at University of Illinois at Urbana–Champaign. In 2016, he became a Fellow of the American Mathematical Society "for contributions to the study of Poisson geometry and Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudogroup
In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s. Definition A pseudogroup imposes several conditions on a sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets ''U'' of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms and compose to a homeomorphism from ''U'' to ''W'', one asks that the pseudogroup is closed under composition and inversion. However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the " noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groupoid
In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :s,t : \operatorname \to \operatorname are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name ''differentiable groupoids''. Definition and basic concepts A Lie groupoid consists of * two smooth ma ... [...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 differential form, closed, nondegenerate form, nondegenerate differential form, 2-form. Symplectic geometry has its origins in the Hamiltonian mechanics, 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 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 wiktionary:Reconstruction:Proto-Indo-European/pleḱ-, *pleḱ- The name reflects the deep connections between complex and sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation Theory
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedded subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class ''Cr''), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dimension of the leaves) is called the dimension of the foliation and is called its codimension. In some papers on general relativity by mathematical physicists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data. Lie theory has been particularly useful in mathematical physics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
