There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Semiconductor equations

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure.



The very general theory is illustrated by typical examples like (semi)classical limits of Schrodinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation. © 1997 John Wiley & Sons, Inc.

Homogenization limits and Wigner transforms

Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation

1. Introduction.- 2. Mathematical Modeling of Semiconductor Devices.- 3. Analysis of the Basic Stationary Semiconductor Device Equations.- 4. Singular Perturbation Analysis of the Stationary Semiconductor Device Problem.- 5. Discretisation of the Stationary Device Problem.- 6. Numerical Simulation - A Case Study.- Notation of Physical Quantities.- Mathematical Notation.- A. Device Geometry.- B. Scalars, Vectors and Matrices.- C. Functions.- D. Landau Symbols.- E. Linear Spaces of Functions and Norms.- F. Functionalanalytic Notations.

The Stationary Semiconductor Device Equations

It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view. In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model).

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Peter A. Markowich

Papers

Semiconductor equations

Homogenization limits and Wigner transforms

Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation

The Stationary Semiconductor Device Equations

On convex sobolev inequalities and the rate of convergence to equilibrium for fokker-planck type equations