Sarah Mitchell

I know how to compute determinants, but I want to understand what they **mean** geometrically. For a $2 \times 2$ matrix, the absolute value of the determinant gives the area of the parallelogram formed by its column vectors. For a $3 \times 3$ matrix, it gives the volume of the parallelepiped.

I'm starting to learn about manifolds. The definition is: A topological space $M$ is an **$n$-dimensional topological manifold** if: 1. $M$ is Hausdorff 2. $M$ is second-countable 3. $M$ is locally Euclidean: each point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$

I understand that for arranging $n$ distinct objects in a line, there are $n!$ ways. But for a circle, the answer is $(n-1)!$. Why? For example, with 4 people at a round table: - Rotational symmetry means arrangements that differ only by rotation are considered the same - So $4! / 4 = 6 = 3!$

The Pigeonhole Principle: If $n$ items are placed into $m$ containers and $n > m$, then at least one container contains at least two items. The generalized form: If $n$ items are placed into $m$ containers, then at least one container contains at least $\lceil n/m \rceil$ items.