Behnam Esmayli - YouTube Channel

Nearly 250 videos already uploaded cover:

Metric Spaces (Basics), Hausdorff Measures (10 videos),
Area Formula (13 videos),
Coarea Formula (14 videos),
Curves in Euclidean and Metric Spaces (12 videos),
Geometry of sets (Fractals and Cantor Sets, Density, Rectifiable Sets, Projection Theorems),
Analysis on Metric Spaces (=Sobolev functions), and
The Heisenberg Group.

Possible future videos:

Lipschitz Maps into Metric Spaces, Measure Theory for Geometry and Analysis, Sobolev and BV Functions, Degree Theory, Quasiconformal Maps

YoungMeasures is named after measures that are named after mathematician Laurence Chisholm Young.

Spring 2024. The Heisenberg Group. A user-friendly introduction to the core constructions in the Heisenberg groups. I begin with their sub-Riemannian side as this requires less of a background, e.g. no need to know Lie groups to be able to understand the sub-Riemannian geometry of the Heisenberg group. Taking advantage of the explicit global coordinate system (x,y,t) of R^3, we prove by hand (and not, say, by Chow's theorem) that any two points can be connected via a "horizontal" curve. Then we use the properties of the projection onto the xy-plane to find explicit bounds for the Carnot-Caratheodory distances...

Summer and Fall 2023. Analysis on metric spaces, Newtonian-Sobolev functions.

Newtonian-Sobolev Functions has been an extremely versatile and effective theory that helps do analysis, potential theory, PDE, etc in very abstract settings. A list of examples of spaces with and without Poincare inequalities. We cover Lebesgue differentiation theorem in doubling metric spaces, Reisz potential, Hajlasz-Sobolev spaces, and more.

April 2023. Geometric Measure Theory Click below to see the list of topics and enjoy!

Geometric Measure Theory (GMT)

Main references: classic books by Mattila and by Falconer.

Examples of fractal sets: Cantor sets in 1-D, four-corner Cantor set, zero and 1-dim Cantor sets

Similitudes (open set condition), self-similar sets, None self-similar sets, finding their Hausdorff dimensions

Koch curve, Sierpinski gasket/Carpet, Viscek set, etc.

Densities. Stated and proved main bounds on density. Gave a proof by Vitali covering.

We mentioned the deeper theorem of Preiss-Marstrand: if density exists, then $s$ has to be an integer, and the set is in fact $s$-rectifiable.

Frostman's lemma. We did several videos on this. We compared it to (local) isodiametric inequality

Frostman's lemma and existence of sets of finite perimeter

Dimension of product of sets. Proof uses Frostman's lemma.

Projection theorems. We followed Falconer and, like him, restricted to case of the plane.

Mentioned the potential theoretic approach originally due to Kaufman.

Rectifiability via behavior of projections: Mattila-Federer-Besicovitch

Rectifiability. We closed by a few videos on rectifiability. Gave definition (and definition of purely unrectifiable sets) and summarized the theorems about how to tell as set is rectifiable or not: Preiss-Marstrand (via density) and Mattila-Federer-Besicovitch (via projections)

At the end I referenced to Jonas Azzam's 51-video playlist where he covers rectifiability in more depth.

May 2022. Curves in Euclidean and Metric Spaces. Arclength, arclength parameterization, the length formula, metric derivative, $\bbbr^n$ with other norms, uniqueness and existence of geodesics, existence theorem for geodesics in compact spaces, geodesic spaces, length spaces, Riemannian manifolds, Hopf-Rinow in Riemannian and metric contexts. intrinsic geodesic problem, characterization of subsets of the plane through which a simple curve can be passed, line integral with application to Sobolev spaces on metric measure spaces, the Banach indicatrix and the area formula.

Feb 2022. The Co-area Formula is completed on my YouTube channel. This amazing formula contains Fubini's theorem and integration in spherical coordinates as its very special cases. Just imagine how powerful it is!

Jan 2022. The Area Formula is complete! A 10-video coverage that included: Linear algebra (Gram determinants), Rademacher's differentiation theorem, Sard's theorem, C^1 approximation of Lipschitz functions, surface measure on embedded submanifolds, and two proofs of the area formula.

Nov 2021. Hausdorff Measures

Hausdorff measure through example
Hausdorff measures -- why we need them
Hausdorff measure -- definition and properties
Borel sets are measurable with respect to Hausdorff measures
Hausdorff measure through closed, or open coverings
Hausdorff measure and Hölder or Lipschitz maps
Hausdorff dimension, definition with proofs
Hausdorff measures are Borel regular
The Hausdorff content
Equality of Hausdorff and Lebesgue measure

Advanced calculus collection contains mostly unrelated videos that deal with fundamental concepts and tools in calculus and analysis. Common themes are compactness arguments, connected sets, Cantor sets, continuity, limit supremum and lim inf, convergence, etc.