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37-494: The school opened in the fall of 1866 as the first teaching institution in the city of Pitești and evolved throughout the years, from the original primary school to a secondary 8 grades school, then to a 12 grades one, and finally became a high-school (9–12 grades) in 1965. Today, "Brătianu" is recognized as one of the best high-schools in Romania. Because of its remarkable selectivity, the high admission percentage of its graduates,

74-402: A tame knot K is the three-dimensional space surrounding the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the 3-sphere ). Let N be a tubular neighborhood of K ; so N is a solid torus . The knot complement is then the complement of N , A related topic is braid theory . Braid theory is an abstract geometric theory studying

111-501: A 'marking' is an isotopy class of homeomorphisms from X to X . The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space. Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it was introduced by Oswald Teichmüller  ( 1940 ). In mathematics ,

148-470: A circle isn't bound to the classical geometric concept, but to all of its homeomorphisms ). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R upon itself (known as an ambient isotopy ); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. Knot complements are frequently-studied 3-manifolds. The knot complement of

185-481: A fixed weekly schedule that is repeated throughout the two semesters of the academic year. They are organized in groups of approximately 30 students and each of these groups has a certain classroom assigned to it. Students attend six – 50 minutes classes daily. AP courses or Honors classes are mainly unknown to the Romanian system but their alternative is found in all highschools: special class profiles. Therefore, although

222-463: A proof of the three-dimensional Poincaré conjecture, using Richard S. Hamilton 's Ricci flow , an idea belonging to the field of geometric analysis . Overall, this progress has led to better integration of the field into the rest of mathematics. A surface is a two-dimensional , topological manifold . The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R —for example,

259-495: A thin part consisting of tubular neighborhoods of closed geodesics and/or ends that are the product of a Euclidean surface and the closed half-ray. The manifold is of finite volume if and only if its thick part is compact. In this case, the ends are of the form torus cross the closed half-ray and are called cusps . Knot complements are the most commonly studied cusped manifolds. Thurston's geometrization conjecture states that certain three-dimensional topological spaces each have

296-542: A trivial tangent bundle . Stated another way, the only characteristic class of a 3-manifold is the obstruction to orientability. Any closed 3-manifold is the boundary of a 4-manifold. This theorem is due independently to several people: it follows from the Dehn – Lickorish theorem via a Heegaard splitting of the 3-manifold. It also follows from René Thom 's computation of the cobordism ring of closed manifolds. The existence of exotic smooth structures on R . This

333-401: A unique geometric structure that can be associated with them. It is an analogue of the uniformization theorem for two-dimensional surfaces , which states that every simply-connected Riemann surface can be given one of three geometries ( Euclidean , spherical , or hyperbolic ). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead,

370-529: 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 that admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds that are homeomorphic but not diffeomorphic ). 4-manifolds are of importance in physics because, in General Relativity , spacetime

407-466: Is a generalization of the Riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. A topological space X is a 3-manifold if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space . The topological, piecewise-linear , and smooth categories are all equivalent in three dimensions, so little distinction

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444-503: Is centered on constructing new versions of Floer homology and applying them to questions in topology. With collaborators, he showed that many Floer-theoretic invariants are algorithmically computable. He also developed a new variant of Seiberg-Witten Floer homology, which he used to prove the existence of non-triangulable manifolds in high dimensions." He has one of the best records ever in mathematical competitions: Low-dimensional topology In mathematics , low-dimensional topology

481-432: Is considered a part of low-dimensional topology or geometric topology . Knot theory is the study of mathematical knots . While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space , R (since we're using topology,

518-468: Is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 2 g . The surfaces in the third family are nonorientable. The Euler characteristic of

555-537: Is diffeomorphic to R . There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in four dimensions. Here are some examples: There are several theorems that in effect state that many of the most basic tools used to study high-dimensional manifolds do not apply to low-dimensional manifolds, such as: Steenrod's theorem states that an orientable 3-manifold has

592-640: Is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as knot theory , geometric group theory , hyperbolic geometry , number theory , Teichmüller theory , topological quantum field theory , gauge theory , Floer homology , and partial differential equations . 3-manifold theory

629-524: Is modeled as a pseudo-Riemannian 4-manifold. An exotic R is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R . The first examples were found in the early 1980s by Michael Freedman , by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson 's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of R , as

666-459: Is not offered but each group of students calculates its own rank and awards diplomas for the highest 3 final grades, and at the end of the 12th grade the school awards the distinction of Chief of graduates to the student with the highest final grade over the four years. The examination all Romanian students take when they graduate the high school studies is the Baccalaureate exam which consists of

703-480: Is the branch of topology that studies manifolds , or more generally topological spaces, of four or fewer dimensions . Representative topics are the structure theory of 3-manifolds and 4-manifolds , knot theory , and braid groups . This can be regarded as a part of geometric topology . It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory . A number of advances starting in

740-404: Is the graduating mark. At the end of each semester and at the end of the year, the final grade is obtained through the arithmetic mean of the grades received throughout the year. Where term papers apply (Romanian, Mathematics, Physics, Informatics), this arithmetic mean is multiplied by 3 added with the term paper’s result and divided by 4 to obtain the final grade. An American style ranking system

777-608: The E. H. Moore Prize from the American Mathematical Society . He was among the recipients of the Clay Research Fellowship (2004–2008). In 2012, he was awarded one of the ten prizes of the European Mathematical Society for his work on low-dimensional topology , and particularly for his role in the development of combinatorial Heegaard Floer homology . He was elected as a member of

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814-521: The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk , the complex plane , or the Riemann sphere . In particular it admits a Riemannian metric of constant curvature . This classifies Riemannian surfaces as elliptic (positively curved—rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their universal cover . The uniformization theorem

851-491: The 1960s had the effect of emphasising low dimensions in topology. The solution by Stephen Smale , in 1961, of the Poincaré conjecture in five or more dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory . Thurston's geometrization conjecture , formulated in

888-590: The 2017 class of Fellows of the American Mathematical Society "for contributions to Floer homology and the topology of manifolds". In 2018, he was an invited speaker at the International Congress of Mathematicians (ICM) in Rio de Janeiro . In 2020, he received a Simons Investigator Award . The citation reads: "Ciprian Manolescu works in low-dimensional topology and gauge theory. His research

925-516: The article on braid groups . Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces . A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously . See also Kleinian model . Its thick-thin decomposition has

962-460: The core curricula is imposed by the legislation of the Ministry, the highschool offers partially different kinds of classes depending on the profile chosen by each of the 30 students groups. The offered profiles refer to the class/classes that group should focus upon as it follows: The grading system used is the Romanian numerical grading system, with grades ranging from 1 to 10, 10 being the maximum; 5

999-639: The direction of Peter B. Kronheimer . He was the winner of the Morgan Prize , awarded jointly by AMS-MAA-SIAM, in 2002. His undergraduate thesis was on Finite dimensional approximation in Seiberg–Witten theory , and his PhD thesis topic was A spectrum valued TQFT from the Seiberg–Witten equations . In early 2013, he released a paper detailing a disproof of the triangulation conjecture for manifolds of dimension 5 and higher. For this paper, he received

1036-400: The everyday braid concept, and some generalizations. The idea is that braids can be organized into groups , in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit presentations , as was shown by Emil Artin  ( 1947 ). For an elementary treatment along these lines, see

1073-631: The following subjects: Romanian – speaking and writing, a foreign language – speaking, Mathematics, a science (the special profile assigned science – if on a special profile class) and a social science. The graduating mark for the Baccalaureate is 6. The statistical data for the 2009 class are: Mathematician Ciprian Manolescu , the only three-time winner of the International Mathematics Olympiad , attended Brătianu High School, as have other 7 international olympiad gold medallists in mathematics, biology, and physics. Other notable alumni in

1110-442: The geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston  ( 1982 ), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture . A 4-manifold is a 4-dimensional topological manifold . A smooth 4-manifold

1147-570: The late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones ' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics . In 2002, Grigori Perelman announced

Ion Brătianu National College (Pitești) - Misplaced Pages Continue

1184-518: The real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is 2 − k . In mathematics , the Teichmüller space T X of a (real) topological surface X , is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism . Each point in T X may be regarded as an isomorphism class of 'marked' Riemann surfaces where

1221-640: The sciences were neurologists Gheorghe Marinescu and Constantin Bălăceanu-Stolnici . Former Romanian President Emil Constantinescu , as well as Prime Minister Armand Călinescu , Foreign Minister Istrate Micescu , and Marshal Ion Antonescu also attended the high school. In arts the high school had many talented students, such as the poets Ion Barbu and Ion Minulescu , the painters Sorin Ilfoveanu , Costin Petrescu , and Rudolf Schweitzer-Cumpăna ,

1258-482: The surface of a ball . On the other hand, there are surfaces, such as the Klein bottle , that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections. The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: The surfaces in the first two families are orientable . It

1295-465: The very high overall GPA of its students and their excellent results in National Olympiads and other contests, in 1997 the "Ion C. Brătianu" High School was awarded the title of "National College" ( Colegiu Național ). The academic program is organized on a 2 shift schedule – the morning shift and the evening shift. In accordance to the Romanian curriculum imposed by the Ministry, the students have

1332-685: The writers Dan Simonescu , Vladimir Streinu , Tudor Teodorescu-Braniște , and George Vâlsan , and the journalist Robert Turcescu . Ciprian Manolescu Ciprian Manolescu (born December 24, 1978) is a Romanian-American mathematician, working in gauge theory , symplectic geometry , and low-dimensional topology . He is currently a professor of mathematics at Stanford University . Manolescu completed his first eight classes at School no. 11 Mihai Eminescu and his secondary education at Ion Brătianu High School in Pitești . He completed his undergraduate studies and PhD at Harvard University under

1369-477: Was shown first by Clifford Taubes . Prior to this construction, non-diffeomorphic smooth structures on spheres— exotic spheres —were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open to this day). For any positive integer n other than 4, there are no exotic smooth structures on R ; in other words, if n ≠ 4 then any smooth manifold homeomorphic to R

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