Time: Wednesday 14-16

Room: Endenicher Allee 60, 2.040

## Schedule and Topics

### 10.10.12 André

**Gradientflows in $$\mathbb{R}^n$$:**Characterization of stationary points, linearization around critical points, convergence by convexity, implicit Euler time-discrete scheme and variational formulation**Basic Riemannian Geometry:**differentiation on manifolds**First part of Otto theorem:**convergence of (finite-dimensional) gradient flows of convex energies to stationary points on a Riemannian manifold

### 17.10.12 André

**Some more Riemannian Geometry:**Riemannian connection/fundamental theorem of Riemannian geometry, geodesics, length/energy of curves, geodesics as energy minimizers, formula for Hessian**Introduction to Fokker Planck equation:**characterization of equilibrium solution, example Ornstein-Uhlenbeck process, behaviour in non-convex potential

### 24.10.12 Leonardo

**Classic results porous medium equation:**Self-similar solutions by rescaling, stationary solutions**Two gradient flow formulations for porous medium equation:**$$L^2$$ and entropic gradient flow

### 31.10.12 Simon

**Wasserstein metric****Time discrete variational formulation of Fokker Planck equation****Existence of time-discrete solutions:**Remarks on weak L¹-convergence

**07.11.12 Angelo**

**Physical**characterzation of unique minimizer and derivation of gradient functional (Darcy’s law, osmotic pressure)**derivation of PME**:**Asymptotic formulation:**energy functional in rescaled equation**Energy-entropy estimate:**characterzation of unique minimizer and derivation of gradient functional**Convergence in induced distance**

### 12.11.12 (8:30) Stefan

**Construction of isometric submersion**of flat manifold of diffeomorphisms on $\mathbb{R}^n$ onto manifold of densities $\mathcal{M}$- Pullback of curves and geodesics under the submersion

### 28.11.12 Simon

- Convergence of time-discrete scheme to Fokker-Planck equation

### 26.11.12 (16:00) Leonardo

**Some results from Optimal Transport:**Gradient and Hessian of convex functions via Alexandrov’s Theorem**Identification of Wasserstein distance as induced distance****Computation of the Hessians:**displacement convexity and lower bounds

### 05.12.12 Dies Academicus

### 12.12.12. Angelo

- Towards rigorous results: smooth setting

### 19.12.12 Stefan

- Porous medium equation on manifolds
- Contraction in Wasserstein space by
**Eulerian calculus**

### 09.01.13 Simon

- A non-local non-linear Fokker-Planck equation
- Example of Dynamic in special regimes
- setting of a constraint gradient flow

### 14.01.13 (16:00) Stefan

- Incompressible Euler equation and Arnolds interpretation
- Breniers relaxation and relation to Wasserstein distance

### 16.01.13 Angelo

- Slow motions of gradient flows: Allen-Cahn equation

### 21.01.13 (16:00) Leonardo

- Rate of convergence for Fokker-Planck equation via transport inequalities (logarithmic Sobolev inequality, HWI-inequality)