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.

Matrix Multiplication, the human way!

Having to do copious calculations by hand when preparing for an exam, I came to realize that there was an alternative way of interpreting a matrix multiplication. This new insight would allow me to instantly guess the following product without … Continue reading

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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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