This is not a usual type of a seminar talk, though I have given several talks with almost the same title lately. Still the talks are different. I made transparencies for more than 20 topics, two to three slides per topic. Necessary definitions, formulations of key results, hints towards proofs, open problems. I chose about 8 topics per talk. The choice depends on the audience, the weather, what I had for breakfast and such. The only thing that unites the topics is that they have been of interest to me in past number of years. They all are related to geometry, PDEs, dynamics, geometric group theory and such.

If the weather cooperates, the breakfast goes well and there are no terrible traffic jams, I probably would do something around there topics:

-- Uniform approximations of metric spaces by graphs of bounded geometry;

-- Graph approximations of the Beltrami-Laplace on Riemannian manifolds; (including a Marriage Lemma for measures);

-- Local generation of metric entropy and surjectivity of the boundary distance function near simple Finsler metrics;

-- Why curvature in Finsler Geometry cannot be responsible for basic geometric phenomena (no rotten tomatoes, please!);

-- On conjugation invariant metrics of unbounded diameter on groups of geometric origin;

-- Area minimizers in normed spaces (including Busemann's Conjecture in D=2 and a formula for the area of a convex polygon in the Euclidean plane);

-- Area spaces: how filling area determines metric;

-- Tomography: imaging vs. volume minimality.

-- maybe something different.