Maximal Spectra and Why Continuous Functions are Important
Richard Haburcak
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Why do we care so much about real-valued functions? We'll show that for a compact Hausdorff space, you can reconstruct the space from the ring of continuous functions to the reals, C(X). After introducing some algebraic geometry to give geometry to rings we'll use a few results in point-set topology to show that we can construct a space out of C(X) which is homeomorphic to the original space. If you know what a ring is, what closed sets are, and are willing to trust point-set topology, you'll get some algebraic geometry for free!
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The Fundamental Theorem of (Semi) Riemannian Geometry
Ryan Maguire
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A connection on a smooth manifold is a means of transporting a tangent space about one point to another. If one has a Riemannian manifold (has an inner product) or a Semi-Riemannian manifold (reduces to non-degeneracy) one would hope there's a connection that is compatible with the metric tensor. The fundamental theorem of Riemannian geometry says that for any semi-Riemannian manifold there is a unique torsion free connection, called the Levi-Civita connection, that preserves the metric tensor. The standard proof, which involves Christoffel symbols and various identities, can be reduced to a single line by means of the Koszul formula since any such connection must satisfy the equation, and this also defines such a connection. With this we can then talk about parallel transport, and even get some real-world applications by studying the Foucault pendulum.
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Fantastic Networks and Where to Find Them
Matt Jones
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Network theory and graph theory study the same objects, collections of vertices and edges, but ask very different questions and use different methods to answer those questions. In this talk, I'll show you some of the entry-level questions that are of interest to people studying networks. I'll spend some time talking about centrality (which vertices are most important in a network), network partitioning (how do I break this network into two smaller networks), and classification (what kinds of networks are there and how do I make them).
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$\lambda$-calculus
Zachary Winkeler
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$\lambda$-calculus is a surprisingly simple model of computation - there are only three rules! Nevertheless, it is just as powerful as any other Turing-complete model, and plays an important role in theoretical computer science. We will use $\lambda$-calculus to build a programming language from scratch, using some fun constructions that mirror those found in set theory. On the way, I'll talk about the Church-Turing thesis and the Curry-Howard isomorphism to give some context as to why $\lambda$-calculus is mathematically interesting.
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Building Character(s)
Alex Wilson
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Representation theory aims to understand algebraic structures by representing their elements as linear transformations on vector spaces. The focus of this talk will be finite-dimensional representations over the complex numbers of finite groups, and the primary goal will be to show how we can decompose an arbitrary representation into irreducible representations. To this end, the theory of characters of representations is amazingly helpful.
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On the Infinitude of Natural Numbers
Grant Molnar
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In this talk, we discuss one of the most important results in number theory, logic, and analysis: there are infinitely many natural numbers. After outlining several corollaries of this theorem, we spend the remainder of the talk proving our main theorem via a careful study of the Riemann zeta function.
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Knots
Samuel Tripp
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I will briefly discuss knot theory, and the construction of the Alexander polynomial, a classical knot invariant, with geometric meaning. This motivates the construction of knot Floer homology, which I will discuss, then present some geometric results.
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How to win arguments and influence people who probably won't want to be your friends anymore
Matt Jones
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Traditional game theory, in which players make choices simultaneously, is a powerful tool for analyzing decision-making. However, these games are also limited in their scope, and often struggle to find the "best" decision. I will introduce a new class of games called extensive games in which players are faced with a sequence of decisions, one after the other. Extensive games have their own way of defining the "best" decision. We will see how extensive games can help resolve arguments that ordinary game theory cannot, albeit in a very toxic way that I do not recommend trying on other people unless you want them to hate you.
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