From Classical Analysis to Analysis on Fractals

Patricia Alonso Ruiz is an Assistant Professor at Texas A&M University in College Station, US. She did her Ph.D. at the University of Siegen, Germany (2013), after getting her licentiate degree from the Universidad Complutense de Madrid, Spain. Her research mainly deals with analysis and probability on fractals, with a focus on function spaces, functional inequalities, semigroups, and Dirichlet forms.Michael Hinz obtained his doctoral degree from Friedrich Schiller University Jena, Germany, and currently works as Wissenschaftliche Mitarbeiter at Bielefeld University, Germany. His research areas are analysis and probability theory, and he is particularly interested in fractal structures and spaces.Kasso A. Okoudjou is a Professor of Mathematics at Tufts University, USA. He received his Ph.D. in Mathematics from the Georgia Institute of Technology and was an H. C. Wang Assistant Professor at Cornell University. He held positions at the University of Maryland–College Park, Technical University of Berlin, MSRI, and MIT. His research interests include applied and pure harmonic analysis especially time-frequency and time-scale analysis, frame theory, and analysis and differential equations on fractals.Luke G. Rogers has a Ph.D. from Yale University and is a Professor of Mathematics at the University of Connecticut.  His research is primarily in harmonic and functional analysis on metric measure spaces, especially those with fractal structure.Alexander Teplyaev is a Professor of Mathematics at the University of Connecticut, USA. He studied probability and mathematical physics in St. Petersburg and at Caltech, has a Ph.D. degree in mathematics from Cornell University, and was a postdoctoral researcher at McMaster University and the University of California with a National Science Foundation fellowship. He also was supported by the Alexander von Humboldt Foundation in Germany and by the Fulbright Program in France. Teplyaev studies irregular structures, such as random or aperiodic non-smooth media, graphs, groups, and fractals. His research deals with spectral, geometric, functional, and probabilistic analysis on singular spaces using symmetric Markov processes and Dirichlet form techniques

• Part I. Introduction to this volume.- From Strichartz Estimates to Differential Equations on Fractals.- Part II. Functional and harmonic analysis on Euclidean spaces.- A new proof of Strichartz estimates for the Schrödinger equation in 2 + 1 dimensions.- Modulational instability of classical water waves.- Convergence Analysis of the Deep Galerkin Method for Weak Solutions.- The 4-player gambler’s ruin problem.- Part III. Analysis on Manifolds.- Product Manifolds with Improved Spectral Cluster and Weyl Remainder Estimates.- A scalar valued Fourier transform for the Heisenberg group.- Asymptotic behavior of the heat semigroup on certain Riemannian manifolds.- Part IV. Intrinsic Analysis on Fractals.- Fourier Series for Fractals in 2 Dimensions.- Blowups and Tops of Overlapping Iterated Function Systems.- Estimates of the local spectral dimension of the Sierpinski gasket.- Heat kernel fluctuations for stochastic processes on fractals and random media.- Index.