If a massive star burns out but cannot support its matter against the force of gravity, a black hole forms. This means that the star collapses to a single point called the singularity. The singularity is a place of infinite gravitational forces and is surrounded by a boundary called the event horizon, inside of which light cannot escape. The animation shows an embedded spacetime diagram of a spherical star shrinking in radius. If the star keeps the same mass, then the geometry outside the star remains the same; what we are seeing is more of the curvature being revealed. In other words, the mass (not the radius) of the star determines the how spacetime is warped around it. We get a black hole when the star is so dense (think small and heavy) that the star can "fit" through its own ripple in the fabric of the universe.

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Consider the sequence: P₁ = 1, P₂ = 5, P₃ = 12, P₄ = 22, ..., and, in general, P_n = P_{n - 1} + 3n - 2 for n ≥ 2. These are called pentagonal numbers because they count points in nested arrangements of pentagons as seen in the picture to the right. Likewise, there are triangular numbers T₁ = 1, T₂ = 3, T₃ = 6, ..., and T_n = T_{n - 1} + n, and square numbers S₁ = 1, S₂ = 4, S₃ = 9, ..., and S_n = S_{n - 1} + 2n - 1. These sequences count points in nested arrangmenets of triangles and squares. It is easy to notice (and prove algebraically) that a pentagonal number is, in fact, the sum of a triangular number and a square number: 5 = 1 + 4, 12 = 3 + 9, 22 = 6 + 16, ..., and P_n = T_{n - 1} + S_n for n ≥ 2. The animation shows a geometric verification of this fact via rays from the corner vertex and stretching angles to deform a pentagon into a triangle and a square. Pentagonal numbers are related to the partition function p(n): if we extend to nonpositive indices by P₀ = 0, P_-1 = 2, P_-2 = 7, P_-3 = 15, ..., and then define f(x) = ··· + x^P_-2 - x^P_-1 + x^P₀ - x^P₁ + x^P₂ - ···, we have 1/f(x) = 1 + p(1)x¹ + p(2)x² + p(3)x³ + ··· .

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The image to the right is the graph of the curve y³ = x² in the real plane. This curve has a "cusp" at the origin, and, in particular, does not have a well-defined tangent line there since the partial derivatives both vanish. One way to resolve this singularity is to view the curve as the shadow of a smooth curve in a higher dimensional space. The animation shows a rotating view of the twisted cubic. Here is another nice visualization and description of the resolution of a cusp via blowup from Donu Arapura at Purdue.

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As explained below in my research section, triangles with some given fixed area and perimeter can be parameterized by an elliptic curve. In particular, since elliptic curves are equipped with a group law, we can add known triangles together to generate new ones. Moreover, rational points on the curve correspond to Heron triangles (i.e., triangles with rational side lengthes), so we can add Heron triangles together to get Heron triangles. The animation shows the family of triangles with area 6 and perimeter 12 and the associated points on the corrseponding elliptic curve; this family includes, for example, the (3, 4, 5) right triangle (which is roughly where the animation begins and ends). It was created from a Geogebra file provided to my advisor Bill McCallum by Tomas Reccio.

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