Tufts Math Graduate Student Seminar ArchiveFall 2019Nov. 20: Surreal Numbers and Combinatorial Game TheorySpeaker: David Tu Abstract: The surreal numbers are an extension of the real numbers that include infinities and infinitesimals, and they are defined by just a few simple axioms. I will present these axioms and then prove how they lead to expected properties of the real numbers and unexpected properties of the surreal numbers. Then we'll play some games and try to determine who has a winning strategy by treating the game as a surreal number. The ultimate goal is to develop winning strategies for mathematical games that you can then unleash upon unsuspecting bystanders. If time permits, I will also cover impartial games and nimbers. Nov. 13: No seminarNov. 6: Counting Number Fields Whose Galois Group is an Iterated Wreath ProductSpeaker: Matt Friedrichsen Abstract: Undisputed fact: the best permutation group is S_{2} . One of the many reasons for this is that every degree two extension of Q has Galois group S_{2}. If you then take a degree two extensions of a degree two extension, you will very likely get a number field whose Galois group is S_{2} wr S_{2}. You can extend this by taking even more degree two extensions of your number fields. A recent paper showed how to count number fields with Galois group S_{2} wr S_{2} (which is really D_{4}) by an invariant called the conductor. I will fill in the background for this by talking about how the conductor comes from representation theory, explain the algebraic relationship used to count D_{4} number fields by conductor, and then talk about if you can generalize this to the iterated wreath product of S_{2}. Oct. 30: Roots of Polynomials, Tschirnhaus Transformations, and Hilbert's 13th ProblemSpeaker: Curtis Heberle Abstract: Everyone knows there is no formula in radicals for the general quintic polynomial. What this talk presupposes is... maybe there is? More precisely, we will see that an arbitrary quintic polynomial can be reduced via a Tschirnhaus Transformation (a polynomial transformation of the roots) to something of the form x^{5}+ax+1. This shows that solving the quintic is intrinsically a "one-parameter problem." What about higher degree polynomials? Using the same methods, a sextic can be reduced to a two-parameter problem, a septic to three parameters, and an octic to four parameters. Hilbert conjectured that these results are the best possible, but showed that in degree nine we can do slightly better than expected, reducing a nonic to only four parameters. We will see how, by studying the space of Tschirnhaus transformations, both the quintic reduction and Hilbert's result for degree nine can be understood in terms of facts from classical algebraic geometry. (For example, the existence of 27 lines on a cubic surface.) Oct. 23: Category Theory and the Yoneda LemmaSpeaker: Rylee Lyman Abstract: The Yoneda Lemma is a fundamental result in category theory. It says, roughly, that objects in (locally small) categories are entirely determined by the network of morphisms from (or to) it. I will attempt to make sense of this statement. I will introduce the notions of a category, functor and natural transformation from scratch and state and prove the Yoneda Lemma. We will use it to construct and understand the tensor product of vector spaces and, time permitting, perhaps another example or two. This talk will be accessible to a wide audience. Oct. 16: Statistics of random surfacesSpeaker: Sunrose Shrestha Abstract: Translation surfaces are orientable surfaces built out of finitely many Euclidean polygons, glued edge-to-edge via translations. In this talk we will use a combinatorial model to explore the topological statistics of certain random translation surfaces. We will also examine certain geometric properties of random square-tiled surfaces (translation surfaces built out of unit squares), and prove results about the frequency of these properties. Oct. 9: The Diffusion Operator and Graph LaplaciansSpeaker: Casey Cavanaugh Abstract: This talk will begin by exploring the Laplace operator in the continuous setting, and why it's used to model diffusion. Then, we will use what we know from the continuous setting to understand why the graph Laplacian is called a Laplacian, and why we can think of this matrix as being a discrete diffusion operator. Time permitting, we'll look at some properties and applications of the graph Laplacian, and why it comes up so frequently in graph theory problems. Oct. 2: The Heisenberg group, sub-Finsler metrics, and how to reach infinitySpeaker: Nate Fisher Abstract: My goal will be to define the words in the title in a coherent way and (maybe) tell you what the horofunction boundary of the Heisenberg group is for a specific type of metric. Sept. 25: Function Field Arithmetic (and some Statistics)Speaker: Daniel Keliher Abstract: Working over function fields over finite fields is often analogous to working over finite extensions of the rationals. In number theory, this provides a useful analogy to prove number-theoretic results in a different setting, often with an enlarged set of tools (for example, the Riemann Hypothesis is true for function fields of curves!). We’ll discuss this analogy and introduce many of the basic objects of interest. Time permitting, we’ll recover an asymptotic formula of Cohen, Diaz y Diaz, and Olivier in the function field setting which enumerates all of the quartic dihedral extensions of a function field with some some discriminant condition. Sept. 18: Handle Decompositions of Smooth ManifoldsSpeaker: Natalie Bohm Abstract: Handle decompositions are used in geometric topology to systematically build smooth manifolds of any dimension, and are useful in determining properties of these manifolds, both geometric and algebraic. In this talk, we will see how to obtain a handle decomposition for a given smooth compact manifold using Morse theory, and examine some of the consequences of handle decompositions. In particular, manifolds with handle decompositions, or handlebodies, can be used to prove Poincaré duality for compact, orientable smooth manifolds, and we will sketch an outline for this proof as well. Sept. 11: The Moduli Space of Graphs and Outer Automorphisms of Free GroupsSpeaker: Rylee Lyman Abstract: To every graph is associated a free group which, in a sense, counts the number of holes or loops in the graph; graphs with the same number of holes have isomorphic associated free groups. Knowing this, it becomes interesting to ask questions about the space of possible graphs within a specified isomorphism type of free group. One way of thinking about this is by "deforming" a graph via a continuous map called a homotopy equivalence. After adding natural restrictions and structures, what results is a topological space whose points are projective classes of "marked metric graphs." This space was first studied by Culler and Vogtmann in 1986, and has come to be called Outer Space. The quotient of Outer Space by its group of symmetries is the Moduli Space of Graphs (with a specified number of holes). We will sketch the construction of Outer Space and discuss some of its properties. |