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52-666: John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology , algebraic K-theory and low-dimensional holomorphic dynamical systems . Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal , the Wolf Prize , the Abel Prize and all three Steele prizes . Milnor was born on February 20, 1931, in Orange, New Jersey . His father

104-463: A differential equation on it. Care must be taken to ensure that the resulting information is insensitive to this choice of extra structure, and so genuinely reflects only the topological properties of the underlying smooth manifold. For example, the Hodge theorem provides a geometric and analytical interpretation of the de Rham cohomology, and gauge theory was used by Simon Donaldson to prove facts about

156-530: A cellular basis for C ∗ ( X ~ ) {\displaystyle C_{*}({\tilde {X}})} and an orthogonal R {\displaystyle \mathbf {R} } -basis for U {\displaystyle U} , then D ∗ := U ⊗ Z [ π ] C ∗ ( X ~ ) {\displaystyle D_{*}:=U\otimes _{\mathbf {Z} [\pi ]}C_{*}({\tilde {X}})}

208-411: A doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox. His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure, classifying Brunnian links up to link-homotopy and introduced new invariants of it, called Milnor invariants . Upon completing his doctorate, he went on to work at Princeton. He was a professor at

260-749: A meromorphic function of s ∈ C {\displaystyle s\in \mathbf {C} } which is holomorphic at s = 0 {\displaystyle s=0} . As in the case of an orthogonal representation, we define the analytic torsion T M ( ρ ; E ) {\displaystyle T_{M}(\rho ;E)} by In 1971 D.B. Ray and I.M. Singer conjectured that T M ( ρ ; E ) = τ M ( ρ ; μ ) {\displaystyle T_{M}(\rho ;E)=\tau _{M}(\rho ;\mu )} for any unitary representation ρ {\displaystyle \rho } . This Ray–Singer conjecture

312-449: A representation of the fundamental group of M {\displaystyle M} on a real vector space of dimension N. Then we can define the de Rham complex and the formal adjoint d p {\displaystyle d_{p}} and δ p {\displaystyle \delta _{p}} due to the flatness of E q {\displaystyle E_{q}} . As usual, we also obtain

364-508: A special sort of distribution (such as a CR structure ), and so on. This distinction between differential geometry and differential topology is blurred, however, in questions specifically pertaining to local diffeomorphism invariants such as the tangent space at a point. Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R n {\displaystyle \mathbb {R} ^{n}} (for example

416-689: A symposium was held at Stony Brook University in celebration of his 60th birthday. Milnor was awarded the 2011 Abel Prize , for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the New Scientist "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning." In 2013 he became a fellow of the American Mathematical Society , for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems". In 2020 he received

468-482: A unimodular representation. M {\displaystyle M} has a smooth triangulation. For any choice of a volume μ ∈ det H ∗ ( M ) {\displaystyle \mu \in \det H_{*}(M)} , we get an invariant τ M ( ρ : μ ) ∈ R + {\displaystyle \tau _{M}(\rho :\mu )\in \mathbf {R} ^{+}} . Then we call

520-1328: Is a contractible finite based free R {\displaystyle \mathbf {R} } -chain complex. Let γ ∗ : D ∗ → D ∗ + 1 {\displaystyle \gamma _{*}:D_{*}\to D_{*+1}} be any chain contraction of D * , i.e. d n + 1 ∘ γ n + γ n − 1 ∘ d n = i d D n {\displaystyle d_{n+1}\circ \gamma _{n}+\gamma _{n-1}\circ d_{n}=id_{D_{n}}} for all n {\displaystyle n} . We obtain an isomorphism ( d ∗ + γ ∗ ) odd : D odd → D even {\displaystyle (d_{*}+\gamma _{*})_{\text{odd}}:D_{\text{odd}}\to D_{\text{even}}} with D odd := ⊕ n odd D n {\displaystyle D_{\text{odd}}:=\oplus _{n\,{\text{odd}}}\,D_{n}} , D even := ⊕ n even D n {\displaystyle D_{\text{even}}:=\oplus _{n\,{\text{even}}}\,D_{n}} . We define

572-449: Is a professor of mathematics at Barnard College and is known for her work in symplectic topology . One of Milnor's best-known works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term exotic sphere , referring to any n -sphere with nonstandard differential structure. Kervaire and Milnor initiated

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624-414: Is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer  ( 1971 , 1973a , 1973b ) as an analytic analogue of Reidemeister torsion. Jeff Cheeger  ( 1977 , 1979 ) and Werner Müller  ( 1978 ) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. Reidemeister torsion

676-453: Is another branch of differential topology, in which topological information about a manifold is deduced from changes in the rank of the Jacobian of a function. For a list of differential topology topics, see the following reference: List of differential geometry topics . Differential topology and differential geometry are first characterized by their similarity . They both study primarily

728-528: Is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics : It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from

780-569: Is the projection of L 2 Λ ( E ) {\displaystyle L^{2}\Lambda (E)} onto the kernel space H q ( E ) {\displaystyle {\mathcal {H}}^{q}(E)} of the Laplacian Δ q {\displaystyle \Delta _{q}} . It was moreover shown by ( Seeley 1967 ) that ζ q ( s ; ρ ) {\displaystyle \zeta _{q}(s;\rho )} extends to

832-689: The American Philosophical Society 1965. He later went on to win the National Medal of Science (1967), the Lester R. Ford Award in 1970 and again in 1984, the Leroy P. Steele Prize for "Seminal Contribution to Research" (1982), the Wolf Prize in Mathematics (1989), the Leroy P. Steele Prize for Mathematical Exposition (2004), and the Leroy P. Steele Prize for Lifetime Achievement (2011). In 1991

884-495: The Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct, using the concept of Reidemeister torsion . In 1984 Milnor introduced a definition of attractor . The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors. Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics

936-715: The Institute for Advanced Study from 1970 to 1990. He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books which are famous for their clarity, presentation, and an inspiration for the research by many mathematicians in their areas even after many decades since their publication. He served as Vice President of the AMS in 1976–77 period. His students have included Tadatoshi Akiba , Jon Folkman , John Mather , Laurent C. Siebenmann , Michael Spivak , and Jonathan Sondow. His wife, Dusa McDuff ,

988-560: The Lomonosov Gold Medal of the Russian Academy of Sciences. Differential topology The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism . Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected ) manifolds in each dimension separately: Beginning in dimension 4,

1040-525: The Poincaré conjecture . Differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on

1092-596: The Reidemeister torsion where A is the matrix of ( d ∗ + γ ∗ ) odd {\displaystyle (d_{*}+\gamma _{*})_{\text{odd}}} with respect to the given bases. The Reidemeister torsion ρ ( X ; U ) {\displaystyle \rho (X;U)} is independent of the choice of the cellular basis for C ∗ ( X ~ ) {\displaystyle C_{*}({\tilde {X}})} ,

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1144-581: The halting problem , it is impossible to classify such groups, so a full topological classification is impossible. Secondly, beginning in dimension four it is possible to have smooth manifolds that are homeomorphic, but with distinct, non-diffeomorphic smooth structures . This is true even for the Euclidean space R 4 {\displaystyle \mathbb {R} ^{4}} , which admits many exotic R 4 {\displaystyle \mathbb {R} ^{4}} structures. This means that

1196-580: The tangent bundle , jet bundles , the Whitney extension theorem , and so forth). The distinction is concise in abstract terms: Analytic torsion In mathematics, Reidemeister torsion (or R-torsion , or Reidemeister–Franz torsion ) is a topological invariant of manifolds introduced by Kurt Reidemeister ( Reidemeister 1935 ) for 3-manifolds and generalized to higher dimensions by Wolfgang Franz  ( 1935 ) and Georges de Rham  ( 1936 ). Analytic torsion (or Ray–Singer torsion )

1248-480: The topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number . Milnor's 1968 book on his theory, Singular Points of Complex Hypersurfaces , inspired the growth of a huge and rich research area that continues to mature to this day. In 1961 Milnor disproved

1300-500: The 4-sphere admit only one smooth structure ? This conjecture is true in dimensions 1, 2, and 3, by the above classification results, but is known to be false in dimension 7 due to the Milnor spheres . Important tools in studying the differential topology of smooth manifolds include the construction of smooth topological invariants of such manifolds, such as de Rham cohomology or the intersection form , as well as smoothable topological constructions, such as smooth surgery theory or

1352-596: The Hodge Laplacian on p-forms Assuming that ∂ M = 0 {\displaystyle \partial M=0} , the Laplacian is then a symmetric positive semi-positive elliptic operator with pure point spectrum As before, we can therefore define a zeta function associated with the Laplacian Δ q {\displaystyle \Delta _{q}} on Λ q ( E ) {\displaystyle \Lambda ^{q}(E)} by where P {\displaystyle P}

1404-720: The Laplacian acting on k -forms is which is formally the product of the positive eigenvalues of the laplacian acting on k -forms. The analytic torsion T ( M , E ) is defined to be Let X {\displaystyle X} be a finite connected CW-complex with fundamental group π := π 1 ( X ) {\displaystyle \pi :=\pi _{1}(X)} and universal cover X ~ {\displaystyle {\tilde {X}}} , and let U {\displaystyle U} be an orthogonal finite-dimensional π {\displaystyle \pi } -representation. Suppose that for all n. If we fix

1456-431: The books ( Turaev 2002 ) and (Nicolaescu  2002 , 2003 ). If M is a Riemannian manifold and E a vector bundle over M , then there is a Laplacian operator acting on the k -forms with values in E . If the eigenvalues on k -forms are λ j then the zeta function ζ k is defined to be for s large, and this is extended to all complex s by analytic continuation . The zeta regularized determinant of

1508-405: The classification becomes much more difficult for two reasons. Firstly, every finitely presented group appears as the fundamental group of some 4-manifold , and since the fundamental group is a diffeomorphism invariant, this makes the classification of 4-manifolds at least as difficult as the classification of finitely presented groups. By the word problem for groups , which is equivalent to

1560-399: The coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer does not need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, they can decide that

1612-421: The coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut. To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure. More mathematically, for example,

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1664-585: The concept of "simple homotopy type", see ( Milnor 1966 ) In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemeister torsion of its knot complement in S 3 {\displaystyle S^{3}} . ( Milnor 1962 ) For each q the Poincaré duality P o {\displaystyle P_{o}} induces and then we obtain The representation of

1716-417: The construction of cobordisms . Morse theory is an important tool which studies smooth manifolds by considering the critical points of differentiable functions on the manifold, demonstrating how the smooth structure of the manifold enters into the set of tools available. Oftentimes more geometric or analytical techniques may be used, by equipping a smooth manifold with a Riemannian metric or by studying

1768-512: The existence of tangent bundles , can be done in the topological setting with much more work, and others cannot. One of the main topics in differential topology is the study of special kinds of smooth mappings between manifolds, namely immersions and submersions , and the intersections of submanifolds via transversality . More generally one is interested in properties and invariants of smooth manifolds that are carried over by diffeomorphisms , another special kind of smooth mapping. Morse theory

1820-461: The fundamental group of knot complement plays a central role in them. It gives the relation between knot theory and torsion invariants. Let ( M , g ) {\displaystyle (M,g)} be an orientable compact Riemann manifold of dimension n and ρ : π ( M ) → G L ⁡ ( E ) {\displaystyle \rho \colon \pi (M)\rightarrow \mathop {GL} (E)}

1872-521: The intersection form of simply connected 4-manifolds. In some cases techniques from contemporary physics may appear, such as topological quantum field theory , which can be used to compute topological invariants of smooth spaces. Famous theorems in differential topology include the Whitney embedding theorem , the hairy ball theorem , the Hopf theorem , the Poincaré–Hopf theorem , Donaldson's theorem , and

1924-443: The orthogonal basis for U {\displaystyle U} and the chain contraction γ ∗ {\displaystyle \gamma _{*}} . Let M {\displaystyle M} be a compact smooth manifold, and let ρ : π ( M ) → G L ( E ) {\displaystyle \rho \colon \pi (M)\rightarrow GL(E)} be

1976-430: The point of view of differential topology, the donut and the coffee cup are the same (in a sense). This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny ( local ) piece of either of them. They must have access to each entire ( global ) object. From the point of view of differential geometry,

2028-403: The positive real number τ M ( ρ : μ ) {\displaystyle \tau _{M}(\rho :\mu )} the Reidemeister torsion of the manifold M {\displaystyle M} with respect to ρ {\displaystyle \rho } and μ {\displaystyle \mu } . Reidemeister torsion

2080-560: The problem of constructing a diffeomorphism between two manifolds of the same dimension is inherently global since locally two such manifolds are always diffeomorphic. Likewise, the problem of computing a quantity on a manifold that is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of R n {\displaystyle \mathbb {R} ^{n}} . Moreover, differential topology does not restrict itself necessarily to

2132-401: The properties of differentiable manifolds, sometimes with a variety of structures imposed on them. One major difference lies in the nature of the problems that each subject tries to address. In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems that are inherently global . Consider the example of a coffee cup and a donut. From

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2184-523: The same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume. On the other hand, smooth manifolds are more rigid than the topological manifolds . John Milnor discovered that some spheres have more than one smooth structure—see Exotic sphere and Donaldson's theorem . Michel Kervaire exhibited topological manifolds with no smooth structure at all. Some constructions of smooth manifold theory, such as

2236-462: The study of diffeomorphism. For example, symplectic topology —a subbranch of differential topology—studies global properties of symplectic manifolds . Differential geometry concerns itself with problems—which may be local or global—that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with a connection , a metric (which may be Riemannian , pseudo-Riemannian , or Finsler ),

2288-400: The study of differential topology in dimensions 4 and higher must use tools genuinely outside the realm of the regular continuous topology of topological manifolds . One of the central open problems in differential topology is the four-dimensional smooth Poincaré conjecture , which asks if every smooth 4-manifold that is homeomorphic to the 4-sphere , is also diffeomorphic to it. That is, does

2340-424: The systematic study of exotic spheres, showing in particular that the 7-sphere has 15 distinct differentiable structures (28 if one considers orientation). Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on

2392-573: The usage of Hopf algebras , theory of quadratic forms and the related area of symmetric bilinear forms , higher algebraic K-theory , game theory , and three-dimensional Lie groups . Milnor was elected as a member of the American Academy of Arts and Sciences in 1961. In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He was elected to the United States National Academy of Sciences in 1963 and

2444-656: The very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston . Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms , which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems. His other significant contributions include microbundles , influencing

2496-555: Was J. Willard Milnor, an engineer, and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Ralph Fox . He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing

2548-430: Was eventually proved, independently, by Cheeger ( 1977 , 1979 ) and Müller (1978) . Both approaches focus on the logarithm of torsions and their traces. This is easier for odd-dimensional manifolds than in the even-dimensional case, which involves additional technical difficulties. This Cheeger–Müller theorem (that the two notions of torsion are equivalent), along with Atiyah–Patodi–Singer theorem , later provided

2600-401: Was first used to combinatorially classify 3-dimensional lens spaces in ( Reidemeister 1935 ) by Reidemeister, and in higher-dimensional spaces by Franz. The classification includes examples of homotopy equivalent 3-dimensional manifolds which are not homeomorphic — at the time (1935) the classification was only up to PL homeomorphism , but later E.J. Brody ( 1960 ) showed that this

2652-420: Was in fact a classification up to homeomorphism . J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes. This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant. Whitehead torsion provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to

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2704-480: Was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic , and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces . Reidemeister torsion is closely related to Whitehead torsion ; see ( Milnor 1966 ). It has also given some important motivation to arithmetic topology ; see ( Mazur ). For more recent work on torsion see

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