Title: Mathematic(s), Structure, and Translation
Abstract: We as Mathematicians live in many ways under the Bourbakian and he has so ingrained himself in our discourse that we have forgotten how to speak mathematics without the influence of Bourbaki. But that the rise of Bourbaki himself is strongly correlated to the movements of structuralism in the humanities and linguistics brings up interesting questions to ask about our relation to the fundamentals of mathematics and its history. And so when we look at modern frameworks to understand mathematics, such as Category Theory, I would like to discuss the fluidity of the foundations of mathematics and perhaps discuss what it means for the future.
Title: The space of pointed hyperbolic n-manifolds.
Abstract: I will give an introduction to the space of pointed hyperbolic n-manifolds. This space is equipped with the geometric topology, in which two pointed manifolds are close if they are almost isometric on large neighborhoods of their basepoints. We'll do some examples in dimensions 2 and 3. Hopefully, I'll end with discussing the proof of the fact that the space of infinite volume pointed hyperbolic 3-manifolds is connected.
Title: Elliptic curves, gap groups, and pairing based cryptography
Abstract: Computers perform arithmetic on an elliptic curve every time they load a webpage. How and why did this happen? We'll survey the landscape of ECC from a mathematician's vantage point and try to understand what those computer scientists are so excited about. Beginning from a introduction to hardness assumptions, we'll see why the group of points on an elliptic curve is a great group for "doing cryptography" and what that even means. If time permits, we'll discuss newer developments in pairing based cryptography and meet some particularly useful curves.
Title: Maximal Systems of Curves
Abstract: Curve complexes and arc complexes are used to understand mapping class groups and Teichmüller spaces. Simplices of curve complexes and arc complexes are labeled by systems of simple loops and systems of simple arcs on surfaces. In particular, the cardinalities of maximal systems of curves determine the dimension of these complexes. In this talk, we explain how to determine the cardinalities of maximal (complete) 1-systems of curves on a non-orientable surface.
Title: Superpermutations and The Haruhi Problem
Abstract: A superpermutation of n symbols is a string that contains every permutation of n symbols as a substring. Historically people are interested in the minimal length of a superpermutation. In this talk I’ll explain why this has something to do with the anime series The Melancholy of Suzumiya Haruhi, and how an anonymous anime forum poster contributed a theorem to the math world.
Title: The Circle Packing Theorem and its Consequences
Abstract: Consider a configuration of disjoint circles tangent to each other in the plane. If one joins the centers of these circles by line segments wherever two circles touch tangentially, then a planar graph is formed. Naturally, one can ask whether the reverse is true: does every planar graph come from an associated arrangement of tangent circles (a "circle packing")? The answer is yes! And better yet, if the graph is a triangulation of S^2, then this circle packing is unique in a suitable sense. We will be proving this theorem, as well as looking at related results and some nice applications to other areas of math.
Title: To IFSinity and beyond!
Abstract: "Given a contraction from a complete metric space to itself, the contraction mapping theorem guarantees the function has a unique fixed point. A consequence is the theory of iterated function systems, which constructs some fractals as the fixed points of particular contracting operators. Since the 80s, dynamicists have made this theory more robust--we will examine some tools in dynamical systems that make the study of self-similarity tractable, with a particular eye to fractal tilings of the plane.
Title: H^1(F,G)
Abstract: When you define a cohomology theory in the traditional way -- say by starting with some functor, choosing an appropriate "resolution" of your object, and applying this functor to your "resolution" -- it can be difficult to understand what the higher cohomology actually *means*. So it is remarkable that in many diverse settings, the first cohomology group H^1(X, G) can be understood very concretely. It consists of the isomorphism classes of G-torsors on X. In this talk I'm going to explain what a G-torsor is, and give several applications of this interpretation of H^1 to questions in number theory.
Title: Sitting is the Opposite of Standing II - Revenge of the Sit.
Abstract: This GIST talk is actually a problem solving session. Inspired by last year's GIST talk "Sitting is the opposite of standing" (organized by the one and only Eric Moss), the idea is to work in small groups to solve some fun and interesting problems/riddles that I have prepared. Of course, only elementary mathematics will be assumed.
Title: All of Math is Linear Algebra. Accept it.
Abstract: Young tableaux are a fun, combinatorial gadget that appear in many different areas of Math. In this talk, we will dive into the world of Young tableaux and their more decorated cousins, the standard Young tableaux. Along the way we will see what these objects can tell us about the structure of the infamous Grassmanian and (if time allows) how they are related to a classic counting problem in algebraic geometry.
Title: TQFTs
Abstract: In what some have described as the greatest 65th birthday present of all time, Michael Atiyah gave to René Thom (and the world) the concept of the topological quantum field theory. This had led to an explosion of beautiful work in topology, algebraic geometry and, most recently, number theory. In this talk I hope to introduce just what these things are, classify them in low dimensions, and give some examples as to where they arise, and where they might be useful.