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See: Math Notebook

I'm writing up my solutions to math exercises that I have done to increase my understanding.

To Do

• Show a bijection between involutions and standard tableau.
• I misplaced my bijective proof of standard tableau and paths on Pascal's triangle, find it or redo it.
• Show that {$SU(2)$} is closed (and thus compact, because it is bounded).
• Show that {$SL(2,\mathbb{C})$} is an analytic manifold.
• Show that {$SU(2)$} is a differentiable manifold but not analytic.
• Show that {$SL(2,\mathbb{R})$} is a differentiable manifold but not analytic.
• Show that the complex solutions {$x$} and {$y$} to the equation {$x^2+y^2=1$} form a sphere with two holes.
• Learn how to calculate the hyperarea of an n-sphere and the hypervolume of the n-ball.

Need to Write Up

• {$SU(N)$} is bounded. Multiplying by the conjugate transpose shows that, for each row, and each column, the sum of squares equal 1, thus for each entry the magnitude is at most 1.
• {$SL(N)$} is not bounded. The determinant is 1, thus the volume is fixed, thus the vectors can be arbitrary large in any dimension.

Solved

• Combinatorial interpretation of {$e^{A} \cdot e^B = e^{A+B}$}
• Square root of identity matrix
• Interpret combinatorially det(A)det(B) = det(AB)
• Why symplectic manifolds are even dimensional
• Confirm that a harmonic oscillator preserves symplectic area
• What is the conjugate of a multidimensional number
• Relating definitions of {$e$}