Lecture Notes!

June 24 2024

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This is the first talk of a lecture series on theorems in geometry. The material should be equally accessible and compelling to first-year students and peer mentors. The first talk of this series explores the types of questions that define Incidence Geometry, focusing on the famous Sylvester-Gallai Theorem and the intriguing questions (including open ones) that stem directly from it. I also introduce projective spaces, which (simply put) are spaces where we include the points where parallel lines intersect (a super useful tool for many areas of geometry), and demonstrate how they can be beautifully put into action to solve problems related to the Sylvester-Gallai Theorem. To conclude, I explain the principle of duality, and we will revisit the Sylvester-Gallai Theorem, but now in its dual form. The concept of duality in a real projective plane is one everybody should explore at least once in their life.

July 5, 2024

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This is the second talk of a lecture series on groundbreaking theorems in geometry! Whether you're a first-year student or a seasoned peer mentor, this lecture should be equally accessible and compelling. This session explores the Ham Sandwich Theorem, a powerful and versatile tool used in incidence geometry and general geometric theory.

The folklore phrasing asks the following: Can you always slice a ham sandwich into two equal halves, no matter how the bread and ham are shaped? What if you add a slice of cheese? How do these cuts look, and how can we calculate where to make them? We will use these questions to start our conversation on how we can cut bounded open sets using hyperplanes and hypersurfaces, two extremely useful tools in geometry that everybody should know how to work with effectively. We will intuitively and rigorously define every notion along the way so do not fret if those words are unfamiliar. This theorem illustrates the beauty of hyperplanes and hypersurfaces in action. But that's not all! For our proof of the Ham Sandwich theorem, we need the Borsuk-Ulam Theorem, which illustrates how some theorems can have many many different rephrasings that seem different but are actually equivalent. Finally, we will venture out into the lesser-known (yet equally intriguing and useful) Polynomial Ham Sandwich theorem. Expect lots of stunning visuals that will bring textbook definitions to life.

July 12, 2024

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This lecture highlights an intriguing exploration of set-theoretic geometry where we take a countable set of points in R^2, split it into two pieces, and through only rigid motions (translation and rotation), end up with two sets identical to the original. Want more than two? Just repeat the process. This is the Sierpiński-Mazurkiewicz Paradox. No, this isn't magic or reliant on controversial principles like the Axiom of Choice. The Banach-Tarski paradox is a similar paradox in some ways that we will also explore and compare to the Sierpiński-Mazurkiewicz Paradox. The Banach-Tarski paradox a relatively famous result that uses the Axiom of Choice to showcase a similar idea about duplicating sets by manipulating them using rigid motions. Those who do not accept the Axiom of Choice do not accept the Banach-Tarski paradox. However, being a consequence of only simple principles, the Sierpiński-Mazurkiewicz Paradox is an unquestionable result of set manipulation that allows us to duplicate some special sets regardless of your stance on the Axiom of Choice, showcasing the surprising and beautiful nature of mathematics. Learn about the Sierpiński-Mazurkiewicz Paradox and how it relates to other duplication paradoxes, as well as the Axiom of Choice. No prior knowledge of any of these concepts is necessary; we define every notion along the way in detail.