**Research Field**

Applied analysis links mathematics and scientific computing to engineering and natural sciences. Modeling in the applied sciences typically leads to systems of nonlinear ordinary or partial differential equations or variational problems. The nonlinearities reflect complex phenomena which result in oscillations, concentrations, or singularities in the mathematical model. Most problems involve the interaction of processes on different time and length scales, leading to so-called multiscale systems. Resolution on all scales is usually impossible, which requires deriving hierarchies of models to describe the problem on the various scales.

Berlin applied analysis is highly visible internationally, with the most important areas being represented by leading scientists.

**Berlin Research Groups**

Several research projects of the **applied analysis** groups in Berlin are motivated by phase transformation phenomena (Dreyer, Kraus, Mielke, Sprekels). For example, the growth of crystals from a melt involves free boundaries, and damage processes in elastic materials have to be understood in their interaction with phase separation in solder alloys. Techniques from the calculus of variations are relevant in the study of local or global minimizers of certain energies. Often small parameters are involved, representing a specific regime of interest for the corresponding applications. The framework of **Γ** - convergence allows one to study the corresponding family of minimization problems and to identify the limit problem. Special interest lies in developing a theory of **Γ** - convergence for evolutionary problems. Since sharp-interface limits lead to lower-dimensional singular structures, there is a strong link to differential geometry, as represented by the former DFG collaborative research centre on Space, Time, Matter.

**Nonlinear dynamical systems** (Fiedler, Mielke, Recke, Yanchuk) are studied in Berlin, for example, in the context of nonlinear processes in opto-electronic devices, in spatiotemporal patterns in nonlinear excitable systems, or in synchronization and cluster formation in coupled neuron systems. Bifurcation and center manifold theory are the key analytical tools for the understanding of these processes. Due to the inherent multiple scales, singular perturbation theory is also relevant.

**Stochastic partial differential equations** (Deuschel, Friz, Imkeller, König, Scheutzow, Stannat) incorporate random terms and variables into the equation, which allows for modeling uncertainties and thereby often enable a much better fit to real-world scenarios. They can hence also be regarded as a particular instance of the novel area of uncertainty quantification (Schneider). The main approach taken towards stochastic partial differential equations (SPDE) in Berlin is via rough paths; a powerful theory allowing for stability analysis. **Evolution equations** (Emmrich) model the evolution of physical systems with time, requiring a rich and deep theory for their analysis, and are nowadays also considered in the SPDE setting by Berlin mathematicians (Emmrich).

**Applied harmonic analysis** (Kutyniok) plays a key role in areas such as imaging sciences - and, more generally, data science -, inverse problems, and numerical analysis of partial differential equations. The main approach consists in the construction and learning of (multiscale) representation systems such as wavelets, curvelets, and shearlets, which on the one hand allow sparse approximation of prescribed function classes aiming to solve, for instance, inverse problems by sparse regularization or data compression tasks, and on the other hand provide efficient decompositions for the analysis of diverse data types.

**Asymptotic analysis and multiscale modeling** play an important role in applications and form a prominent source of mathematical challenges. Homogenization, averaging techniques, and asymptotic expansions are used to extract reduced models for the statistics of large systems. A few examples are the following (many of those research activities are also linked to the competence center PVcomB on nanotechnology for photovoltaics)::

- Solid-liquid phase transitions occur in metal soldering or in crystals used in wafer fabrication. The early stage of such a phase transformation requires a microscopic description on the atomic scale. Here there is a strong link to stochastic analysis, where the corresponding particle models are studied. The second stage, growth, is often modeled by nonlinear parabolic systems for the respective fields (temperature, concentration, potentials), whereas the late stage, coarsening, is best described as a free boundary problem.
- In the context of thin liquid films (Wagner), the separation of length scales allows for the derivation of simplified models derived from the basic equations of fluid mechanics.Corresponding reduced theories result in degenerate higher-order equations for which a general mathematical theory is still lacking.
- In semiconductor physics, the importance of active interfaces leads to new mathematical questions concerning stable and thermodynamically correct bulk-interface couplings (Glitzky, Mielke). Moreover, the usage of organic semiconductor materials requires new stochasic models for solar cells (Wagner).
- Multiple scale asymptotics has recently opened highly innovative routes of theoretical research in meteorology and climate science (Klein). It provides a unifying framework for the large variety of simplified models known in theoretical meteorology, and it paves the way for the systematic investigation and robust numerical simulation of the ubiquitous scale interactions in the atmosphere and the oceans.

Asymptotic analysis also plays an important role in analyzing financial models, and there are applications of rough paths to multiscale analysis (Friz).

**Basic Courses**

The three Basic Courses here provide a thorough introduction to the theory of ordinary differential equations and dynamical systems, that of partial differential equations, and functional analysis.

Outline of the contents: **Dynamical systems**

◦ Flow properties of dynamical systems

◦ Omega limit sets

◦ Stability of fixed points and periodic orbits

◦ Invariant manifolds

◦ Examples of local and global bifurcations

◦ Chaotic behavior**Partial differential equations**

◦ Scalar first order equation

◦ Elementary PDEs: heat equation, wave equation, Laplace equation

◦ Solutions of linear problems via orthogonal series (separation)

◦ Elliptic problems via Lax–Milgram theory, maximum principles

◦ Existence and smoothing properties of parabolic equations

◦ Hyperbolic equations in several space dimensions

◦ Semilinear equations

** Functional analysis**

◦ Metric and Normed Spaces

◦ Linear Operators and Dual Spaces

◦ Fundamental Theorems of Functional Analysis such as Hahn-Banach and Banach-Steinhaus

◦ Weak Convergence and Weak Topology

◦ Hilbert Spaces and the Riesz Representation Theorem

◦ Spectral Theory for Compact and Self-Adjoint Operators

**Spectrum of Advances Courses**

Regularly offered advanced courses include Nonlinear Functional Analysis, Evolution Equations, Calculus of Variations, Nonlinear Partial Differential Equations, Nonlinear Dynamical Systems, and Bifurcation Theory, and Rough paths and Regularity Structures. More specialized courses on current topics complement these choices.