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.
|
| 10/10/2012 |
Cliff Blakestad |
p-Adics, Polynomials, and Polygons
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.
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