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70-604: The index problem for elliptic differential operators was posed by Israel Gel'fand . He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants . Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem , and the Hirzebruch signature theorem . Friedrich Hirzebruch and Armand Borel had proved

140-407: A dynamical system , then the cotangent bundle T ∗ M {\displaystyle \!\,T^{*}\!M} can be thought of as the set of possible positions and momenta . For example, this is a way to describe the phase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass

210-502: A phase space on which Hamiltonian mechanics plays out. The cotangent bundle carries a canonical one-form θ also known as the symplectic potential , Poincaré 1 -form, or Liouville 1 -form. This means that if we regard T * M as a manifold in its own right, there is a canonical section of the vector bundle T *( T * M ) over T * M . This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that x are local coordinates on

280-401: A pullback sheaf ϕ ∗ T ∗ N {\displaystyle \phi ^{*}T^{*}N} on M . There is an induced map of vector bundles ϕ ∗ ( T ∗ N ) → T ∗ M {\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M} . The tangent bundle of

350-657: A smooth manifold and let M × M be the Cartesian product of M with itself. The diagonal mapping Δ sends a point p in M to the point ( p , p ) of M × M . The image of Δ is called the diagonal. Let I {\displaystyle {\mathcal {I}}} be the sheaf of germs of smooth functions on M × M which vanish on the diagonal. Then the quotient sheaf I / I 2 {\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}} consists of equivalence classes of functions which vanish on

420-447: A (non-unique) parametrix (or pseudoinverse ) D ′ such that DD′ -1 and D′D -1 are both compact operators. An important consequence is that the kernel of D is finite-dimensional, because all eigenspaces of compact operators, other than the kernel, are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic pseudodifferential operator .) As

490-471: A canonical symplectic 2-form on it, as an exterior derivative of the tautological one-form , the symplectic potential . Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on R n × R n {\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}} . But there

560-488: A certain sense with the Chern-character construction above). If X is a compact submanifold of a manifold Y then there is a pushforward (or "shriek") map from K( TX ) to K( TY ). The topological index of an element of K( TX ) is defined to be the image of this operation with Y some Euclidean space, for which K( TY ) can be naturally identified with the integers Z (as a consequence of Bott-periodicity). This map

630-560: A function f ∈ C ∞ ( R n ) , {\displaystyle f\in C^{\infty }(\mathbb {R} ^{n}),} with the condition that ∇ f ≠ 0 , {\displaystyle \nabla f\neq 0,} the tangent bundle is where d f x ∈ T x ∗ M {\displaystyle df_{x}\in T_{x}^{*}M}

700-470: A generalization of it to all complex manifolds: Hirzebruch's proof only worked for projective complex manifolds X . The Hirzebruch signature theorem states that the signature of a compact oriented manifold X of dimension 4 k is given by the L genus of the manifold. This follows from the Atiyah–Singer index theorem applied to the following signature operator . The bundles E and F are given by

770-558: A map from Λ even {\displaystyle \Lambda ^{\text{even}}} to Λ odd {\displaystyle \Lambda ^{\text{odd}}} . Then the analytical index of D {\displaystyle D} is the Euler characteristic χ ( M ) {\displaystyle \chi (M)} of the Hodge cohomology of M {\displaystyle M} , and

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840-403: A rather complicated way under coordinate transforms (see jet bundle ); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundles E and F is a section of the pullback of the bundle Hom( E , F ) to

910-427: A result X is always an orientable manifold (the tangent bundle TX is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates . Because cotangent bundles can be thought of as symplectic manifolds , any real function on the cotangent bundle can be interpreted to be a Hamiltonian ; thus the cotangent bundle can be understood to be

980-417: A structure which is unique (up to isotopy close to identity). The quasiconformal structures ( Connes, Sullivan & Teleman 1994 ) and more generally the L -structures, p > n ( n +1)/2, introduced by M. Hilsum ( Hilsum 1999 ), are the weakest analytical structures on topological manifolds of dimension n for which the index theorem is known to hold. Suppose that M {\displaystyle M}

1050-512: A unique vector v ∈ T x M {\displaystyle v\in T_{x}M} for which v ∗ ( u ) = v ⋅ u , {\displaystyle v^{*}(u)=v\cdot u,} for an arbitrary u ∈ T x M , {\displaystyle u\in T_{x}M,} Since the cotangent bundle X = T * M

1120-419: Is a vector bundle , it can be regarded as a manifold in its own right. Because at each point the tangent directions of M can be paired with their dual covectors in the fiber, X possesses a canonical one-form θ called the tautological one-form , discussed below. The exterior derivative of θ is a symplectic 2-form , out of which a non-degenerate volume form can be built for X . For example, as

1190-520: Is a compact oriented manifold of dimension n = 2 r {\displaystyle n=2r} . If we take Λ even {\displaystyle \Lambda ^{\text{even}}} to be the sum of the even exterior powers of the cotangent bundle, and Λ odd {\displaystyle \Lambda ^{\text{odd}}} to be the sum of the odd powers, define D = d + d ∗ {\displaystyle D=d+d^{*}} , considered as

1260-781: Is a point, then we recover the statement above. Here K ( X ) {\displaystyle K(X)} is the Grothendieck group of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by the Chow ring of a smooth variety, and the Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles. Due to ( Teleman 1983 ), ( Teleman 1984 ): The proof of this result goes through specific considerations, including

1330-504: Is based on a signature operator S , defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare ( Donaldson & Sullivan 1989 )). Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of ( Connes, Sullivan & Teleman 1994 )). The work ( Connes, Sullivan & Teleman 1994 ) "provides local constructions for characteristic classes based on higher dimensional relatives of

1400-552: Is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah–Singer index theorem implies some deep integrality properties, as it implies that the topological index is integral. The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold X has odd dimension, though there are pseudodifferential elliptic operators whose index does not vanish in odd dimensions. The Grothendieck–Riemann–Roch theorem

1470-618: Is called elliptic if the symbol is nonzero whenever at least one y is nonzero. Example: The Laplace operator in k variables has symbol y 1 2 + ⋯ + y k 2 {\displaystyle y_{1}^{2}+\cdots +y_{k}^{2}} , and so is elliptic as this is nonzero whenever any of the y i {\displaystyle y_{i}} 's are nonzero. The wave operator has symbol − y 1 2 + ⋯ + y k 2 {\displaystyle -y_{1}^{2}+\cdots +y_{k}^{2}} , which

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1540-412: Is given by in other words the value of the top dimensional component of the mixed cohomology class ch ⁡ ( D ) Td ⁡ ( X ) {\displaystyle \operatorname {ch} (D)\operatorname {Td} (X)} on the fundamental homology class of the manifold X {\displaystyle X} up to a difference of sign. Here, In some situations, it

1610-418: Is given in these coordinates by Intrinsically, the value of the canonical one-form in each fixed point of T*M is given as a pullback . Specifically, suppose that π : T*M → M is the projection of the bundle. Taking a point in T x * M is the same as choosing of a point x in M and a one-form ω at x , and the tautological one-form θ assigns to the point ( x , ω)

1680-463: Is independent of the embedding of X in Euclidean space. Now a differential operator as above naturally defines an element of K( TX ), and the image in Z under this map "is" the topological index. As usual, D is an elliptic differential operator between vector bundles E and F over a compact manifold X . The index problem is the following: compute the (analytical) index of D using only

1750-478: Is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be with the differential given by ∂ ¯ {\displaystyle {\overline {\partial }}} . Then the i' th cohomology group is just the coherent cohomology group H( X , V ), so the analytical index of this complex is the holomorphic Euler characteristic of V : Since we are dealing with complex bundles,

1820-444: Is not elliptic if k ≥ 2 {\displaystyle k\geq 2} , as the symbol vanishes for some non-zero values of the y s. The symbol of a differential operator of order n on a smooth manifold X is defined in much the same way using local coordinate charts, and is a function on the cotangent bundle of X , homogeneous of degree n on each cotangent space. (In general, differential operators transform in

1890-503: Is possible to simplify the above formula for computational purposes. In particular, if X {\displaystyle X} is a 2 m {\displaystyle 2m} -dimensional orientable (compact) manifold with non-zero Euler class e ( T X ) {\displaystyle e(TX)} , then applying the Thom isomorphism and dividing by the Euler class,

1960-807: Is the directional derivative d f x ( v ) = ∇ f ( x ) ⋅ v {\displaystyle df_{x}(v)=\nabla \!f(x)\cdot v} . By definition, the cotangent bundle in this case is where T x ∗ M = { v ∈ T x R n   :   d f x ( v ) = 0 } ∗ . {\displaystyle T_{x}^{*}M=\{v\in T_{x}\mathbb {R} ^{n}\ :\ df_{x}(v)=0\}^{*}.} Since every covector v ∗ ∈ T x ∗ M {\displaystyle v^{*}\in T_{x}^{*}M} corresponds to

2030-485: Is the function of 2 k variables x 1 , … , x k , y 1 , … , y k {\displaystyle x_{1},\dots ,x_{k},y_{1},\dots ,y_{k}} , given by dropping all terms of order less than n and replacing ∂ / ∂ x i {\displaystyle \partial /\partial x_{i}} by y i {\displaystyle y_{i}} . So

2100-486: Is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even. In dimension 4 this result implies Rochlin's theorem that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4

2170-427: Is the signature of the manifold X , and its topological index is the L genus of X , so these are equal. The  genus is a rational number defined for any manifold, but is in general not an integer. Borel and Hirzebruch showed that it is integral for spin manifolds, and an even integer if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the  genus for spin manifolds

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2240-540: The Chern-Weil homomorphism ). Take X to be a complex manifold of (complex) dimension n with a holomorphic vector bundle V . We let the vector bundles E and F be the sums of the bundles of differential forms with coefficients in V of type (0, i ) with i even or odd, and we let the differential operator D be the sum restricted to E . This derivation of the Hirzebruch–Riemann–Roch theorem

2310-467: The United States in 1989. Gelfand is known for many developments including: Gelfand ran a seminar at Moscow State University from 1943 [1] until May 1989 (when it continued at Rutgers University ), which covered a wide range of topics and was an important school for many mathematicians. The Gelfand–Tsetlin (also spelled Zetlin) basis is a widely used tool in theoretical physics and

2380-399: The dual space of covectors, linear functions v ∗ : R n → R {\displaystyle v^{*}:\mathbb {R} ^{n}\to \mathbb {R} } . Given a smooth manifold M ⊂ R n {\displaystyle M\subset \mathbb {R} ^{n}} embedded as a hypersurface represented by the vanishing locus of

2450-524: The +1 and −1 eigenspaces of the operator on the bundle of differential forms of X , that acts on k -forms as i k ( k − 1 ) {\displaystyle i^{k(k-1)}} times the Hodge star operator . The operator D is the Hodge Laplacian restricted to E , where d is the Cartan exterior derivative and d * is its adjoint. The analytic index of D

2520-583: The 20th century", having exerted a tremendous influence on the field both through his own works and those of his students. Gelfand died at the Robert Wood Johnson University Hospital near his home in Highland Park, New Jersey . He was less than five weeks past his 96th birthday. His death was first reported on the blog of his former collaborator Andrei Zelevinsky and confirmed a few hours later by an obituary in

2590-767: The Chern roots x i ( E ⊗ C ) = c 1 ( l i ) {\displaystyle x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})} , x r + i ( E ⊗ C ) = c 1 ( l i ¯ ) = − x i ( E ⊗ C ) {\displaystyle x_{r+i}(E\otimes \mathbb {C} )=c_{1}{\mathord {\left({\overline {l_{i}}}\right)}}=-x_{i}(E\otimes \mathbb {C} )} , i = 1 , … , r {\displaystyle i=1,\,\ldots ,\,r} . Using Chern roots as above and

2660-499: The Russian online newspaper Polit.ru . Cotangent bundle In mathematics , especially differential geometry , the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle . This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds , or (in

2730-472: The United States shortly before his 76th birthday, at Rutgers University . Gelfand is also a 1994 MacArthur Fellow . His legacy continues through his students, who include Endre Szemerédi , Alexandre Kirillov , Edward Frenkel , Joseph Bernstein , David Kazhdan , as well as his own son, Sergei Gelfand. A native of Kherson Governorate , Russian Empire (now, Odesa Oblast , Ukraine ), Gelfand

2800-409: The adjoint operator). In other words, This is sometimes called the analytical index of D . Example: Suppose that the manifold is the circle (thought of as R / Z ), and D is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λ x ) if λ is an integral multiple of 2π i and is 0 otherwise, and

2870-417: The base manifold M . In terms of these base coordinates, there are fibre coordinates p i  : a one-form at a particular point of T * M has the form p i   dx ( Einstein summation convention implied). So the manifold T * M itself carries local coordinates ( x , p i ) where the x 's are coordinates on the base and the p's are coordinates in the fibre. The canonical one-form

Atiyah–Singer index theorem - Misplaced Pages Continue

2940-501: The computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by e ( T X ) = ∏ i n x i ( T X ) {\textstyle e(TX)=\prod _{i}^{n}x_{i}(TX)} and Applying the index theorem, we obtain the Hirzebruch-Riemann-Roch theorem : In fact we get

3010-411: The cotangent space of X . The differential operator is called elliptic if the element of Hom( E x , F x ) is invertible for all non-zero cotangent vectors at any point x of X . A key property of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible. More precisely, an elliptic operator D on a compact manifold has

3080-545: The diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M : By Taylor's theorem , this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of M . Thus it defines a vector bundle on M : the cotangent bundle . Smooth sections of the cotangent bundle are called (differential) one-forms . A smooth morphism ϕ : M → N {\displaystyle \phi \colon M\to N} of manifolds, induces

3150-400: The elliptic differential operator D has a pseudoinverse, it is a Fredholm operator . Any Fredholm operator has an index , defined as the difference between the (finite) dimension of the kernel of D (solutions of Df = 0), and the (finite) dimension of the cokernel of D (the constraints on the right-hand-side of an inhomogeneous equation like Df = g , or equivalently the kernel of

3220-521: The extension of Hodge theory on combinatorial and Lipschitz manifolds ( Teleman 1980 ), ( Teleman 1983 ), the extension of Atiyah–Singer's signature operator to Lipschitz manifolds ( Teleman 1983 ), Kasparov's K-homology ( Kasparov 1972 ) and topological cobordism ( Kirby & Siebenmann 1977 ). This result shows that the index theorem is not merely a differentiability statement, but rather a topological statement. Due to ( Donaldson & Sullivan 1989 ), ( Connes, Sullivan & Teleman 1994 ): This theory

3290-403: The form of cotangent sheaf) algebraic varieties or schemes . In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs . Let M be

3360-427: The index theorem shows that we can usually at least evaluate their difference .) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data. Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index

3430-508: The index, given by the difference of their dimensions, does indeed vary continuously, and can be given in terms of topological data by the index theorem. The topological index of an elliptic differential operator D {\displaystyle D} between smooth vector bundles E {\displaystyle E} and F {\displaystyle F} on an n {\displaystyle n} -dimensional compact manifold X {\displaystyle X}

3500-538: The integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961). The Atiyah–Singer theorem was announced in 1963. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that

3570-405: The kernel of the adjoint is a similar space with λ replaced by its complex conjugate. So D has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the same, so

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3640-542: The measurable Riemann mapping in dimension two and the Yang–Mills theory in dimension four." These results constitute significant advances along the lines of Singer's program Prospects in Mathematics ( Singer 1971 ). At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper ( Teleman 1985 ) provides a link between Thom's original construction of

3710-423: The one form defined is the sum of y i d x i {\displaystyle y_{i}\,dx_{i}} , and the differential is the canonical symplectic form, the sum of d y i ∧ d x i {\displaystyle dy_{i}\land dx_{i}} . If the manifold M {\displaystyle M} represents the set of possible positions in

3780-517: The rational Pontrjagin classes ( Thom 1956 ) and index theory. It is important to mention that the index formula is a topological statement. The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique. Sullivan's result on Lipschitz and quasiconformal structures ( Sullivan 1979 ) shows that any topological manifold in dimension different from 4 possesses such

3850-439: The result of Gelfand's work on the representation theory of the unitary group and Lie groups in general. Gelfand also published works on biology and medicine. For a long time he took an interest in cell biology and organized a research seminar on the subject. He worked extensively in mathematics education, particularly with correspondence education. In 1994, he was awarded a MacArthur Fellowship for this work. Gelfand

3920-583: The standard properties of the Euler class, we have that e ( T M ) = ∏ i r x i ( T M ⊗ C ) {\textstyle e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )} . As for the Chern character and the Todd class, Applying the index theorem, which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying

3990-430: The symbol s and topological data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states: In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually;

4060-424: The symbol is homogeneous in the variables y , of degree n . The symbol is well defined even though ∂ / ∂ x i {\displaystyle \partial /\partial x_{i}} does not commute with x i {\displaystyle x_{i}} because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator

4130-1080: The topological index is the integral of the Euler class over the manifold. The index formula for this operator yields the Chern–Gauss–Bonnet theorem . The concrete computation goes as follows: according to one variation of the splitting principle , if E {\displaystyle E} is a real vector bundle of dimension n = 2 r {\displaystyle n=2r} , in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles l 1 , … , l r {\displaystyle l_{1},\,\ldots ,\,l_{r}} such that E ⊗ C = l 1 ⊕ l 1 ¯ ⊕ ⋯ l r ⊕ l r ¯ {\displaystyle E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus \dotsm l_{r}\oplus {\overline {l_{r}}}} . Therefore, we can consider

4200-413: The topological index may be expressed as where division makes sense by pulling e ( T X ) − 1 {\displaystyle e(TX)^{-1}} back from the cohomology ring of the classifying space B S O {\displaystyle BSO} . One can also define the topological index using only K-theory (and this alternative definition is compatible in

4270-425: The value That is, for a vector v in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ to v at ( x , ω) is computed by projecting v into the tangent bundle at x using d π : T ( T * M ) → TM and applying ω to this projection. Note that the tautological one-form is not a pullback of a one-form on the base M . The cotangent bundle has

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4340-694: The vector space R n {\displaystyle \mathbb {R} ^{n}} is T R n = R n × R n {\displaystyle T\,\mathbb {R} ^{n}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}} , and the cotangent bundle is T ∗ R n = R n × ( R n ) ∗ {\displaystyle T^{*}\mathbb {R} ^{n}=\mathbb {R} ^{n}\times (\mathbb {R} ^{n})^{*}} , where ( R n ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}} denotes

4410-856: The  genus is minus one eighth of the signature. Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just the Fourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions. Israel Gel%27fand Israel Moiseevich Gelfand , also written Israïl Moyseyovich Gel'fand , or Izrail M. Gelfand ( Yiddish : ישראל געלפֿאַנד , Russian : Изра́иль Моисе́евич Гельфа́нд , Ukrainian : Ізраїль Мойсейович Гельфанд ; 2 September [ O.S. 20 August] 1913 – 5 October 2009)

4480-513: Was a prominent Soviet -American mathematician . He made significant contributions to many branches of mathematics, including group theory , representation theory and functional analysis . The recipient of many awards, including the Order of Lenin and the first Wolf Prize , he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to

4550-588: Was an advocate of animal rights . He became a vegetarian in 1994 and vegan in 2000. Gelfand held several honorary degrees and was awarded the Order of Lenin three times for his research. In 1977 he was elected a Foreign Member of the Royal Society . He won the Wolf Prize in 1978, Kyoto Prize in 1989 and MacArthur Foundation Fellowship in 1994. He held the presidency of the Moscow Mathematical Society between 1968 and 1970, and

4620-514: Was born into a Jewish family in the small southern Ukrainian town of Okny . According to his own account, Gelfand was expelled from high school under the Soviets because his father had been a mill owner. Bypassing both high school and college, he proceeded to postgraduate study at the age of 19 at Moscow State University , where his advisor was the preeminent mathematician Andrei Kolmogorov . He received his PhD in 1935. Gelfand immigrated to

4690-776: Was elected a foreign member of the U.S. National Academy of Science , the American Academy of Arts and Sciences , the Royal Irish Academy , the American Mathematical Society and the London Mathematical Society . In an October 2003 article in The New York Times , written on the occasion of his 90th birthday, Gelfand is described as a scholar who is considered "among the greatest mathematicians of

4760-627: Was held in Paris simultaneously with the seminar led by Richard Palais at Princeton University . The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the cobordism theory of the first proof with K-theory , and they used this to give proofs of various generalizations in another sequence of papers. If D is a differential operator on a Euclidean space of order n in k variables x 1 , … , x k {\displaystyle x_{1},\dots ,x_{k}} , then its symbol

4830-447: Was married to Zorya Shapiro , and their two sons, Sergei and Vladimir both live in the United States. The third son, Aleksandr, died of leukemia . Following the divorce from his first wife, Gelfand married his second wife, Tatiana; together they had a daughter, Tatiana. The family also includes four grandchildren and three great-grandchildren. Memories about I. Gelfand are collected at a dedicated website handled by his family. Gelfand

4900-397: Was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map f : X → Y {\displaystyle f:X\to Y} of compact stably almost complex manifolds, then there is a commutative diagram if Y = ∗ {\displaystyle Y=*}

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