Conclusions and Open Problems Conclusions and Open Problems 905

Conclusions and open problems Conclusions and open problems 905

In these notes I have tried to present a consistent picture of the theory of optimal transport, with a dynamical, probabilistic and geometric point of view, insisting on the notions of displacement interpolation, probabilistic representation, and curvature effects. The qualitative description of optimal transport, developed in Part I, now seems to be more or less under control. Even the smoothness of the transport map in curved geometries starts to be better understood, thanks in particular to the recent works of Grégoire Loeper, Xinan Ma, Neil Trudinger and Xu-Jia Wang which were described in Chapter 12. Among issues which seem to be of interest I shall mention: • find relevant examples of cost functions with nonnegative, or positive c-curvature (Definition 12.27), and theorems guaranteeing that the optimal transport does not approach singularities of the cost function — so that the smoothness of the transport map can be established; • get a precise description of the singularities of the optimal transport map when the latter is not smooth; • further analyze the displacement interpolation on singular spaces, maybe via nonsmooth generalizations of Mather’s estimates (as in Open Problem 8.21). For the applications of optimal transport to Riemannian geometry, a consistent picture is also emerging, as I have tried to show in Part II. The main regularity problems seem to be under control here, but there remain several challenging “structural” problems: • How can one best understand the relation between plain displacement convexity and distorted displacement convexity, as described in Chapter 17? Is there an Eulerian counterpart of the latter concept? See Open Problems 17.38 and 17.39 for more precise formulations. • Optimal transport seems to work well to establish sharp geometric inequalities when the “natural dimension of the inequality” coincides with the dimension bound; on the other hand, so far it has failed to establish for instance sharp logarithmic Sobolev or Talagrand inequalities (infinite-dimensional) under a CD(K, N) condition for N<∞ (Open Problems 21.6 and 22.44). The sharp L2-Sobolev inequality (21.9) has also escaped investigations based on optimal transport (Open Problems 21.11). Can one find a more precise strategy to attack such problems by a displacement convexity approach? A seemingly closely related question is whether one can mimick (maybe by changes of unknowns in the transport problem?) the changes of variables in the Γ2 formalism, which 906 Conclusions and open problems are often at the basis of the derivation of such sharp inequalities, as in the recent papers of Jérôme Demange. To add to the confusion, the mysterious structure condition (25.10) has popped out in these works; it is natural to ask whether this condition has any interpretation in terms of optimal transport. • Are there interesting examples of displacement convex functionals apart from the ones that have already been explored during the ⊗k past ten years — basically all of the form M U(ρ) dν + M k Vdµ ?It is frustrating that so few examples of displacement convex functionals are known, in contrast with the enormous amount of plainly convex functionals that one can construct. Open Problem 15.11 might be related to this question. • Is there a transport-based proof of the famous Lévy–Gromov isoperimetric inequalities (Open Problem 21.16), that would not involve so much “hard analysis” as the currently known arguments? Besides its intrinsic interest, such a proof could hopefully be adapted to nonsmooth spaces such as the weak CD(K, N) spaces studied in Part III. • Caffarelli’s log concave perturbation theorem (alluded to in Chapter 2) is another riddle in the picture. The Gaussian space can be seen as the infinite-dimensional version of the sphere, which is the Riemannian “reference space” with positive constant (sectional) curvature; and the space Rn equipped with a log concave measure is a space of nonnegative Ricci curvature. So Caffarelli’s theorem can be restated n as follows: If the Euclidean space (R ,d2) is equipped with a probability measure ν that makes it a CD(K, ∞) space, then ν can be realized as a 1-Lipschitz push-forward of the reference Gaussian measure with curvature K. This implies almost obviously that isoperimetric inequalities in n (R ,d2,ν) are not worse than isoperimetric inequalities in the Gaussian space; so there is a strong analogy between Caffarelli’s theorem on the one hand, and the Lévy–Gromov isoperimetric inequality on the other hand. It is natural to ask whether there is a common framework for both results; this does not seem obvious at all, and I have not been able to formulate even a decent guess of what could be a geometric generalization of Caffarelli’s theorem. • Another important remark is that the geometric theory has been almost exclusively developed in the case of the optimal transport with quadratic cost function; the exponent p = 2 here is natural in the context of Riemannian geometry, but working with other exponents (or Conclusions and open problems 907 with radically different Lagrangian cost functions) might lead to new geometric territories. An illustration is provided by the recent work of Shin-ichi Ohta in Finsler geometry. A related question is Open Problem 15.12. In Part III of these notes, I discussed the emerging theory of weak Ricci curvature lower bounds in metric-measure spaces, based on displacement convexity inequalities. The theory has grown very fast and is starting to be rather well-developed; however, some challenging issues remain to be solved before one can consider it as mature. Here are three missing pieces of the puzzle: • A globalization theorem that would play the role of the Toponogov–Perelman theorem for Alexandrov spaces with a lower bound on the curvature. This result should state that a weak local CD(K, N) space is automatically a weak CD(K, N) space. Theorem 30.37 shows that this is true at least if K =0,N<∞ and X is nonbranching; if Conjecture 30.34 turns out to be true, the same result will be available for all values of K. • The compatibility with the theory of Alexandrov spaces (with lower curvature bounds). Alexandrov spaces have proven their flexibility and have gained a lot of popularity among geometers. Since Alexandrov bounds are weak sectional curvature bounds, they should in principle be able to control weak Ricci curvature bounds. The natural question here can be stated as follows: Let (X ,d) be a finite-dimensional Alexandrov space with dimension n and curvature bounded below by κ, and let Hn be the n-dimensional Hausdorff measure on X ;is(X ,d,Hn) aweakCD((n − 1)κ, n) space? • A thorough discussion of the branching problem: Find examples of weak CD(K, N) spaces that are branching; that are singular but nonbranching; identify simple regularity conditions that prevent branching; etc. It is also of interest to enquire whether the nonbranching assumption can be dispensed with in Theorems 30.26 and 30.37 (recall Remarks 30.27 and 30.39). More generally, we would like to have more information about the structure of weak CD(K, N) spaces, at least when N is finite. It is known from the work of Jeff Cheeger and others that metric-measures spaces in which the measure is (locally) doubling and satisfies a (local) Poincaré inequality have at least some little bit of regularity: There is a tangent space defined almost everywhere, varying in a measurable way. 908 Conclusions and open problems

In the context of Alexandrov spaces with curvature bounded below, some rather strong structure theorems have been established by Grigori Perelman and others; it is natural to ask whether similar results hold true for weak CD(K, N) spaces. Another relevant problem is to check the compatibility of the CD(K, N) condition with the operations of quotient by Lie group actions, and lifting to the universal covering. As explained in the bibliographical notes of Chapter 30, only partial results are known in these directions. Besides these issues, it seems important to ﬁnd further examples of weak CD(K, N) spaces, apart from the ones presented in Chapter 29, mostly constructed as limits or quotients of manifolds. It was realized in a recent Oberwolfach meeting, as a consequence of discussions between Dario Cordero-Erausquin, Karl-Theodor Sturm and myself, that the Euclidean space Rn, equipped with any norm ·,isaweakCD(0,n) space:

Theorem. Let · be a norm on Rn (considered as a distance on n n R × R ), and let λn be the n-dimensional Lebesgue measure. Then n the metric-measure space (R , ·,λn) is a weak CD(0,n) space in the sense of Deﬁnition 29.8.

I did not include this theorem in the body of these notes, because it appeals to some results that have not yet been adapted to a genuinely geometric context, and which I preferred not to discuss. I shall sketch the proof at the end of this text, but first I would like to explain why this result is at the same time motivating, and a bit shocking: (a) As pointed out to me by John Lott, if · is not Euclidean, n then the metric-measure space (R , ·,λn) cannot be realized as a limit of smooth Riemannian manifolds with a uniform CD(0,N) bound, because it fails to satisfy the splitting principle. (If a nonnegatively curved space admits a line, i.e. a geodesic parametrized by R, then the space can be “factorized” by this geodesic.) Results by Jeff Cheeger, Toby Colding and Detlef Gromoll say that the splitting principle holds for CD(0,N) manifolds and their measured Gromov–Hausdorff limits. (b) If · is not the Euclidean norm, the resulting metric space is very singular in certain respects: It is in general not an Alexandrov space, and it can be extremely branching. For instance, if ·is the ∞ norm, then any two distinct points are joined by an uncountable Conclusions and open problems 909

Rn · infinity of geodesics. Since ( , ∞ ,λn) is the (pointed) limit of the Rn · →∞ nonbranching spaces ( , p ,λn)asp , we also realize that weak CD(K, N) bounds do not prevent the appearance of branching in measured Gromov-Hausdorff limits, at least if K ≤ 0. On the other hand, the study of optimal Sobolev inequalities in Rn which I performed together with Bruno Nazaret and Dario Cordero- Erausquin shows that optimal Sobolev inequalities basically do not depend on the choice of the norm on Rn. In a Riemannian context, Sobolev inequalities strongly depend on Ricci curvature bounds; so, our result suggests that it is not absurd to decide that Rn isaweakCD(0,n) space independently of the norm. Shin-ichi Ohta has developed this point of view by studying curvature-dimension conditions in certain classes of Finsler spaces. One can also ask whether there are additional regularity conditions that might be added to the definition of weak CD(K, N) space, in order to enforce nonbranching, or the splitting principle, or both, and in particular rule out non-Euclidean norms. As a side consequence of point (a) above, we realize that smooth CD(K, N) manifolds are not dense in the spaces CDD(K, N, D, m, M) introduced in Theorem 29.32. The interpretation of dissipative equations as gradient flows with respect to optimal transport, and the theory reviewed in Chapters 23 to 25, also lead to fascinating issues that are relevant in smooth or nonsmooth geometry as well as in partial differential equations. For instance, (a) Can one define a reasonably well-behaved heat flow on weak CD(K, N) spaces by taking the gradient flow for Boltzmann’s H functional? The theory of gradient flows in abstract metric spaces has been pushed very far, in particular by Luigi Ambrosio, Giuseppe Savaréand collaborators; so it might not be so difficult to define an object that would play the role of a heat semigroup. But this will be of limited value unless one can prove relevant theorems about this object. Shin-ichi Ohta, and independently Giuseppe Savaré, recently made progress in this direction by constructing gradient flows in the Wasser- stein space over a finite-dimensional Alexandrov space of curvature bounded below, or over more general spaces satisfying very weak regularity assumptions expressed in terms of distances and angles. In the particular case when the energy functional is the Boltzmann entropy, this provides a natural notion of heat equation and heat semigroup. 910 Conclusions and open problems

Savaré uses a very elegant argument, based on properties of Wasserstein distances and entropy, to prove the linearity of this semigroup, and other properties as well (positivity, contraction in Wp for 1 ≤ p ≤ 2, contraction in Lp, some regularizing effect). This problem might be related to the possibility of defining a Laplace operator on a singular space, an issue which has been addressed in particular by Jeff Cheeger and Toby Colding, for limits of Riemannian manifolds. However, their construction is strongly based on regularity properties enjoyed by such limits, and breaks down, e.g., for Rn equipped with a non-Euclidean norm ·. In fact, as noted by Karl- N Theodor Sturm, the gradient flow of the H functional in P2((R , ·)) yields a nonlinear evolution. The fact that this equation has the same fundamental solution (in the sense of a solution evolving from a Dirac mass) as the Euclidean one is one argument to believe that this is a natural notion of heat equation on the non-Euclidean RN —andmay reinforce us in the conviction that this space deserves its status of weak CD(0,N) space. (b) Can one extend the theory of dissipative equations to other equations, which are of Hamiltonian, or, even more interestingly, of dissipative Hamiltonian nature? As explained in the bibliographical notes of Chapter 23, there has been some recent work in that direction by Luigi Ambrosio, Wilfrid Gangbo and others, however the situation is still far from clear. A loosely related issue is the study of the semi-geostrophic system, which in the simplest situations can formally be written as a Hamil- tonian flow, where the Hamiltonian function is the square Wasserstein distance with respect to some uniform reference measure. I think that the rigorous qualitative understanding of the semi-geostrophic system is one of the most exciting problems that I am aware of in theoretical fluid mechanics; and discussions with Mike Cullen convinced me that it is very relevant in applications to meteorology. Although the theory of the semi-geostrophic system is still full of fundamental open problems, enough has already been written on it to make the substance of a complete monograph. On a much more theoretical level, the geometric understanding of the Wasserstein space P2(X ), where X is a Riemannian manifold or just a geodesic space, has been the object of several recent studies, and still retains many mysteries. For instance, there is a neat statement accord- ingtowhichP2(X ) is nonnegatively curved, in the sense of Alexandrov, Conclusions and open problems 911 if and only if X itself is nonnegatively curved. But there is no similar statement for nonzero lower bounds on the curvature! In fact, if x is a point of negative curvature, then the curvature of P2(X ) seems to be unbounded in both directions (+∞ and −∞) in the neighborhood of δx. Also it is not clear what exactly is “the right” structure on, say, n P2(R ); recent works on the subject have suggested differing answers. Another relevant open problem is whether there is a natural “volume” measure on P2(M). Karl-Theodor Sturm and Max-Kostja von Renesse have recently managed to construct a natural one-parameter family of 1 “Gibbs” probability measures on P2(S ). A multi-dimensional generalization would be of great interest. In their book on gradient flows, Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré make an intriguing observation: One may define “gen- n eralized geodesics” in P2(R ) by considering the law of (1−t) X0 +tX1, where (X0,Z)and(X1,Z) are optimal couplings. These generalized geodesics have intriguing properties: For instance, they still satisfy the characteristic displacement interpolation inequalities; and they provide curves of “nonpositive curvature”, that can be exploited for various pur- poses, such as error estimates for approximate gradient flow schemes. It is natural to further investigate the properties of these objects, which are reminiscent of the c-segments considered in Chapter 12. The list above provides but a sample among the many problems that remain open in the theory of optimal transport. As I already mentioned in the preface, another crucial issue which I did not address at all is the numerical analysis of optimal transport. This topic also has a long and complex history, with some famous schemes such as the old simplex algorithm, described for instance in Alexander Schrijver’s monograph Combinatorial Optimization: Polyhe- dra and Efficiency; or the more recent auction algorithm developed by Dimitri Bertsekas. Numerical schemes based on Monge–Ampère equations have been suggested, but hardly implemented yet. Recent works by Uriel Frisch and collaborators in cosmology provide an example where one would like to efficiently solve the optimal transport problem with huge sets of data. To add to the variety of methods, continuous schemes based on partial differential equations have been making their way lately. All in all, this subject certainly deserves a systematic study on its own, with experiments, comparisons of algorithms, benchmark problems and so forth. By the way, the optimum matching problem 912 Conclusions and open problems is one of the topics that Donald Knuth has planned to address in his long-awaited Volume 4 of The Art of Computer Programming. Needless to say, the theory might also decide to explore new horizons which I am unable to foresee.

Sketch of proof of the Theorem. First consider the case when N = · is a uniformly convex, smooth norm, in the sense that

2 2 λIn ≤∇ N ≤ ΛIn for some positive constants λ and Λ. Then the cost function c(x, y)= N(x − y)2 is both strictly convex and C1,1, i.e. uniformly semiconcave. This makes it possible to apply Theorem 10.28 (recall Example 10.35) and deduce the following theorem about the structure of optimal maps: If µ0 and µ1 are compactly supported and absolutely continuous, then there is a unique optimal transport, and it takes the form

T (x)=x −∇(N 2)∗(−∇ψ(x)),ψa c-convex function.

Since the norm is uniformly convex, geodesic lines are just straight lines; so the displacement interpolation takes the form (Tt)#(ρ0 λn), where 2 ∗ Tt(x)=x − t ∇(N ) (−∇ψ(x)) 0 ≤ t ≤ 1. Let θ(x)=∇(N 2)∗(−∇ψ(x)). By [814, Remark 2.56], the Jacobian matrix ∇θ, although not symmetric, is pointwise diagonalizable, with eigenvalues bounded above by 1 (this remark goes back at least to a 1996 preprint by Otto [666, Proposition A.4]; a more general statement 1/n is in [30, Theorem 6.2.7]). It follows easily that t → det(In − t∇θ) is a concave function of t [814, Lemma 5.21], and one can reproduce ∈DC the proof of displacement convexity for Uλn ,assoonasU n [814, Theorem 5.15 (i)]. n This shows that (R ,N,λn) satisﬁes the CD(0,n) displacement convexity inequalities when N is a smooth uniformly convex norm. Now if N is arbitrary, it can be approximated by a sequence (Nk)k∈N of smooth n uniformly convex norms, in such a way that (R ,N,λn, 0) is the pointed n measured Gromov–Hausdorﬀ limit of (R ,Nk,λn, 0) as k →∞.Then the general conclusion follows by stability of the weak CD(0,n) criterion (Theorem 29.24).

n Remark. In the above argument the spaces (R ,Nk,λn) satisfy the property that the displacement interpolation between any two absolutely Conclusions and open problems 913 continuous, compactly supported probability measures is unique, while n the limit space (R ,N,λn) does not necessarily satisfy this property. For · instance, if N = ∞ , there is an enormous number of displacement interpolations between two given probability measures; and most of them do not satisfy the displacement convexity inequalities that are used to define CD(0,n) bounds. This shows that if in Definition 29.8 one requires the inequality (29.11) to hold true for any Wasserstein geodesic, rather thanforsomeWassersteingeodesic,thentheresultingCD(K, N)property is not stable under measured Gromov–Hausdorff convergence. References

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Coupling...... 5 Deterministiccoupling...... 6 Existence of an optimal coupling ...... 43 Lower semicontinuity of the cost functional ...... 43 Tightness of transference plans ...... 44 Optimality is inherited by restriction ...... 46 Convexity of the optimal cost ...... 47 Cyclical monotonicity ...... 52 c-convexity ...... 54 c-concavity...... 56 Alternative characterization of c-convexity...... 57 Kantorovich duality ...... 57 Restriction of c-convexity...... 75 Restriction for the Kantorovich duality theorem ...... 75 Stability of optimal transport ...... 77 Compactness of the set of optimal plans ...... 77 Measurable selection of optimal plans ...... 78 Stability of the transport map ...... 79 Dual transport inequalities ...... 80 Criterion for solvability of the Monge problem ...... 84 Wasserstein distances ...... 93 Kantorovich–Rubinsteindistance...... 94 Wassersteinspace...... 94 Weak convergence in Pp ...... 96 Wp metrizes Pp ...... 96 Continuity of Wp ...... 97 Metrizability of the weak topology ...... 97 Cauchy sequences in Wp are tight ...... 99

957 958 List of short statements

Wasserstein distance is controlled by weighted total variation ...... 103 Topology of the Wasserstein space ...... 104 Classical conditions on a Lagrangian function ...... 118 Lagrangian action ...... 121 Coercive action ...... 122 Properties of Lagrangian actions ...... 123 Dynamical coupling ...... 126 Dynamical optimal coupling ...... 126 Displacement interpolation ...... 127 Displacement interpolation as geodesics ...... 127 Uniqueness of displacement interpolation ...... 128 Wp-Lipschitz continuity of p-moments ...... 137 Interpolation from intermediate times and restriction ...... 138 Nonbranching is inherited by the Wasserstein space ...... 139 Hamilton-Jacobi-Hopf-Lax-Oleinik evolution semigroup ...... 143 Elementary properties of Hamilton–Jacobi semigroups ...... 144 Interpolation of prices ...... 146 Mather’sshorteninglemma...... 166 Mather’s shortening lemma again ...... 167 The transport from intermediate times is locally Lipschitz ...... 168 Absolute continuity of displacement interpolation ...... 170 Focalization is impossible before the cut locus ...... 180 Lipschitzgraphtheorem...... 182 Useful transport quantities describing a Lagrangian system ...... 188 Mather critical value and stationary Hamilton–Jacobi equation . . . . . 189 Aroughnonsmoothshorteninglemma...... 194 Shortening lemma for power cost functions ...... 195 Conditions for single-valued subdifferentials ...... 206 Solution of the Monge problem, I ...... 208 Monge problem for quadratic cost, first result ...... 209 Non-connectedness of the c-subdifferential ...... 210 Differentiability ...... 217 Approximate differentiability ...... 218 Lipschitzcontinuity...... 220 Subdifferentiability, superdifferentiability ...... 220 Sub- and superdifferentiability imply differentiability ...... 221 Regularity and differentiability almost everywhere ...... 222 Semiconvexity...... 228 Local equivalence of semiconvexity and subdifferentiability ...... 229 Properties of Lagrangian cost functions ...... 235 c-subdifferentiability of c-convex functions ...... 239 List of short statements 959

Subdifferentiability of c-convex functions ...... 239 Differentiability of c-convex functions ...... 239 Solution of the Monge problem II ...... 243 Solution of the Monge problem without conditions at infinity ...... 247 Solution of the Monge problem for the square distance ...... 249 Solution of the Monge problem with possibly infinite total cost . . . . 250 Generalized Monge problem for the square distance ...... 256 Tangentcone...... 257 Countable rectifiability ...... 257 Sufficient conditions for countable rectifiability...... 257 Clarke subdifferential ...... 261 Nonsmooth implicit function theorem...... 261 Implicit function theorem for two subdifferentiable functions...... 262 Jacobian equation ...... 274 Changeofvariables ...... 275 An example of discontinuous optimal transport ...... 283 A further example of discontinuous optimal transport ...... 285 Smoothness needs Assumption (C) ...... 288 c-segment...... 291 c-convexity...... 291 regular cost function ...... 292 Reformulation of regularity ...... 293 Nonregularity implies nondensity of differentiable c-convex functions ...... 296 Nonsmoothness of the Kantorovich potential ...... 297 c-secondfundamentalform...... 300 Ma–Trudinger–Wang tensor, or c-curvature operator ...... 300 c-exponential ...... 301 Loeper’sidentity...... 301 Differential formulation of c-convexity...... 303 Differential formulation of regularity ...... 304 Equivalence of regularity conditions ...... 311 Smoothness of the optimal transport needs nonnegative curvature . . 312 Differential criterion for c-convexity...... 313 Control of c-subdifferential by c-convexityoftarget...... 316 Caffarelli’s regularity theory ...... 318 Urbas–Trudinger–Wang regularity theory...... 318 Loeper–Ma–Trudinger–Wangregularitytheory...... 318 Caffarelli’s interior a priori estimates ...... 320 Loeper–Ma–Trudinger–Wang interior a priori estimates ...... 320 Smoothness of optimal transport on Sn−1 ...... 322 960 List of short statements

Standard approximation scheme ...... 337 Regularization of singular transport problems ...... 338 C2-small functions are d2/2-convex...... 341 Representation of Lipschitz paths in P2(M) ...... 344 Second differentiability of semiconvex functions ...... 363 CD(K, N)curvature-dimensionbound...... 386 One-dimensional CD(K, N)modelspaces...... 387 Integral reformulation of curvature-dimension bounds ...... 389 Curvature-dimension bounds with direction of motion taken out . . . 390 Curvature-dimension bounds by comparison ...... 392 Barycenters...... 393 Distortion coefficients ...... 393 Computation of distortion coefficients ...... 395 Reference distortion coefficients ...... 396 Distortion coefficients and concavity of Jacobian determinant ...... 398 Ricci curvature bounds in terms of distortion coefficients ...... 398 Alexandrov’s second differentiability theorem...... 402 One-dimensional comparison for second-order inequalities ...... 409 Jacobi matrices have symmetric logarithmic derivatives ...... 412 Cosymmetrization of Jacobi matrices ...... 412 Jacobi matrices with positive determinant ...... 413 GradientformulainWassersteinspace...... 424 HessianformulainWassersteinspace...... 425 Convexityinageodesicspace...... 435 Convexity and lower Hessian bounds ...... 437 Λ-convexity...... 439 Displacementconvexity...... 441 Displacementconvexityclasses...... 449 Behavior of functions in DCN ...... 451 Moment conditions make sense of Uν (µ) ...... 461 Localdisplacementconvexity...... 464 CD bounds read off from displacement convexity ...... 465 CD(K, ∞) and CD(0,N) bounds via optimal transport ...... 466 Necessary condition for displacement convexity ...... 473 Finiteness of time-integral in displacement convexity inequality . . . . 473 Distorted Uν functional ...... 476 β Domain of definition of Uπ,ν ...... 477 (K,N) βt Definition of Uπ,ν ...... 478 β Definition of Uπ,ν in the limit cases...... 479 Distorteddisplacementconvexity ...... 480 CD bounds read off from distorted displacement convexity ...... 481 List of short statements 961

One-dimensional CD bounds and displacement convexity ...... 486 Intrinsicdisplacementconvexity ...... 486 Doublingproperty...... 493 Doublingmeasureshavefullsupport ...... 494 Distorted Brunn–Minkowski inequality...... 495 Brunn–Minkowski inequality in nonnegative curvature ...... 496 Bishop–Gromov inequality ...... 499 CD(K, N)impliesdoubling...... 501 Dimension-free control on the growth of balls ...... 502 Local Poincaré inequality ...... 505 CD(K, N) implies pointwise bounds on displacement interpolants. . . 507 Preservation of uniform bounds in nonnegative curvature ...... 507 Jacobian bounds revisited ...... 512 Intrinsic pointwise bounds on the displacement interpolant ...... 513 Democratic condition ...... 515 CD(K, N)impliesDm...... 515 Doubling + democratic imply local Poincaré ...... 517 CD(K, N) implies local Poincaré...... 517 Prékopa–Leindler inequalities ...... 520 Finite-dimension distorted Prékopa–Leindler inequality ...... 521 Differentiating an energy along optimal transport ...... 526 Generalized Fisher information ...... 531 Fisher information ...... 531 Distorted HWI inequality ...... 531 HWI inequalities ...... 533 Logarithmic Sobolev inequality ...... 546 Bakry–Emerytheorem...... 547´ Sobolev-L∞ interpolation inequalities ...... 548 Sobolev inequalities from CD(K, N)...... 550 Sobolev inequalities in Rn ...... 551 CD (K, N) implies L1-Sobolev inequalities...... 553 Poincaré inequality ...... 556 Exponential measure ...... 556 Lichnerowicz’s spectral gap inequality ...... 557 Tp inequality ...... 569 Dual formulation of Tp ...... 570 Dual formulation of T1 ...... 570 Tensorization of Tp ...... 570 T2 inequalities tensorize exactly ...... 570 Additivity of entropy ...... 574 Gaussian concentration ...... 575 962 List of short statements

CKP inequality ...... 582 CD (K, ∞) implies T2(K) ...... 583 Some properties of the quadratic Hamilton–Jacobi semigroup ...... 584 Logarithmic Sobolev ⇒ T2 ⇒ Poincaré ...... 585 T2 sometimes implies log Sobolev ...... 590 T2 and dimension free Gaussian concentration ...... 590 Quadratic-linearcost...... 593 Reformulations of Poincaré inequalities ...... 593 From generalized log Sobolev to transport to generalized Poincaré . . 594 Measure concentration from Poincaré inequality ...... 600 Product measure concentration from Poincaré inequality ...... 601 Finite-dimensional transport-energy inequalities ...... 605 Further finite-dimensional transport-energy inequalities ...... 606 Some properties of the Hamilton–Jacobi semigroup on a manifold. . . 610 Reformulations of gradient flows ...... 631 Locally absolutely continuous paths ...... 635 Gradientflowsinageodesicspace...... 635 Derivative of the Wasserstein distance ...... 636 Computation of subdifferentials in Wasserstein space ...... 649 Displacement convexity of H: above-tangent formulation ...... 650 Diffusion equations as gradient flows in the Wasserstein space . . . . . 672 Heat equation as a gradient flow ...... 674 Stability of gradient flows in the Wasserstein space ...... 675 Differentiation through doubling of variables ...... 678 Computations for gradient flow diffusion equations ...... 694 Integrated regularity for gradient flows...... 698 Equilibration in positive curvature ...... 699 Short-time regularization for gradient flows ...... 705 Infinite-dimensional Sobolev inequalities from Ricci curvature ...... 721 Bakry–Emery´ theorem again ...... 721 Generalized Sobolev inequalities under Ricci curvature bounds . . . . . 722 Sobolev inequalities ...... 722 From Sobolev-type inequalities to concentration inequalities ...... 723 From Log Sobolev to Talagrand ...... 724 Metric couplings as semi-distances ...... 748 Metricgluinglemma ...... 748 Approximate isometries converge to isometries ...... 751 Gromov–Hausdorff convergence ...... 752 Convergence of geodesic spaces ...... 754 Compactness criterion in Gromov–Hausdorff topology ...... 754 Local Gromov–Hausdorff convergence ...... 755 List of short statements 963

Geodesic local Gromov–Hausdorff convergence ...... 755 Pointed Gromov–Hausdorff convergence ...... 756 Blow-up...... 757 Ascoli theorem in Gromov–Hausdorff converging sequences ...... 759 Prokhorov theorem in Gromov–Hausdorff converging sequences . . . . 759 Compactness of locally finite measures ...... 760 Doubling lets metric and metric-measure approaches coincide ...... 764 d GP convergence and doubling imply d GHP convergence ...... 765 Doubling implies uniform total boundedness ...... 766 Measured Gromov–Hausdorff topology ...... 767 Compactness in measured Gromov–Hausdorff topology ...... 768 Gromov’sprecompactnesstheorem...... 769 Kinetic energy ...... 774 Regularityofthespeedfield...... 775 If Xk converges then P2(Xk)also...... 777 If f is an approximate isometry then f# also...... 778 Optimal transport is stable under Gromov–Hausdorff convergence . . 780 Gromov–Hausdorff stability of the dual Kantorovichproblem...... 790 Pointed convergence of Xk implies local convergence of P2(Xk).....791 Integral functionals for singular measures ...... 798 Rewriting of the distorted Uν functional ...... 799 Rescaled subadditivity of the distorted Uν functionals ...... 799 Weak curvature-dimension condition ...... 801 Smooth weak CD(K, N) spaces are CD(K, N) manifolds ...... 801 Consistency of the CD(K, N) conditions ...... 802 Bonnet–Myers diameter bound for weak CD(K, N)spaces...... 803 Sufficient condition to be a weak CD(K, N)space...... 803 Legendre transform of a real-valued convex function ...... 808 Legendre representation of Uν ...... 809 β Continuity and contraction properties of Uν and Uπ,ν ...... 809 Another sufficient condition to be a weak CD(K, N)space ...... 825 Stability of weak CD(K, N) under MGH ...... 826 Stability of weak CD(K, N) under pMGH ...... 826 Smooth MGH limits of CD(K, N) manifolds are CD(K, N) ...... 832 Compactness of the space of weak CD(K, N)spaces...... 833 Regularizingkernels...... 834 Separability of L1(C)...... 836 Elementary consequences of weak CD(K, N) bounds...... 848 Restriction of the CD(K, N)propertytothesupport...... 848 β Domain of definition of Uν and Uπ,ν on noncompact spaces ...... 851 964 List of short statements

Displacement convexity inequalities in weak CD(K, N) spaces . . . . . 852 Lower semicontinuity of Uν again ...... 853 Brunn–Minkowski inequality in weak CD(K, N)spaces...... 862 Nonatomicity of the support...... 863 Exhaustion by intermediate points ...... 863 Bishop–Gromov inequality in metric-measure spaces ...... 864 Measure of small balls in weak CD(K, N)spaces...... 865 Dimension of weak CD(K, N)spaces...... 865 Weak CD(K, N) spaces are locally doubling ...... 865 Unique geodesics in nonbranching CD(K, N)spaces...... 866 Regularity of interpolants in weak CD(K, N)spaces...... 868 Uniform bound on the interpolant in nonnegative curvature ...... 868 HWI and log Sobolev inequalities in weak CD(K, ∞)spaces...... 871 Sobolev inequality in weak CD(K, N)spaces ...... 872 Global Poincaré inequalities in weak CD(K, N)spaces...... 873 Local Poincaré inequalities in nonbranching CD(K, N) spaces . . . . . 874 Talagrand inequalities and weak curvature bounds ...... 875 Hamilton–Jacobi semigroup in metric spaces ...... 876 Equivalent definitions of CD(K, N) in nonbranching spaces ...... 877 Local-to-global CD(K, N) property along a path ...... 888 Local CD(K, N)space...... 889 From local to global CD(K, N) ...... 889 From local to global CD(K, ∞) ...... 894 Cutoff functions ...... 897 List of figures

1.1 Construction of the Knothe–Rosenblatt map ...... 9

3.1 Monge’s problem of d´eblais and remblais ...... 30 3.2 Economic illustration of Monge’s problem ...... 30

4.1 Monge approximation of a genuine Kantorovich optimal plan . . 48

5.1 An attempt to improve the cost by a cycle ...... 52 5.2 c-convex function ...... 55

8.1 Monge’sshorteninglemma ...... 164 8.2 The map from the intermediate point is well-deﬁned ...... 169 8.3 Principle of the proof of Mather’s shortening lemma ...... 171 8.4 ShortcutsinMather’sproof...... 174 8.5 Oscillations of a pendulum ...... 189

9.1 How to prove that the subdiﬀerential is single-valued ...... 208 9.2 Nonexistence of Monge transport, nonuniqueness of optimal coupling...... 210

10.1 Singularities of the distance function ...... 217 10.2 k-dimensionalgraph...... 261

12.1 Caﬀarelli’s counterexample ...... 284 12.2 Principle of Loeper’s counterexample ...... 286 12.3 Regular cost function ...... 294

14.1 The Gauss map ...... 358 14.2 Parallel transport ...... 365

965 966 List of ﬁgures

14.3 Jacobi ﬁelds ...... 366 14.4 Distortion by curvature ...... 394 14.5 Distortion by curvature, again ...... 395 14.6 Model distortion coeﬃcients ...... 397

16.1 The one-dimensional Green function ...... 437 16.2 The lazy gas experiment ...... 446

17.1 Approximation of an element of DCN ...... 453

18.1 An example of a measure that is not doubling ...... 494

26.1 Triangles in a nonnegatively curved world ...... 738

27.1 Principle of the definition of the Hausdorff distance ...... 744 27.2 An example of Gromov–Hausdorff convergence ...... 753 27.3 Approximate isometries cannot in general be continuous ...... 753 27.4 An example of reduction of support ...... 761 Index

absolute continuity contraction principle, 810 of a curve, 115, 635 convergence of a measure, XXI geodesic Gromov–Hausdorff, 755 action, 114, 121 Gromov–Hausdorff, 752 coercive, 122, 127 local Gromov–Hausdorff, 755 Alexandrov space, 193, 737, 739 measured Gromov–Hausdorff, 767 Alexandrov theorem, 363, 402 pointed Gromov–Hausdorff, 756 Aronson–Bénilan estimates, 708, 716 Wasserstein space, 96 Assumption (C), 205, 212, 288, 311 weak, 97 Aubry set, 86, 188 convexity c-concavity, 56 ´ Bakry–Emery theorem, 547, 559, 721, c-convexity, 54, 291 727 c-transform, 55, 56 barycenter, 393 d2/2-convexity, 341 Bishop–Gromov inequality, 377, 499, in Rn,55 864 in P , 441 Bochner formula, 374, 419 2 in a geodesic space, 435 generalized, 383 in a Riemannian manifold, 437 Bonnet–Myers theorem, 378, 872 semiconvexity, 228 Brunn–Minkowski inequality, 495, 503 correspondence, 744 distorted, 495, 862 cost function, 10 Caffarelli perturbation theorem, 26, 28 quadratic, 163, 209, 336 change of variables formula, 12, 18, 274 quadratic-linear, 593, 594 Cheng–Toponogov theorem, 492 recapitulative table, 971 compactness counterexamples (to regularity) in Gromov–Hausdorff topology, 754 Caffarelli, 283 in measured Gromov–Hausdorff Loeper, 285, 297 topology, 768, 833 without Assumption (C), 288 competitivity (of price functions), 54 coupling, 5, 17 concentration of measure, 567, 600, 618 deterministic, 6 Gaussian, 575, 590 dynamical, 126 conservation of mass, 14, 19 exact (classical), 9 contact set, contact point, 88 Holley, 8, 18

967 968 Index

increasing rearrangement, 7 minimal, 107 Knothe–Rosenblatt, 8, 18 Toscani, 98 measurable, 7 total variation, 103 Moser, 7, 16, 18 Wasserstein, 93, 106 optimal, 10 weak-∗,98 trivial, 6 distortion coefficient, 393, 395 covariant derivative, 150 in model space, 396 critical value (Mather’s), 188 divergence, 362 Csiszár–Kullback–Pinsker inequality, doubling of variables, 678 582, 620 doubling property, 493, 501, 764 weighted, 580 curvature, 357 entropy, 424, 433, 622, 627, 695 c-curvature, 300 additivity, 574 Gauss, 358 generalized Ricci, 381 Euler equation (pressureless), 340, 373 Ricci, 357, 376 Euler–Lagrange equation, 116, 153 sectional, 193, 359 Eulerian point of view, 14, 373 curvature-dimension bound, 385, 389 exponential map, 153 and displacement convexity, 465, 481, Jacobian determinant, 364, 418 487, 852 exponential measure, 556, 600, 605 stability, 826 weak, 801, 838 fast diffusion equation, 694 cut locus, 181 Finsler structure, 155 and optimal transport, 349 FKG inequalities, 9, 18 cutoff function, 897 focal point, 181, 395 cyclical monotonicity, 52, 57, 89 and cut locus, 180 Fokker–Planck equation, 709 democratic condition, 515 formula differentiability, 217 Bochner, 374, 383 almost everywhere, 222, 363 change of variables, 18, 275 approximate, 13, 218, 269, 271 conservation of mass, 14, 19 sub- and superdifferentiability, 220 diffusion, 15, 20 diffusion equations, 672, 694 first variation, 152 and curvature-dimension bounds, 489 displacement convexity, 441, 446, 487, 852 Gaussian measure, 379, 567, 575, 590 class, 443, 449, 473, 487 geodesic distorted, 480 curve, 116, 153 intrinsic, 486 distance, 148 local, 464, 889 space, 120, 156 displacement interpolation, 126, 158 uniqueness, 154, 244, 866 equations of, 340 geodesic space, 754 distance gluing lemma, 18 bounded Lipschitz, 97 metric, 748 Fortet–Mourier, 97 gradient flow, 629, 631, 681 Gromov–Hausdorff, 746 in a geodesic space, 635 Hausdorff, 743 in Wasserstein space, 672, 694 Kantorovich–Rubinstein, 94 granular media, 711 Lévy–Prokhorov, 97, 744 Green function, 436, 441 Index 969

Gromov–Hausdorff topology, 746, 752, lazy gas, 445 754, 769 length, 119, 148 local, 755, 770 length space, 120, 155 measured, 767, 770 Li–Yau estimates, 378, 708, 716 pointed, 756, 770 linear programming, 31, 34 Lipschitz continuity, 220 Hamilton–Jacobi semigroup, 143, 375, Lipschitz graph theorem, 182 584, 876 locality, 888, 889, 899 Hamiltonian, 117, 145 log Sobolev inequality, 546, 558, 585, Hamiltonian equations, 689, 717 721, 871, 875 heat equation, 16, 674 generalized, 594 Hessian, 361, 437 Holley–Stroock theorem, 547 Ma–Trudinger–Wang tensor, 300, 311 HWI inequality, 529, 871 marginal, 5 hyperbolic space, 360, 738 Mather set, 86, 188 metric (Riemannian), 148 implicit functions, 261 metric coupling, 748 information metric-measure space, 760 Fisher, 433, 530, 685, 695, 871 midpoint, 156 Kullback (Cf. entropy), 81 Monge coupling, 10 interpolation Monge problem, 10 of laws, 126, 127, 158 for quadratic cost, 209 of prices, 143, 146 original, 29 isometry, 746 smoothness, 281 approximate, 750 solvability, 84, 208, 243, 247, 250, 267 isoperimetry Monge–Ampère equation, 282, 323 Euclidean, 23 Monge–Kantorovich problem, 10 Gaussian, 564 Moser coupling, 16, 18 isoperimetric inequality, 23, 27, 522, 545, 558, 563 no-crossing property, 139, 163, 167 Lévy–Gromov, 555, 564, 618 nonbranching property, 154, 877 Jacobi equation, 366 Jacobi field, 366 optimal coupling, 10 Jacobian determinant, 13, 274 compactness, 77 dynamical, 126, 779 Kantorovich duality, 57, 60, 85, 95, 146 existence, 43 Kantorovich–Rubinstein distance, 60, stability, 77, 780 94 Otto calculus, 421 kinetic energy, 774 kinetic theory, 35, 201, 433, 542, 689, parallel transport, 149, 365 710 Poincaré inequality, 593 Knothe–Rosenblatt coupling, 8, 18 generalized, 594 global, 378, 556, 585, 600, 873 Lagrangian, 114, 235 local, 505, 517, 874 classical conditions, 118 polar factorization theorem, 35 Lagrangian point of view, 14 Polish space, XX Langevin process, 21 porous medium equation, 694 Laplacian, 361 Prékopa–Leindler inequality, 520 970 Index pressure and iterated pressure, 422 Talagrand inequality, 569, 583, 585, Prokhorov theorem, 43, 759 723, 875 tangent cone, 257 Rademacher’s theorem, 222, 898 total variation, 582 rearrangement, 7 transference plan, 10 rectifiability, 257 dynamical, 126, 779 regular cost function, 292, 311 generalized optimal, 250 regularity theory, 317 optimal, 10 regularizing kernel, 834, 845 transform restriction property, 46 c-transform, 55, 56 dual side, 75 Legendre, 55, 809 Riemannian manifold, 148 transport inequality, 80, 569, 605 transport map, 6 stability, 79 second fundamental form, 300, 304 twist condition, 216, 267 selection theorem, 92 strong, 299 semi-distance, 747 semi-geostrophic system, 35 Vlasov equation, 27 shortening principle, 163, 166, 199 volume (Riemannian), 152 Sobolev inequality, 378, 545, 548, 553, 722, 872 Wasserstein distance, 93, 106 logarithmic, 546 Wasserstein space, 94, 96, 104 Spectral gap inequality, 378 differential structure, 343, 636, 682 speed, 118, 775 geodesics, 127 stochastic mechanics, 159 gradient flows, 672 subdifferential Gromov–Hausdorff convergence, 777 c-subdifferential, 55, 88 representation of curves, 344 Clarke, 261 weak CD(K, N) space, 801, 838 synthetic point of view, 736 weak KAM theory, 189, 201 Some notable cost functions

In the following table I have compiled a few remarkable cost functions which have appeared in various applications. The list is by no means exhaustive and the suggested references should be considered just as entry points to the corresponding literature.

971 972 Some notable cost functions [484] [328] [501] [269] [660] [659] [712] [268] [164] [399] [20, 636] [419, 834] [520, 788] Chapter 6 Chapter 6 [39, 40, 164] [493, 669, 671] [814, Chapter 6] Where quoted [814, Appendix 1.4] [814, Appendix 1.4] [777] [814, Section 7.5] [24, 154, 156, 159] [814, Section 3.2] n R Use relativistic theory -Wasserstein distances modeling in economy p relativistic heat equation (flat) conformal geometry Strassen’s duality theorem semi-geostrophic equations semi-geostrophic equations Kantorovich’s cost function Monge’s original cost function representation of total variation Far-field reflector antenna design Near-field reflector antenna design Rubinstein–Wolansky’s design of lens prescribed integral curvature problem Tanaka’s study of Boltzmann equation Brenier’s study of incompressible fluids definition of Kantorovich distance/norm diffusion equations of Fokker–Planck type most useful for geometric applications in Hsu–Sturm’s maximal coupling of Brownian paths shape optimization, sandpile growth, compression molding 2 S ∈ 3 y 2 R R 3 2 3 3 2 n n n R or S R R R R S R R or 2 R Setting R surface, Polish space Polish space Polish space Polish space ∈ x 2 ) 2 1 ) 2 y | y ) y 2 · ) y | − | | ) · ) y y − 2 y ε + 3 x 2 p | x ) x y | ( ≥ − x ) − y <β< y − ) y − x x − = 0 x | | − +( −| x, y | x, y − x − , x,y x, y 3 ( ( 1 2 α Cost ( x 1 β x x d | ) | d d | log ( 1 1 y 1+ log f log(1 y − erf( log(1 − − − − x − 1 | 1 x ( Some notable cost functions 973 [576, 782] [24, 25, 155] Where quoted [774] and Chapter 22 [246, 616] and Part II [577, 762, 763] and Part III [104, 105, 601] and Chapter 8 Use Study of Ricci flow Riemannian geometry incompressible Euler equation Mather’s theory of Lagrangian mechanics Talagrand’s study of exponential concentration Lott–Sturm–Villani’s nonsmooth Ricci curvature bounds n R Setting subset of geodesic space product metric space Riemannian manifold Riemannian manifold Riemannian manifold ) 2 dt ) ) i dt ) ± ,y ) i L dt t, x x , ) ( 0 t, x ( L 2 2 ,d p ) ) ) i x, v, t − +Scal( ( ,y x, y x, y i 2 2 L ( ( Cost | | x 2 d d v ( v | | d inf min( and variants ( i inf inf 1 2 Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics A Selection

247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. Chow/Hale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampère Equations 253. Dwork: Lectures on ρ-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Hörmander: The Analysis of Linear Partial Differential Operators I 257. Hörmander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems 261. Bosch/Güntzer/Remmert: Non Archimedian Analysis – A System Approach to Rigid Analytic Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel’skiˇı/Zabreˇıko: Geometrical Methods of Nonlinear Analysis 264. Aubin/Cellina: Differential Inclusions 265. Grauert/Remmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds 267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I 268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. II 269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, Large Deviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 273. Nikol’skiˇı: Treatise on the shift Operator 274. Hörmander: The Analysis of Linear Partial Differential Operators III 275. Hörmander: The Analysis of Linear Partial Differential Operators IV 276. Liggett: Interacting Particle Systems 277. Fulton/Lang: Riemann-Roch Algebra 278. Barr/Wells: Toposes, Triples and Theories 279. Bishop/Bridges: Constructive Analysis 280. Neukirch: Class Field Theory 281. Chandrasekharan: Elliptic Functions 282. Lelong/Gruman: Entire Functions of Several Complex Variables 283. Kodaira: Complex Manifolds and Deformation of Complex Structures 284. Finn: Equilibrium Capillary Surfaces 285. Burago/Zalgaller: Geometric Inequalities 286. Andrianaov: Quadratic Forms and Hecke Operators 287. Maskit: Kleinian Groups 288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes 289. Manin: Gauge Field Theory and Complex Geometry 290. Conway/Sloane: Sphere Packings, Lattices and Groups 291. Hahn/O’Meara: The Classical Groups and K-Theory 292. Kashiwara/Schapira: Sheaves on Manifolds 293. Revuz/Yor: Continuous Martingales and Brownian Motion 294. Knus: Quadratic and Hermitian Forms over Rings 295. Dierkes/Hildebrandt/Küster/Wohlrab: Minimal Surfaces I 296. Dierkes/Hildebrandt/Küster/Wohlrab: Minimal Surfaces II 297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators 298. Berline/Getzler/Vergne: Heat Kernels and Dirac Operators 299. Pommerenke: Boundary Behaviour of Conformal Maps 300. Orlik/Terao: Arrangements of Hyperplanes 301. Loday: Cyclic Homology 302. Lange/Birkenhake: Complex Abelian Varieties 303. DeVore/Lorentz: Constructive Approximation 304. Lorentz/v. Golitschek/Makovoz: Construcitve Approximation. Advanced Problems 305. Hiriart-Urruty/Lemaréchal: Convex Analysis and Minimization Algorithms I. Fundamentals 306. Hiriart-Urruty/Lemaréchal: Convex Analysis and Minimization Algorithms II. Advanced Theory and Bundle Methods 307. Schwarz: Quantum Field Theory and Topology 308. Schwarz: Topology for Physicists 309. Adem/Milgram: Cohomology of Finite Groups 310. Giaquinta/Hildebrandt: Calculus of Variations I: The Lagrangian Formalism 311. Giaquinta/Hildebrandt: Calculus of Variations II: The Hamiltonian Formalism 312. Chung/Zhao: From Brownian Motion to Schrödinger’s Equation 313. Malliavin: Stochastic Analysis 314. Adams/Hedberg: Function spaces and Potential Theory 315. Bürgisser/Clausen/Shokrollahi: Algebraic Complexity Theory 316. Saff/Totik: Logarithmic Potentials with External Fields 317. Rockafellar/Wets: Variational Analysis 318. Kobayashi: Hyperbolic Complex Spaces 319. Bridson/Haefliger: Metric Spaces of Non-Positive Curvature 320. Kipnis/Landim: Scaling Limits of Interacting Particle Systems 321. Grimmett: Percolation 322. Neukirch: Algebraic Number Theory 323. Neukirch/Schmidt/Wingberg: Cohomology of Number Fields 324. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes 325. Dafermos: Hyperbolic Conservation Laws in Continuum Physics 326. Waldschmidt: Diophantine Approximation on Linear Algebraic Groups 327. Martinet: Perfect Lattices in Euclidean Spaces 328. Van der Put/Singer: Galois Theory of Linear Differential Equations 329. Korevaar: Tauberian Theory. A Century of Developments 330. Mordukhovich: Variational Analysis and Generalized Differentiation I: Basic Theory 331. Mordukhovich: Variational Analysis and Generalized Differentiation II: Applications 332. Kashiwara/Schapira: Categories and Sheaves. An Introduction to Ind-Objects and Derived Categories 333. Grimmett: The Random-Cluster Model 334. Sernesi: Deformations of Algebraic Schemes 335. Bushnell/Henniart: The Local Langlands Conjecture for GL(2) 336. Gruber: Convex and Discrete Geometry 337. Maz'ya/Shaposhnikova: Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators 338. Villani: Optimal Transport: Old and New