Grant Molnar
Nearing Nearrings
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In this talk, we motivate and define nearrings, an algebraic structure that generalizes rings. We describe some of the properties of nearrings, and draw parallels between these properties and those of other, more familiar structures from abstract algebra. The material in this talk is drawn from James R. Clay's "Nearrings: Geneses and Applications".
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Richard Haburcak
Triangles. Boundaries. And...?
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In mathematics, we are interested in a variety of objects, and it can be helpful to arrange them into nice families or spaces so we can study their properties. We'll consider one of the simplest mathematical objects, triangles, and see how intricate structures naturally show up when we study the space of triangles. We'll discuss the idea of moduli spaces and see hints that we need to allow for more general spaces than just manifolds if we want to have moduli spaces that reflect all of the structure of triangles. Come find out how all triangles form a triangle!
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Ben Adembaum
Lattice Path Lattices
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This talk will focus on a natural partial order defined on lattice paths. Some natural questions to ask are when is this associated partial order a lattice, or more specifically when is it a distributive lattice. We will discuss a characterization for paths with small horizontal steps, and what the isomorphism classes are in certain special cases.
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Matthew Ellison
The abc Conjecture
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This talk will be an elementary introduction to the abc conjecture. We'll begin by stating and proving the Mason-Stothers Theorem, which is the abc conjecture for polynomials, and see how it implies Fermat's Last Theorem for polynomials. Then we'll translate the Mason-Stothers Theorem to integers to obtain the statement of the abc conjecture, and go into some background and a few of its consequences. This talk follows one from Lang's book Math Talks for Undergraduates --- though it should be interesting for non-specialist math graduates too!
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Brian Mintz
Probability Paradoxes: Eleven Enlightening Examples
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Have you heard about Simpson's Paradox? What about Non-transitive Dice, or the Sleeping Beauty Problem? In this talk, we'll explore these ideas and many more as we ponder unexpected subtleties that arise in basic probability. While people often think probability is well defined and uncontroversial, even simple questions can have answers so counterintuitive that they've been called paradoxes. From divergent bets to interpreting medical results, come and hear about the many ways probability isn't as straightforward as it seems!
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Luke Askew
Skeletons
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The skeleton of a category categorifies the notion of selecting a representative up to equivalence. Assuming no prior knowledge of categories, we will introduce the basics of category theory, see how skeletons can give a better definition for the fundamental group of a topological space, and have fun with groupoids.
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Steve Fan
Roth's Theorem on Arithmetic Progressions
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One big pearl in arithmetic combinatorics is the beautiful theorem of Szemeredi, which asserts that any "dense" subset of natural numbers contains infinitely many arithmetic progressions of length k for every k>=1. The first interesting case of this theorem, namely the case k=3, was established in 1953 by Roth using an adaptation of the Hardy-Littlewood circle method. In this talk, I will sketch an elegant proof of Roth's theorem based on Fourier analysis and a density increment argument. The essential ideas in the proof have been greatly generalized by Gowers in 2001 to handle all k>=3.
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Alina Glaubitz
Indirect Reciprocity
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Many aspects of human society are defined by our ability to cooperate in large groups. But what brings about our unique capacity to work together? In this talk, we will argue that indirect reciprocity is a determining factor that enables human society to function. We will discuss image scoring, reputation dynamics, and the 'leading eight' and use these concepts to understand better human morals, social norms, and cooperation, all with the power of simple mathematical models.
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