Author Archives: Behnam Esmayli

About Behnam Esmayli

I started PhD in Mathematics at Pitt in Fall 2015. My top interests are Differential Topology, Analysis, Geometric Analysis, Algebraic Topology, Differential Geometry, in that order.

What is a Manifold? (6/6)

In posts 1-3 we were able to reduce all of the geometry of a curve in 3-space to an interval along with two or three real-valued functions. We also discussed when two sets of such data give equivalent (overlapping) curves. This … Continue reading

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What is a Manifold? (5/6)

In our last post, we invented a new geometry by re-scaling the inner product of the usual Euclidean plane. This modification did not change any of the angles in our geometry, in the sense that if two curves intersected in a particular Euclidean … Continue reading

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What is a Manifold? (4/6)

After our luxurious treatment of 1-d manifolds, we turn to 2-d manifolds. My story of surfaces starts in a beautifully weird morning when I got up to realize that life in the usual Euclidean plane had changed dramatically. Vectors had shortened, areas … Continue reading

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What is a Manifold? (3/6)

Intrinsic descriptions One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not a priori embedded in . In other words, we don’t have to start with a subset of to … Continue reading

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What is a Manifold? (2/6)

We continue from Part One of this journey our attempt to illustrate how one can start with calculus and arrive at the definition of a 1-dimensional manifold. In the previous segment, we concluded with the fact that a curve in  may … Continue reading

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