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49-689: Tata Institute of Fundamental Research ( TIFR ) is an Indian Research Institute under the Department of Atomic Energy of the Government of India . It is a public deemed university located at Navy Nagar , Colaba in Mumbai . It also has a campus in Bangalore , International Centre for Theoretical Sciences (ICTS), and an affiliated campus in Serilingampally near Hyderabad . TIFR conducts research primarily in

98-445: A bijective holomorphic function from M to N whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent Riemann surfaces are for all practical purposes identical. Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f ( g ( z )) , h can be considered as

147-491: A projective space . Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space . This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry . The corresponding statement for higher-dimensional objects

196-553: A description. The geometric classification is reflected in maps between Riemann surfaces, as detailed in Liouville's theorem and the Little Picard theorem : maps from hyperbolic to parabolic to elliptic are easy, but maps from elliptic to parabolic or parabolic to hyperbolic are very constrained (indeed, generally constant!). There are inclusions of the disc in the plane in the sphere: Δ ⊂ C ⊂ ^ C , but any holomorphic map from

245-555: A map from an open set of R to R whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h ′( z ). However, the real determinant of multiplication by a complex number α equals | α | , so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas. Every non-compact Riemann surface admits non-constant holomorphic functions (with values in C ). In fact, every non-compact Riemann surface

294-572: A unique complete 2-dimensional real Riemann metric with constant curvature equal to −1, 0 or 1 that belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of isothermal coordinates . In complex analytic terms, the Poincaré–Koebe uniformization theorem (a generalization of the Riemann mapping theorem ) states that every simply connected Riemann surface

343-645: Is a Stein manifold . In contrast, on a compact Riemann surface X every holomorphic function with values in C is constant due to the maximum principle . However, there always exist non-constant meromorphic functions (holomorphic functions with values in the Riemann sphere C ∪ {∞} ). More precisely, the function field of X is a finite extension of C ( t ), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see Siegel (1955) . Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and

392-472: Is a complex algebraic curve by Chow's theorem and the Riemann–Roch theorem . There are several equivalent definitions of a Riemann surface. A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to X by means of the charts. Showing that a conformal structure determines a complex structure is more difficult. On

441-503: Is a surface : a two-dimensional real manifold , but it contains more structure (specifically a complex structure ). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable . Given this, the sphere and torus admit complex structures but the Möbius strip , Klein bottle and real projective plane do not. Every compact Riemann surface

490-514: Is because holomorphic and meromorphic maps behave locally like z ↦ z for integer n , so non-constant maps are ramified covering maps , and for compact Riemann surfaces these are constrained by the Riemann–Hurwitz formula in algebraic topology , which relates the Euler characteristic of a space and a ramified cover. For example, hyperbolic Riemann surfaces are ramified covering spaces of

539-418: Is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N , the map h ∘ f ∘ g is holomorphic (as a function from C to C ) wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces M and N are called biholomorphic (or conformally equivalent to emphasize the conformal point of view) if there exists

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588-405: Is compact. Then its topological type is described by its genus g ≥ 2 . Its Teichmüller space and moduli space are (6 g − 6 -dimensional. A similar classification of Riemann surfaces of finite type (that is homeomorphic to a closed surface minus a finite number of points) can be given. However in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such

637-464: Is conformally equivalent to one of the following: A Riemann surface is elliptic, parabolic or hyperbolic according to whether its universal cover is isomorphic to P ( C ), C or D . The elements in each class admit a more precise description. The Riemann sphere P ( C ) is the only example, as there is no group acting on it by biholomorphic transformations freely and properly discontinuously and so any Riemann surface whose universal cover

686-724: Is elliptic. With one puncture, which can be placed at infinity, it is the complex plane, which is parabolic. With two punctures, it is the punctured plane or alternatively annulus or cylinder, which is parabolic. With three or more punctures, it is hyperbolic – compare pair of pants . One can map from one puncture to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity), but all maps from zero punctures to one or more, or one or two punctures to three or more are constant. Continuing in this vein, compact Riemann surfaces can map to surfaces of lower genus, but not to higher genus, except as constant maps. This

735-425: Is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem . As an example, consider the torus T  := C / ( Z + τ Z ) . The Weierstrass function ℘ τ ( z ) belonging to the lattice Z + τ Z is a meromorphic function on T . This function and its derivative ℘ τ ′( z ) generate

784-1039: Is involved in building India's first gravity wave detector. The High Energy Physics Department, TIFR has been involved in major accelerator projects like the KEK , Tevatron , LEP and the LHC . TIFR also runs the Pelletron particle accelerator facility. Bhabha's motivation resulted in the development of an NMR spectrometer for solid state studies. The Department of Condensed Matter Physics and Material Sciences also conducts experimental research in high-temperature superconductivity, nanoelectronics and nanophotonics. The School of Technology and Computer Science grew out of early activities carried out at TIFR for building digital computers. Today, its activities cover areas such as Algorithms, Complexity Theory, Formal Method, Applied Probability, Learning Theory, Mathematical Finance, Information Theory, Communications, etc. The Department Of Biological Sciences

833-407: Is isomorphic to P ( C ) must itself be isomorphic to it. If X is a Riemann surface whose universal cover is isomorphic to the complex plane C then it is isomorphic to one of the following surfaces: Topologically there are only three types: the plane, the cylinder and the torus . But while in the two former case the (parabolic) Riemann surface structure is unique, varying the parameter τ in

882-424: Is typically used by complex analysts. It employs a different definition for "parabolic" and "hyperbolic". In this alternative classification scheme, a Riemann surface is called parabolic if there are no non-constant negative subharmonic functions on the surface and is otherwise called hyperbolic . This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than

931-544: The Abel–Jacobi map of the surface. All compact Riemann surfaces are algebraic curves since they can be embedded into some CP . This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve. The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a projective variety , i.e. can be given by polynomial equations inside

980-685: The Rockefeller Institute , Carnegie Institution of Washington and the Institute for Advanced Study . Research was advanced in both theory and application. This was aided by substantial private donation. As of 2006, there were over 14,000 research centres in the United States. The expansion of universities into the faculty of research fed into these developments as mass education produced mass scientific communities . A growing public consciousness of scientific research brought public perception to

1029-424: The complex plane : locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued functions such as √ z or log( z ), e.g. the subset of pairs ( z , w ) ∈ C with w = log( z ) . Every Riemann surface

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1078-428: The embedding of open Riemann surfaces in C 3 {\displaystyle \mathbb {C} ^{3}} , C. S. Seshadri 's work on projective modules over polynomial rings and M. S. Narasimhan 's results in the theory of pseudo differential operators. Narasimhan and Seshadri wrote a seminal paper on stable vector bundles , work which has been recognised as one of the most influential articles in

1127-404: The mapping class group . In this case it is the modular curve . In the remaining cases, X is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface). The topological type of X can be any orientable surface save the torus and sphere . A case of particular interest is when X

1176-560: The 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. The earliest research institute in Europe was Tycho Brahe 's Uraniborg complex on the island of Hven , a 16th-century astronomical laboratory set up to make highly accurate measurements of

1225-542: The 1950s, TIFR gained prominence in the field of cosmic ray physics, with the setting up of research facilities in Ooty and in the Kolar gold mines . In 1957, India's first digital computer, TIFRAC was built in TIFR. Acting on the suggestions of British physiologist Archibald Hill , Bhabha invited Obaid Siddiqi to set up a research group in molecular biology. This ultimately resulted in

1274-489: The International Centre for Theoretical Physics and the research complex Elettra Sincrotrone Trieste, the biology project EMBL, and the fusion project ITER which in addition to technical developments has a strong research focus. Research institutes came to emerge at the beginning of the twentieth century. In 1900, at least in Europe and the United States, the scientific profession had only evolved so far as to include

1323-584: The Islamic world. The first of these was the 9th-century Baghdad observatory built during the time of the Abbasid caliph al-Ma'mun , though the most famous were the 13th-century Maragheh observatory , and the 15th-century Ulugh Beg Observatory . The Kerala School of Astronomy and Mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala , India . The school flourished between

1372-522: The area. M. S. Raghunathan started research at TIFR on algebraic and discrete groups , and was recognised for his work on rigidity . The School of Natural Sciences is further split into seven departments working in several areas of physics , chemistry and biology . Within physics, the Department of Theoretical Physics (DTP) was set up by Bhabha, who conducted research in high energy physics and Condensed Matter Physics . The department worked on

1421-487: The centre for all large-scale projects in nuclear research . The first theoretical physics group was set up by Bhabha's students B.M. Udgaonkar and K.S. Singhvi . In December 1950, Bhabha organised an international conference at TIFR on elementary particle physics . Several world-renowned scientists attended the conference, including Rudolf Peierls , Léon Rosenfeld , William Fowler as well as Meghnad Saha , Vikram Sarabhai and others providing expertise from India. In

1470-671: The establishment of the National Centre for Biological Sciences (NCBS) , Bangalore twenty years later. In 1970, TIFR started research in radio astronomy with the setting up of the Ooty Radio Telescope . Encouraged by the success of ORT, Govind Swarup persuaded J. R. D. Tata to help set up the Giant Metrewave Radio Telescope near Pune , India . TIFR attained the official deemed university status in June 2002. To meet

1519-481: The ever-growing demand of space needed for research labs and accommodation institute is coming up with a new campus at Hyderabad . Research at TIFR is distributed across three schools, working over the mathematical sciences, natural sciences, technology and computer science. Since its birth in the 1950s, several contributions to mathematics have come from TIFR School of Mathematics. Notable contributions from TIFR mathematicians include Raghavan Narasimhan 's proof of

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1568-539: The fore in driving specific research developments. After the Second World War and the atom bomb specific research threads were followed: environmental pollution and national defence . Riemann surface In mathematics , particularly in complex analysis , a Riemann surface is a connected one-dimensional complex manifold . These surfaces were first studied by and are named after Bernhard Riemann . Riemann surfaces can be thought of as deformed versions of

1617-596: The function field of  T . There is an equation where the coefficients g 2 and g 3 depend on τ , thus giving an elliptic curve E τ in the sense of algebraic geometry. Reversing this is accomplished by the j -invariant j ( E ), which can be used to determine τ and hence a torus. The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature . That is, every connected Riemann surface X admits

1666-460: The intersection of these two open sets, composing one embedding with the inverse of the other gives This transition map is holomorphic, so these two embeddings define a Riemann surface structure on S . As sets, S = C ∪ {∞} . The Riemann sphere has another description, as the projective line CP = ( C ∖ {0}) / C . As with any map between complex manifolds, a function f  : M → N between two Riemann surfaces M and N

1715-640: The major advances in this period such as Quantum Field Theory , string theory , and superconductivity . The current faculty includes Sandip Trivedi , Shiraz Minwalla , Abhijit Gadde , and Gautam Mandal . Several early faculty members at the institution were renowned in their fields. These include Ashoke Sen , who conducted seminal work on String Theory , specifically S-Duality , while at this institution. Other distinguished members were Spenta Wadia , Sunil Mukhi , Deepak Dhar and Nandini Trivedi . The Department of Astrophysics works in areas like stellar binaries , gravitational waves and cosmology . TIFR

1764-571: The natural sciences, the biological sciences and theoretical computer science. Homi J. Bhabha , known for his role in the development of the Indian atomic energy programme, wrote to the Sir Dorabji Tata Trust requesting financial assistance to set up a scientific research institute. With support from J.R.D. Tata , then chairman of the Tata Group , TIFR was founded on 1 June 1945, and Homi Bhabha

1813-417: The negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc. To avoid confusion, call the classification based on metrics of constant curvature the geometric classification , and the one based on degeneracy of function spaces

1862-733: The principles of mass production and large-scale teamwork to the process of invention in the late 1800s, and because of that, he is often credited with the creation of the first industrial research laboratory. From the throes of the Scientific Revolution came the 17th century scientific academy. In London, the Royal Society was founded in 1660, and in France Louis XIV founded the Académie royale des sciences in 1666 which came after private academic assemblies had been created earlier in

1911-512: The seventeenth century to foster research. In the early 18th century, Peter the Great established an educational-research institute to be built in his newly created imperial capital, St Petersburg . His plan combined provisions for linguistic, philosophical and scientific instruction with a separate academy in which graduates could pursue further scientific research. It was the first institution of its kind in Europe to conduct scientific research within

1960-411: The sphere (they have non-constant meromorphic functions), but the sphere does not cover or otherwise map to higher genus surfaces, except as a constant. The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group ) reflects its geometry: The classification scheme above is typically used by geometers. There is a different classification for Riemann surfaces that

2009-458: The sphere to the plane is constant, any holomorphic map from the plane into the unit disk is constant (Liouville's theorem), and in fact any holomorphic map from the plane into the plane minus two points is constant (Little Picard theorem)! These statements are clarified by considering the type of a Riemann sphere ^ C with a number of punctures. With no punctures, it is the Riemann sphere, which

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2058-458: The stars. In the United States there are numerous notable research institutes including Bell Labs , Xerox Parc , The Scripps Research Institute , Beckman Institute , RTI International , and SRI International . Hughes Aircraft used a research institute structure for its organizational model. Thomas Edison , dubbed "The Wizard of Menlo Park", was one of the first inventors to apply

2107-789: The structure of a university. The St Petersburg Academy was established by decree on 28 January 1724. At the European level, there are now several government-funded institutions such as the European Space Agency (ESA), the nuclear research centre CERN , the European Southern Observatory (ESO) (Grenoble), the European Synchrotron Radiation Facility (ESRF) (Grenoble), EUMETSAT , the Italian-European Sistema Trieste with, among others,

2156-473: The summer season by the Tata Institute of Fundamental Research. VSRP is offered in the subjects Physics and Astronomy, Chemistry, Mathematics, Biology and Computer Science. 18°54′27″N 72°48′22″E  /  18.90757°N 72.80601°E  / 18.90757; 72.80601  ( TIFR ) Research institute In the early medieval period, several astronomical observatories were built in

2205-421: The theoretical implications of science and not its application. Research scientists had yet to establish a leadership in expertise. Outside scientific circles it was generally assumed that a person in an occupation related to the sciences carried out work which was necessarily "scientific" and that the skill of the scientist did not hold any more merit than the skill of a labourer. A philosophical position on science

2254-462: The third case gives non-isomorphic Riemann surfaces. The description by the parameter τ gives the Teichmüller space of "marked" Riemann surfaces (in addition to the Riemann surface structure one adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic moduli space (forgetting the marking) one takes the quotient of Teichmüller space by

2303-628: Was appointed its first director. The institute initially operated within the campus of the Indian Institute of Science , Bangalore before relocating to Mumbai later that year. TIFR's new campus in Colaba was designed by Chicago -based architect Helmuth Bartsch and was inaugurated by Prime Minister Jawaharlal Nehru on 15 January 1962. Shortly after Indian Independence , in 1949, the Council of Scientific and Industrial Research (CSIR) designated TIFR to be

2352-597: Was not thought by all researchers to be intellectually superior to applied methods. However any research on scientific application was limited by comparison. A loose definition attributed all naturally occurring phenomena to "science". The growth of scientific study stimulated a desire to reinvigorate the scientific discipline by robust research in order to extract "pure" science from such broad categorisation. This began with research conducted autonomously away from public utility and governmental supervision. Enclaves for industrial investigations became established. These included

2401-439: Was set up by Obaid Siddiqi in early 1960s as a molecular biology group. Over the years has expanded to encompass various other branches of modern biology. The department has fourteen labs covering various aspects of modern molecular and cell biology. TIFR also includes institutes outside its main campus in Colaba and Mumbai : The Visiting Students Research Programme ( VSRP ) is a summer programme conducted annually during

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