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We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e., on the standard sphere and Gromoll-Meyer exotic sphere . In this case, and are homeomorphic but not diffeomorphic. The standard sphere can be constructed as the quotient manifold with the so-called -action of S3, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S3. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism . The stochastic flow can be regarded as the same stochastic flow on S7, but viewed in Gromoll-Meyer differential structure. The flow on and the corresponding flow on constructed in this paper have the same regularities. There is no difference between the stochastic flow's appearance on S7 viewed in standard differential structure and the appearance of the same stochastic flow viewed in the Gromoll-Meyer differential structure. Furthermore, since the inverse mapping h-1 is differentiable on , the Riemannian metric tensor on , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow and the Fokker-Planck equation associated with the stochastic differential equation have the same regularities provided that the function β is C1-differentiable. Therefore both differential structures on S7 give the same description of the dynamics of the distribution function of the stochastic process understudy on seven spheres.

Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which S^n has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which S^n has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4., Comment: 52 pages. Revised version includes some additional recommendations of referee

Milnor’s fibration theorem is a landmark in singularity theory; it allowed to deepen the study of the geometry and topology of analytic maps near their critical points. In this chapter we revisit the classical theory and we glance at some areas of current research. We start with a glimpse at the origin of the fibration theorem, which is motivated by the study of exotic spheres. We then discuss an elementary example where all the ingredients of the fibration theorem are described in simple terms, and we use this as a guideline all along the chapter. The first part concerns complex singularities, which is a fairly mature area of mathematics; we survey some of the main steps in this line of research and indicate a wide bibliography as well as relations with other chapters in this book. The second part concerns real singularities, a theory that still is in its youth, though it springs also from Milnor’s seminal work.

We study the interplay between geodesics on two non-holonomic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.

We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently close in the sense of $L^1$-norm to those of $M$ at $p$. Our result hence not only produces a weak version of the Cartan--Ambrose--Hicks theorem in the case where underlying manifolds admit a point with a single cut point, but also is a kind of a weak version of the Blaschke conjecture for spheres proved by Berger. In particular that result generalizes one of theorems in Cheeger's Ph.D. Thesis in that case. Remark that every exotic sphere of dimension $> 4$ admits a metric such that there is a point whose cut locus consists of a single point., 16 page, minor corrections and revision

In this article, a six-parameter family of highly connected 7-manifolds which admit an S O ( 3 ) -invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an S O ( 3 ) -invariant metric of non-negative curvature.

We construct a codimension 3 completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp ( 2 ) -principal bundle with the structure group Sp ( 1 ) . The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a ( 4 n + 3 ) -dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S 7 .

The spaces (sets equipped with some structures) involved in a mathematical structure appearing in a physical theory has been discussed. The spaces are called spaces of theory. Some structures which may be involved in a space of theory (such as topological, algebraic, geometrical, and differential structures) have been introduced and their roles in the formulation of a mathematical structure in a physical theory have been discussed. The equivalence of two mathematical structures appearing in a physical theory and the equivalence of two spaces of theory have been introduced. The possibility that a topological space admits several differential structures and that those structures may not be equivalent has also been exposed. Some implications of the existence of exoticisms of topological spaces on the formulations of physical theory have been discussed. Some explicit or concrete example of exotic spheres having constructed have been exposed and analysed. Some relative regularities on Milnor spheres have also been discussed.

One of the greatest challenges in the exploration of spatial complexity consists in measuring it on 4d surfaces and objects. Some interesting results are already available for hypercubes and, in this respect, Hamiltonian cycles and cubical complexes appear promising for solving some problems of 4d spatial complexity on hypercubes. Topologically surprising objects, such as the exotic spheres, abound in 4d. Yet, in some cases, higher-dimensional topologies can be easier to work out calculations on manifolds, thus leaving 3d and 4d objects as likely more difficult to examine topologically, and, by consequence, with respect to the spatial complexity of surfaces and objects that are in them. We need to explore what algorithmic measures of spatial complexity might apply to objects in 4d spaces and, more ambitiously perhaps, decide whether the 3d-and-4d-spaces are the ones capable of sustaining the highest spatial complexity among all n-dimensional spaces.

We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S' are symplectomorphic only if the classes defined by S and S' agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section., 19 pages, no figures. Version 2: Theorem 1.4 added. Version 3: Clarified attributions, typographical corrections

The history and solution of the Kervaire invariant problem is discussed, along with some of the future prospects raised by its solution.

The study of n-dimensional manifolds has seen great advances in the last half century. In dimensions greater than four, surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial, where serious algebraic issues remain. In dimension 3, the proof by Perelman of Thurston’s Geometrization Conjecture (1) allows an algorithmic classification of 3-manifolds. The work of Freedman (2) classifies topological 4-manifolds if the fundamental group is not too large. Also, gauge theory in the hands of Donaldson (3) has provided invariants leading to proofs that some topological 4-manifolds have no smooth structure, that many compact 4-manifolds have countably many smooth structures, and that many noncompact 4-manifolds, in particular 4D Euclidean space R 4 , have uncountably many. However, the gauge theory invariants run into trouble with small 4-manifolds, such as those with the same homology groups as the 4D sphere, S 4 . In particular, the smooth 4D Poincare Conjecture, the last remaining case of that hallowed conjecture, is still open. (In higher dimensions, the smooth Poincare Conjecture is sometimes true in the following sense. In dimensions 3, 5, 6, 12, and 61, a homotopy sphere is diffeomorphic to the standard one, and in all other known cases, there are increasingly many exotic smooth structures on the topological sphere; however, it is possible that there may be more high-dimensional cases with no exotic spheres.) The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic. What is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions. The s-cobordism theorem states that, if M 0 m and M 1 m are the two boundary components of an m + 1 -dimensional manifold W and if … [↵][1]1Email: kirby{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Given a closed simply connected manifold $M$ of dimension $2n\ge6$, we compare the ring of characteristic classes of smooth oriented bundles with fibre $M$ to the analogous ring resulting from replacing $M$ by the connected sum $M\sharp\Sigma$ with an exotic sphere $\Sigma$. We show that, after inverting the order of $\Sigma$ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of $\Sigma$ is necessary., Comment: 16 pages, to appear in Mathematische Annalen

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres., LaTeX2e, 28 pages, final version

For each nonnegative integer we find an open (4m+9)-dimensional simply-connected manifold admitting complete nonnegatively curved metrics whose souls are non-diffeomorphic, homeomorphic, and have codimension 2. We give a diffeomorphism classification of the pairs (N, soul) when N is a nontrivial complex line bundle over the product of 7-sphere and complex projective plane: up to diffeomorphism there are precisely three such pairs, distinguished by their non-diffeomorphic souls., 49 pages, to appear in Advances in Mathematics

Short report on a lecture on the occasion of the 60th birthday of Egbert Brieskorn, Oberwolfach, 16.07.1996.

We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of $\mathbb{CP}^{n/2}$, $\mathbb{HP}^{n/4}$, $S^{n-k} \times S^k$ ($2 \leq k \leq n-2$), $S^{n-2} \times \mathbb{H}^2$, $S^{n-2} \times \mathbb{R}^2$ all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} \times \mathbb{R}$ lies in the interior of our curvature cone., Comment: Ann. of Math. 187, 263-299 (2018)

Diese Doktorarbeit befasst sich mit den Diffeomorphismengruppen exotischer Sphären. Eine glatte Homotopiesphäre ist eine glatte Mannigfaltigkeit, die homotopieäquivalent aber nicht notwendigerweise diffeomorph zu einer Standardsphäre ist. Falls sie nicht diffeomorph ist, so wird sie als exotische Sphäre bezeichnet. Es gibt eine Auswertungsabbildung von der Diffeomorphismengruppe einer exotischen Sphäre zur exotischen Sphäre selbst, gegeben durch Auswertung an einem Basispunkt. Die Existenzfrage eines Schnittes zu dieser Auswertungsabbildung wird unter Zuhilfenahme von Methoden aus Homotopietheorie und Orthogonalkalkül untersucht. Das Hindernis zur Existenz eines solchen Schnittes ist durch ein Element in einer bestimmten Homotopiegruppe gegeben. Das Hauptresultat ist die Nichtexistenz eines solchen Schnittes in Dimension sieben im Fall des Erzeugers der Kervaire-Milnor Gruppe der Homotopiesphären. This thesis is concerned with the diffeomorphism groups of exotic spheres. A smooth homotopy sphere is a smooth manifold, which is homotopy equivalent to a sphere, but not necessarily diffeomorphic. If it is not diffeomorphic, it is referred to as an exotic sphere. There is an evaluation map from the diffeomorphism group of an exotic sphere to the exotic sphere itself, given by evaluation at a basepoint. Methods from homotopy theory and orthogonal calculus are used to investigate the existence question of a section of this evaluation map. The obstruction to the existence of such a section is given by an element in a certain homotopy group. The main result is the non-existence of such a section in dimension seven in the case of the generator of the Kervaire-Milnor group of homotopy spheres.

Consider cotangent bundles of exotic spheres with their canonical symplectic structure. They admit automorphisms that preserve the part at infinity of one fiber and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behavior)., Simons Foundation (Simons Investigator Grant), National Science Foundation (U.S.) (Grant DMS-1005288)

Every A 1 \mathbb {A}^{1} -bundle over A ∗ 2 , \mathbb {A}_{\ast }^{2}, the complex affine plane punctured at the origin, is trivial in the differentiable category, but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3 3 -sphere S C 3 , \mathbb {S}_{\mathbb {C}}^{3}, given by z 1 2 + z 2 2 + z 3 2 + z 4 2 = 1 , z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=1, admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous A 1 \mathbb {A}^{1} -bundles over A ∗ 2 \mathbb {A}_{\ast }^{2} are classified up to G m \mathbb {G}_{m} -equivariant algebraic isomorphism, and a criterion for nonisomorphy is given. In fact S C 3 \mathbb {S}_{\mathbb {C}}^{3} is not isomorphic as an abstract variety to the total space of any A 1 \mathbb {A}^{1} -bundle over A ∗ 2 \mathbb {A}_{\ast }^{2} of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a byproduct, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomorphism type of a new manifold of positive sectional curvature., Comment: 54 pages, LaTeX

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres $${\mathbb{S}^k}$$ , the forgetful map $${F_{\mathbb{S}^k}}$$ is not one-to-one. This result follows from Theorem A, which proves that the quotient map $${\mathcal{MET}^{\,\,sec

For a smooth manifold M we define the Teichmuller space 2T(M) of all Riemannian metrics on M and the Teichmuller space 2T€(M) of € -pinched negatively curved metrics on M, where 0 < € < oo. We prove that if M is hyperbolic, the natural inclusion ?f€(M)

The possibility of the incompleteness of differentiable vector fields in stochastic differential equations on a manifold related to exotic differential structures is discussed. It is shown that the incompleteness of vector fields due to the chosen differentiable structure (exotica). In this work, the incompleteness of vector fields will be realized from the fact that the isometric stochastic flows in an exotic differential structure are not equivalent to that of (standard) isometric stochastic flows. Isometric stochastic flows with the same one point motion in 7-dimensional standard spheres S7 and 7-dimensional Gromoll-Meyer exotic spheres ΣGM7 are studied, these manifolds are homeomorphic but not diffeomorphic. Therefore, the incompleteness of vector fields will have an impact on inequivalence between the isometric stochastic flows on standard sphere S7 and on Gromoll-Meyer sphere ΣGM7 in which the isometric stochastic flows is the solution of the stochastic differential equation itself. In this work, is...

We show that, for certain families $\phi_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $\phi^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the $A_m$-Milnor fibre. The nontriviality is detected by considering the action of symplectomorphisms on the space of parametrised Lagrangian submanifolds. We find related examples of symplectic mapping classes for $T^*(S^n\times S^1)$ and of an exotic symplectic structure on $T^*(S^n\times S^1)$ standard at infinity., Comment: 17 pages, 3 figures; v2 streamlined version. Accepted for publication by Journal of Topology

We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl structures., Comment: 31 pages, minor changes made, to appear in Commun. Math. Phys

We use the geometry of the geodesics of a certain left-invariant metric on the Lie group S p ( 2 ) Sp(2) to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of π 6 ( S 3 ) \pi _6(S^3) ) and an exotic (i.e. not isotopic to the identity) diffeomorphism of S 6 S^6 (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of π 14 ( S 7 ) \pi _{14}(S^{7}) and exotic diffeomorphisms of S 14 S^{14} , thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.

We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\scriptstyle{\Sigma^{2n+1}}$ the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the known $\scriptstyle{2^{2m}}$ distinct diffeomorphism types of homotopy real projective spaces in dimension $4m+1$., Comment: 22 pages, revised version with some clarifications and added references

In the surgery theory of smooth smooth manifolds, it is often difficult to determine the existence or the non-existence of a smooth normal map with nontrivial surgery obstructions. We present and prove a simple relation over the mod 2 Steenrod algebra between two smooth Kervaire classes in different dimensions. This formula enables us to compute the Kervaire surgery invariants for various manifolds.

We use an explicit generator of π6(S3) to construct a homotopy 10-sphere with an effective SO(4) action that does not bound a parallelizable manifold.

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere.

Using a Kaluza–Klein-type procedure, an explicit metric h on an exotic sphere Σ7 is constructed, satisfying the Wiedersehen condition at a set of points diffeomorphic to S 1. The formulas for the geodesics allows the writing down of formulas for an explicit degree 1 diffeomorphism σ: S 6 → S 6 that is not isotopic to the identity.

A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian p p -groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture.

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense, then they are diffeomorphic. We have three applications of the main theorem: The first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones. The third one is that a compact $n$-manifold $M$ is a twisted sphere and there exists a bi-Lipschitz homeomorphism between $M$ and the unit $n$-sphere $S^n(1)$ which is a diffeomorphism except for a single point, if $M$ satisfies certain two conditions with respect to critical points of its distance function in the Clarke sense. Moreover, we have three corollaries from the third theorem; the first one is that for any twisted sphere $\Sigma^n$ of general dimension $n$, there exists a bi-Lipschitz homeomorphism between $\Sigma^n$ and $S^n(1)$ which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic $n$-sphere $\Sigma^n$ of dimension $n>4$ and $S^n(1)$; the second one is that if an exotic $4$-sphere $\Sigma^4$ exists, then $\Sigma^4$ does not satisfy one of the two conditions above; the third one is that for any Grove-Shiohama type $n$-sphere $N$, there exists a bi-Lipschitz homeomorphism between $N$ and $S^n(1)$ which is a diffeomorphism except for one of points that attain their diameters., Comment: 39 pages, no figures, minor corrections

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.

We suggest that exotic sphere fibrations can be mapped to band topologies in condensed matter systems. These fibrations can correspond to geometric phases of two double bands or state vector bases with second Chern numbers m+n and -n respectively. They can be related to topological insulators, magneto-electric effects, and photonic crystals with special edge states. We also consider time-reversal symmetry breaking perturbations of topological insulator, and heterostructures of topological insulators with normal insulators and with superconductors. We consider periodic TI/NI/TI/NI' heterostuctures, and periodic TI/SC/TI/SC' heterostuctures. They also give rise to models of Weyl semimetals which have thermal and electrical transports., 23 pages; journal version

This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres \cite{St96} ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of B��rard-Bergery \cite{BB77} ( see also pp.234-235 of \cite{Be78} )., 18 pages. References are updated, and some typos are corrected due to the referees. To appear in Adv. Math

We propose a new notion of positivity for topological field theories (TFTs), based on S. Eilenberg's concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. For instance, we derive Polya's counting theory in combinatorics from state sum identities in a suitable positive TFT. Several other concrete examples are discussed, among them Novikov signatures of fiber bundles over spacetimes and arithmetic functions in number theory. In the future, we will employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres.

There are mathematicians able to solve incredibly difficult problems devising amazingly new ideas. There are mathematicians with a sure grasp of entire subjects, able to single out the more promising research directions. There are mathematicians with a crystal clear vision, able to explain and clarify any subject they talk about. And then there is John Milnor. He is all three: he solves, understands and explains, at an exceptional level on all counts. And he is a nice guy too.