Slow Pitch Colloquium

Wednesdays at 4:15 PM in MATH 350

The Slow Pitch Colloquium is aimed at both undergraduate and graduate students in mathematics. Students and faculty are encouraged to give talks. Topics can vary from research to amusements in recreational mathematics. If you are interested in giving a talk, please contact Camilo Mesa, Ryan Rosenbaum

Past Colloquia

Date Speaker Title/Abstract
5/1/13 Matt Moore

A general commutator

In group theory, the commutator $[x,y]$ induces a binary operation on the lattice of normal subgroups, which is used to define the important concepts of abelian group, solvable group, nilpotent group, etc. Together with the lattice operations, the commutator thus carries much of the information of the structure of a group. In this talk we will define a commutator for congruence lattices of general algebras and study some basic properties of the operation.

4/24/13 Amy Feaver

Classy!

In 1801, Gauss proposed three conjectures about the class numbers of quadratic fields. Two of these conjectures have been proven. However, the conjecture that there are infinitely many real quadratic fields of class number 1, still remains open today. In this talk, we will explore some progress that has been made toward the understanding of this conjecture, including a series of more recent conjectures, brought about by the Cohen-Lenstra heuristics.

4/17/13 Bryce Chriestenson

An introduction to group (co)homology

For a fixed group $G$ I will define the group (co)homology of $G$ with coefficients in an arbitrary $G$-module. After this I will calculate a few examples, and compute the low-dimensional (co)homology groups of $G$. These low-dimensional groups are related to: Group extensions of $G$, derivations of $G$-modules, the abelianization of $G$, universal central extensions of $G$, crossed $G$-modules, Schur multipliers, and more. I will say as much about these topics as I can fit into one hour.

3/20/13 Matthew Grimes

Sheaves and Cech Cohomology

Answering global questions with local data is a common theme in geometry, and sheaves are a powerful tool for organizing such problems. Cech cohomology aids in the study of sheaves on topological spaces, and helps us understand when local data is enough to answer a global question. We will discuss the basics of sheaves, and then go over the construction of Cech cohomology. We will examine basic theorems with an emphasis on motivation, examples, and accessibility. Applications to topology and geometry will also be covered, and so a cursory review of simplicial homology and complex analysis prior to the talk may be useful.

2/27/13 Bryce Chriestenson

The real homotopy type of singular spaces via the Whitney-deRham Complex

This talk will present certain homological invariants attached to a stratified space $X$, called the Whitney-deRham cohomology. This cohomology is defined as the cohomology of a chain complex, $\Omega^{ * }_{ \mathrm{ W } } \left( X \right)$, associated to $X$ in a somewhat ad-hoc way. The main result is to show that though the definition of $\Omega^{ * }_{ \mathrm{ W } } \left( X \right)$ depends on several choices, when certain conditions are imposed on $X$, the Whitney-deRham cohomology only depends on the homotopy type of $X$. This is achieved by showing that $\Omega^{ * }_{ \mathrm{ W } } \left( X \right)$ can be realized as a fine complex of sheaves which is a resolution of the locally constant sheaf on $X$. An application of this work is in the area of homotopy theory. One can canonically define a commutative differential graded algebra(CDGA), $A_{PL}\left(X\right)$, on $X$ in such a way that any CDGA which is quasi-isomorphic to it determines the real homotopy type of $X$. It will be shown that the complex $\Omega^{ * }_{ \mathrm{ W } } \left( X \right)$ is quasi-isomorphic to $A_{PL}\left(X\right)$, and thus determines the real homotopy type of $X$.

2/13/13 Amy Feaver

Mahler measure

The Mahler measure of a complex-valued polynomial $P(z)$ is the geometric mean of the modulus of the polynomial evaluated over the unit circle. At first glance, this quantity appears to be a harmless integral, but it turns out that it invades several areas of number theory. In this talk, we will discuss the relationship between Mahler measure, the golden ratio and algebraic integers. We will also extend the definition of Mahler measure to multivariable functions, and will conclude with evaluating the Mahler measure of a polynomial in three variables, finding that it can be expressed in terms of the Riemann zeta function evaluated at 3.

1/30/13 Nathan Wakefield

Canonical Heights in Generalized Iterations

Heights are an important component of the number theorist's toolbox. This presentation will describe the basic elements of heights, and how they play a role in many major areas of number theory; ranging from elliptic curves to dynamical systems. The canonical height is a particularly important tool in arithmetic dynamics. In this talk, we will discuss recent developments in the area of canonical heights, and explain an extension to generalized iteration. We will conclude with an open conjecture and discuss how heights may contribute to its solution.

11/28/2012 Ryan Rosenbaum

An Introduction to Transcendental Number Theory

A transcendental number is a complex number which is not the root of a polynomial in $\mathbb{Q} [x]$. We will give an accessible introduction to the theory of real transcendental numbers. We will touch on basic properties of transcendental numbers (like they actually exist), Liouville's theorem, irrationality measure and Roth's theorem. Time permitting we will include a construction of transcendental numbers relating to the undecidability of the halting problem over Turing machines.

11/7/2012 Trubee Davison

Constructing the Fractal Set with an Application to the Music of Bach

Given a complete and separable metric space (X,d), and a finite family, S, of contractions on X, one can construct a compact subset K of X that is invariant under S. This set K is called the fractal set, or attractor set, associated to S. In this talk, we will discuss the construction of the fractal set. As a somewhat loose, but fun, application of fractals, we will then look at one example of the 'fractal-type' phrasing that occurs in many of Bach's compositions. The speaker will conclude with a performance on the violin of a Bach composition.

10/31/2012 Boramey Chhay

Fractal Sets in $\mathbb{R}^n$

Fractals are sets which have fractional Hausdorff dimension. We will construct some of these sets and see how $n$-dimensional Hausdorff measure relates to Lebesgue measure. Time permitting we will discus probability.

10/24/2012 Bryce Chriestenson

de Rham currents on an open subset of $\mathbb R^n$

In this talk I will describe the construction of the complex of de Rham currents. This complex is dual to the complex of compactly supported differential forms. Time permitting I will explain how to generalize this construction to a smooth manifold. I will also try to describe how currents are related to non commutative geometry.

We will introduce p-adic numbers and some of their basic properties including Hensel's lemma. We will then explore consequences of these properties for polynomial factorization via Newton polygons.

10/3/2012 Nathan Wakefield

Primitive Divisors in Generalized Iterations of Chebyshev Polynomials

Let $(g_i)_{i \geq 1}$ be a sequence of Chebyshev polynomials, each with degree at least two, and define $(f_i)_{i \geq 1}$ by the following recursion: $f_1=g_1$, $f_n=g_n\circ f_{n-1}$ for $n \geq 2$. Choose $\alpha \in \mathbb{Q}$ such that $\{ g_1^n (\alpha) : n \geq 1 \}$ is an infinite set. The main result of this talk is as follows: If $f_n(\alpha)=\frac{A_n}{B_n}$ is written in lowest terms, then for all but finitely many $n >0$ the numerator $A_n$ has a primitive divisor; that is, there is a prime $p$ which divides $A_n$ but does not divide $A_i$ for any $i \lt n$. I will also talk about some further directions of my future work at the end.

9/26/2012 Matt Grimes

Roots of Algebraic Geometry

My talk will examine a special class of topological spaces: varieties over an algebraically closed field. Varieties are (loosely speaking) zero sets of polynomials, and we will leverage this description to answer various geometric and topological questions by studying the associated ring of polynomial functions. Along the way, we will explore the Zariski topology and the make explicit the translation between geometric and algebraic ideas.

9/12/2012 Justin Keller

Introduction to Hopf Algebras

We characterize Hopf algebras over a field $k$ as those $k$-algebras whose representation theory is "nice". That is to say, as those algebras whose category of modules is monoidal with repsect to the tensor product and where every module has a corresponding dual module, motivating the definition of Hopf algebra. We'll assume a knowledge of vector spaces, but we will briefly recall the definitions of the other necessary structures, including algebras over a field, modules over an algebra, monoids, and categories. We'll also recall the tensor product for vector spaces. Time permitting, we'll look at some interesting examples of Hopf algebras and discuss other motivations.

9/5/2012 Jason Hill

Introduction to Sage

Sage is an open-source mathematical software system providing a functional, procedural, and object-oriented environment that gives simultaneous access to many scientific computing resources under a single interface. For instance, Sage includes:

- Python and Cython (programming)
- GSL, SciPy, NumPy, ATLAS (numerical methods)
- BLAS, LAPACK, LinBox, IML, GSL (linear algebra)
- GAP, NetworkX (group and graph theory)
- PARI/GP, FLINT, NTL (number theory)
- R (statistical computing)
- Maxima (calculus)
- mwrank, ecm, Singular (arithmetic and algebraic geometry)

In this talk, I'll give an introduction to Sage via the web/notebook interface using sage.colorado.edu. This talk is designed for those new to Sage, but all are welcome to attend. Please bring a laptop, or share with a neighbor, and register an account on sage.colorado.edu before the talk. Optionally, you may download Sage for your own computer (Mac OS-X or Linux) at sagemath.org.

8/21/2012 Jason Hill

A Brief Introduction to Intransitive Permutation Groups

I work with intransitive permutation group algorithms. In this 20-minute introduction, I'll provide some examples of why these groups are interesting, why their structure makes them challenging to work with, and what my research has found.

8/21/2012 Ryan Rosenbaum

Fourier Expansions of Automorphic Forms

We will discuss Fourier expansions of $SL(2,\mathbb Z)$-periodic functions on the complex upper half-plane, highlighting similarities with the case of Fourier expansions of periodic functions on the reals. Time permitting, we will discuss number theoretic properties of the Fourier coefficients.