By: Zachary Berens
Abstract: I will introduce models of first and second order partial differential equations and their general solutions. In generalizing solutions to PDEs, one often cares about so-called "weak solutions." This leads to the notion of a "weak derivative" which we'll define. Then we'll talk about two types of function spaces (vector spaces of functions) where solutions to second order equations can be found: Lp spaces and Sobolev spaces. We are mainly concerned with the Sobolev embedding theorems which provide a powerful tool for studying solutions to elliptic PDEs. Some analysis background would be helpful (compactness, completeness and uniform convergence). But it is not necessary. This is the first talk in a two-part series.
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By: Zachary Berens
Abstract: Category theory is a great tool for understanding structural similarities between different classes of mathematical objects (sets, vector spaces, groups, etc.). This talk is an accessible introduction to category theory with a focus on two important constructions: limits (no relation to calculus) and colimits. The null space of a linear map is an example of a limit. The direct sum of vector spaces is an example of a colimit. We’ll get our hands dirty with lots of examples. No algebra background is assumed (but it would make some things easier).
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By: Ben Goldstein
Abstract: Polytopes generalize polyhedra to higher dimensions. In this talk, we will explore the metric geometry of these objects, ultimately proving that any convex polytope admits a non-overlapping unfolding. No advanced math background is assumed.
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By: Nathanael Ong
Abstract: The goal of this talk is to give an introduction to Lie Groups. We will cover the definition of a Lie group and give lots of examples. We will examine algebraic, geometric, and topological perspectives.
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By: Noah Harris
Abstract: We will define a manifold and give an intrinsic definition of the tangent bundle. Notes by Adam Kern: differential-geometry-Download
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By: Zachary Berens
Abstract: The goal of this talk is to give an advanced, yet accessible introduction to algebraic geometry. We will work up to the definition of an affine scheme.
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