# Graduate Student Seminar 05-06

## Past Talks

Abstract: In this talk I would be talking about different methods in Statistics that are applicable in datasets in real life situations, in particular, medicine. I will be discussing some examples from my experience as biostatistician as well as consultant.

Abstract: In this talk we will analyze the dynamics of a one-spike solution to the one dimensional Gierer-Meinhardt (GM) Model. For this problem, we will show that the equilibrium location of a one-spike solution depends on certain global properties of the inhibitor decay rate over the domain.

Abstract: My talk is aimed at those students who already
have a grasp on the basics of LaTeX but are looking for some help on
more advanced formatting. I will be going through how to import
graphs, how to do graphs and pictures in LaTeX, how to use the
"minipage" environment, spacing as well as some other small
LaTeX "tricks" to get things looking like you want
them.

Since I may not have time to cover all of the useful
LaTeX concepts that I want to, I have created a website with these
and more:

Abstract: ''We teach linear algebra because it all works out so nicely.'' -- Dr. Katherine Hare

A finitely generated graded module over a polynomial ring is an
algebraic construct containing a sequence of vector spaces. Simple
questions, like the dimension of each vectorspace, can be hard to
answer.

In this talk, we use graph theory to recursively
compute Betti numbers which answers the above question for a small
class of modules. In more complicated cases, we use the combinatorics
of simplicial complexes. Concrete examples, pretty diagrams, and
simple linear algebra will be used throughout.

Keywords: edge
ideal, free resolution, Betti numbers, Hilbert function, graph,
simplicial complex, splitting ideal/edge/face/facet, dominating
vertex.

References:

1. Huy Tai Ha and Adam Van Tuyl,
''Splittable ideals and the resolutions of monomial ideals'',
Preprint, math.AC/0503203, 2005.

2. D. Eisenbud, *The
geometry of syzygies*, Graduate Texts in Math. **229**,
Springer, New York, 2005.

3. D. Eisenbud, *Commutative
algebra*, Graduate Texts in Math. **150**, Springer, New York,
1995.

Abstract: The collection of polynomials with rational coefficients which take integer values when evaluated at integers form a ring with some interesting algebraic properties. This talk will describe this ring and its generalizations which have applications in number theory and in algebraic topology.

Abstract: A somewhat overlooked 1970 paper by Edgar Asplund established a decomposition for a maximal monotone operator T on a general Banach space. The operator T is decomposed as the pointwise sum of a subdifferential mapping and an 'acyclic' component. I will reproduce a modern version of this result, and summarize current knowledge of the properties of this acyclic part, as well as providing examples of acyclic operators.

Abstract: In the early 20th century, there were many competing approaches to the foundations of mathematics. One of the more colorful ones is the formalism of "combinators", which was developed by Schoenfinkel in the 1920's and later refined by Curry. It is based on the idea that "everything is a function" (as opposed to set theory, where "everything is a set"). While the theory of combinators was not very successful as a foundation for mathematics, it later turned out to be quite useful in computer science. Combinators are fun, and satisfy many strange and counterintuitive properties. One of them is "every function has a fixed point".

Abstract: I will introduce the graph-theoretic firefighting problem.
Suppose a fire breaks out on a vertex of a given graph. Then
firefighters protect f vertices. At each subsequent time interval the
fire spreads to all unprotected neighbours then firefighters protect
f vertices. This continues until the fire can no longer spread. The
goal of the firefighters is to contain the fire.

I will talk
about the minimum number of firefighters needed to contain a fire
several specific infinite graphs. The notion of fractional
firefighting will also be introduced, where the number of vertices
protected at each time interval is allowed to vary. The ?new? minimum
number of firefighters needed to contain a fire has also been
determined for these specific graphs.

Abstract: A phylogeny is a tree representing the evolutionary relationships between a set of species. I discuss a method (conditioned genome reconstruction) for estimating a phylogeny from patterns of gene presence and absence. There are two main difficulties: the rates of gene gain and loss are variable; and genes absent from every species may not be observable. Conditioned genome reconstruction aims to overcome these difficulties, but calculates distances conditional on presence of genes in an additional species known as the conditioning genome. I show that this method is consistent for any choice of conditioning genome, but can perform badly on finite data for some choices. I describe improved algorithms that have much better performance.

Abstract: Can a function and its Fourier Transform both be well localized? Not really. I will make this more precise in a variety of ways. Can we find a different orthonormal basis for L^2 so that a function and its transform relative to our new basis are both well localized? It's not a wave decomposition; it's... wavelets! This possibility will be discussed a little at the end of my talk, after I say a bunch about Fourier transforms.

Abstract: For D f_{xx} + F(f) = f_t, spike solutions are shown to be unstable. Altering the problem slightly can result in stability, and some results of previous work are presented