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Fractal Dimension and Cell Colony Boundaries
Gabriela Rodriguez
The boundary of a
colony of cells growing in vitro can be quite rough and irregular.
Its fractal dimension is typically between 1 and 2. Fractal
dimension and other critical exponents have proven useful in
classifying colony growth dynamics.
This talk will
focus on the theoretical concept of fractal dimension and its
practical approximation using the box-counting method. The method
will be illustrated by computing the fractal dimension of the Koch
curve and by estimating the fractal dimension of a tumor boundary.
This talk will be accessible to anyone who has a background in
calculus.
Exploring the Property of Intrinsic Knottedness in Spatial Graphs
Jonathan
Miller
A
graph is said to be intrinsically knotted (IK) if every spatial
embedding of it in 3-dimensional space contains a nontrivial knotted
cycle. It's known that there are finitely many such minor-minimal
graphs (that is, one in which no IK “sub-graph” exists) - but
because establishing this property is difficult and requires a great
deal of ingenuity, a full characterization of this set remains an
important open problem in this field. In this talk, I’ll be
presenting some methods I employed as part of a research team with
Prof. Naimi of the Math Department. I'll discuss some new IK graphs
we discovered, as well as the tools we developed to approach this
problem. While it's helpful to have
some background in Graph Theory, any student with an interest in
graphs should be able to follow along.
Crystallographic Point Groups
Elizabeth
Mojarro
Crystals are
beautiful and fascinating substances that have captivated the
attention of the general public and mathematicians alike for
hundreds of years. Their unique characteristics have been the
subject of continual study and led to the development of
crystallography (i.e. the study of the structures of crystals).
Group theory is an essential tool used in classifying these
structures. In particular, crystals can be classified into 32
distinct crystallographic point groups, which are crystallographic
groups that leave a common point fixed. I will discuss the role of
Group Theory in crystallography and its role in determining the
point group a crystal belongs to.
The
Game of Nim and Its Winning Strategy
Jason
Jebbia
Do games and
puzzles involving numbers and strategy interest you? The game of
Nim is just that kind of game. It has been played for hundreds of
years under different names. Nim is a simple, mathematical game in
which two-players take turns removing objects from piles until no
objects are left. For such a simple concept, this game has quite a
fascinating theory. In this talk, modular arithmetic and binary
numbers are used to develop a winning strategy for this game and
variations of the game. The importance of Nim to the Sprague-Grundy
Theory for impartial games will also be discussed.
Macroscopic ODE Models of Traffic Flow
Zhengyi
Zhou
Have
you ever wondered how traffic lights are synchronized to minimize
congestion? Macroscopic models of traffic flow using ordinary
differential equations may help. It is a relatively new and
unexplored way of modeling effects of traffic lights on traffic
flow. In this study, a few models are constructed, numerically
analyzed, and applied to a sequence of lights and a traffic junction
in search for optimal traffic control strategies. The main math
tool used in this study is ordinary differential equations, but
anyone with some calculus knowledge can understand the talk.
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