#### Mathematics Department Events

**Spring 2015 Colloquia**

*Tangled up in Red, Green and Blue (with Apologies to Bob Dylan): An Introduction to
the Mathematical Theory of Knots.* Ron Taylor of Berry College

**Tuesday March 24, 2:30pm in A 215 (Student Center)**

**Abstract:** Knot theory as a scientific endeavor dates back to the 1800s when chemists thought
that the elements could be in the shape of knots. Mathematically the study of knot
theory is also relatively young, but it is a vibrant area of study. In this talk we
will discuss some of the historical development of the mathematics of knots and the
central questions that knot theorists ask. There are a variety of techniques called
knot invariants that are used to answer these questions. We will discuss a few of
these invariants along with the foundational ideas that go along with validating these
techniques. We will also prove that one particular method for studying knots is an
invariant and talk about some open questions in knot theory.

**Cycles, triangles, and colorings: The power of Paul Erdos to shape Discrete Mathematics.
****Jennifer Vandenbussche of Kennesaw State University**

**Friday March 13 (π-day-eve), 3:14pm in Q202**

**Abstract: **The Cycle-Plus-Triangles graph coloring problem was made popular by Paul Erdos, one
of the leading discrete mathematicians of the 20th century. In this talk, I will
briefly introduce Paul Erdos, talk about his Cycle-Plus-Triangles coloring problem,
and discuss some extensions of it. I will keep the talk accessible to all undergraduates,
so everyone can celebrate Pi Day by coming to see an example of what research in mathematics
looks like. (Hint: It doesn't have to involve calculus!)

**Translation Lengths of Point-Pushing Pseudo-Anosov Maps of Riemann Surfaces. ****Chaohui Zhang of Morehouse College**

**Thursday February 19, 2:30pm in A 215 (Student Center)**

**Abstract: **Let

*S*be a hyperbolic Riemann surface of genus

*p*> 1, which has no free boundary curve and contains only one puncture

*x*. Let

*C(S)*denote the curve complex of S, which is equipped with a path metric

*d*. Let

_{C}*C*denote the set of vertices of

_{0}(S)*C(S)*. Masur-Minsky showed that there exists a constant

*c*> 0 such that for any pseudo-Anosov map

*f : S →S*, any integer

*n*and

*u*in

*C*we have

_{0}(S),*d*

_{C}(u, f^{n}(u)) > c|n|.

In this presentation, we first prove that for any multi twist *f*, *d _{C}(u, f^{m}(u)) *remains bounded. Then we consider the subgroup

*F*of the mapping class group Mod(

*S*) which consists of elements isotopic to the identity on

*S*U {

*x*}. We show that for any pseudo-Anosov element

*f*in

*F*,

*d*for

_{C}(u, f^{m}(u)) ≥ m*m*= 1, 2, ..., 7

*; and*(2

*d*_{C}(u, f^{m}(u))*≥**m*+ 5)/3 for

*7*

*m >**It follows that the action of any pseudo-Anosov element*

*.**in*

*f**on*

*F**is hyperbolic (in the sense of Gromov). Furthermore, we obtain a positive lower bound on the translation length for the action of*

*C(S)**f*on

*C(S)*.

**Fall 2014 Colloquia**

**Computational Graph Theory. Brian Patterson, Oglethorpe University**

**Thursday October 23, 2:30pm in A 216 (Student Center)**

**Abstract: **Dr. Patterson will present two areas of graph theory in mathematics of interest to
computer scientists: the optimal weighted Hamiltonian cycle problem (also called traveling
salesperson) and the graph coloring problem. The traveling salesperson problem is
sometimes used in Oglethorpe’s core mathematics class to explain an intractable but
useful mathematical problem that computer science provides approximate solutions for.
Two algorithms will be presented. The graph coloring problem is a challenging and
ongoing area of mathematical research with many applications to scheduling problems.
It is presented briefly in the Oglethorpe core mathematics course but explored more
thoroughly in the upper-level graph theory class and independent study. Depth-first
and genetic algorithms approaches will be presented.

**How small is too small? Modeling the effects of habitat fragmentation via reaction
diffusion equations. Jerome Goddard II, Auburn University**

**Monday September 22, 2:30pm Ballroom B (Student Center)**

**Abstract: **Habitat fragmentation occurs when an organism’s preferred habitat is divided or broken
into smaller fragments (called patches) and can be caused by natural events, such
as geological processes, or human activity, such as land conversion. Habitat fragmentation
is often cited as a contributor to animal species becoming threatened or endangered.
Two important aspects of habitat fragmentation are the size of fragmented patches
of preferred habitat and the inferior habitat surrounding the patches, called the
matrix. Ecological field studies have indicated that an organism’s survival in a patch
is often linked to both the size of the patch and the quality of its surrounding matrix.
In this talk, we will focus on modeling the effects of habitat fragmentation via the
reaction diffusion framework. The reaction diffusion framework has been extensively
employed in population dynamics providing important biological insight into the patch-level
consequences of various assumptions made on individual behavior in ecological systems.
Such models have seen enormous success both in their empirical validation with actual
spatio-temporal distribution data and their ability to yield general conclusions about
an eco-system based on the analytical results of these theoretical models. First,
we will introduce the reaction diffusion framework and a specific reaction diffusion
model with logistic growth and Robin boundary condition (which will model the negative
effects of the patch matrix). Second, we will use mathematics to explore the dynamics
of the model via the well-known quadrature method and ultimately obtain a causal relationship
between the size of the patch and the quality of the matrix versus the maximum population
density sustainable by that patch. This important example regarding habitat fragmentation
will hopefully serve to illustrate the usefulness of mathematical models in helping
to understand complex biological relationships.

**Spring 2014 Colloquia**

**Wavelets, the Molecules and Atoms of the Mathematical Universe****Josip Derado, Kennesaw State University**

**TuesdayMarch 25 2:30pm Q207**

Wavelets are the building blocks of a mathematical universe. Specifically they are an orthonormal basis for the function spaces. In this talk we will present an introduction into wavelet theory. We will start with Haar wavelets, and we will see how their structure is generalized by Yves Meyer and Stephan Mallat. We will then see how to construct Daubechies wavelets. At the end we will talk about some open problems in the wavelet theory.

**The Life of Pi: On the Life and Contributions of Ramanujan Zhu Cao, SPSU**

Q202, 3:14 PM Piday March 14

Although he died at the age of 32, legendary Indian mathematician Srinivasa Ramanujan independently established nearly 3900 results (mostly identities and equations) that are recorded in his three notebooks and his "lost" notebook. During his short lifetime, he made remarkable contributions to number theory, combinatorics, and analysis, among them fascinating formulas that can be used to calculate digits of pi in unusual ways. In this talk, I’ll give an introduction of some formulas for pi and the life of Ramanujan.

**New Problems in Matrix TheoryUlrica Wilson, Morehouse College/ICERM**

Tuesday, February 18 2:30 PM Q207

Refreshments 2 PM Building D Second Floor

An eventual property of a matrix M is a property that holds for all powers M^k, for some positive integer k. Eventually positive and eventually nonnegative matrices have been studied extensively since their introduction by Friedland in 1978. I will introduce some eventual properties of matrices and student projects that have come from this work. I will also describe an annual summer workshop for faculty at primarily undergraduate institutions jointly sponsored by the American Institute for Mathematics and the Institute for Computational and Experimental Research in Mathematics; and the EDGE Program for women entering PhD programs in mathematics.

**Fall 2013 Colloquia**

**Sum-Product Inequalities****Albert Bush, Georgia Institute of Technology**

Tuesday, October 22 3 PM Q207

Refreshments 2:30 PM Building D Second Floor

Erdos and Szemeredi conjectured that if one has a set of n integers, one must have either the sumset or product set be of nearly maximal size, n^(2-epsilon). In this talk, I will introduce the sum-product problem, show previous, beautiful geometric proofs by Solymosi and Elekes, and discuss some recent progress by myself and some collaborators.

**Periods of sequences given by linear recurrence relations mod p**

**Alan Koch, Agnes Scott College**

Tuesday, September 17 3 PM Q207

Reception and snacks at 2:30pm on 2nd floor of Building D

Let p>2 be prime. Any integer sequence which satisfies a linear recurrence relation,
for example the Fibonacci sequence, becomes periodic when reduced mod p. If w is the
order of the recurrence relation then the period length of the sequence is bounded
by p^{w}-1. The length of the period also depends on the integer coefficients of the recurrence
relation and the initial conditions for the sequence. Using linear algebra and some
number theory we will determine which positive integers arise as period lengths for
some sequence. In the case where w=2 we will give a formula for the probability that
randomly chosen integer coefficients give a period length of p^{2}-1.

** **

**Spring 2013 Colloquia**

**Symmetry Groups and Temari Balls PlusCarolyn Yackel, Mercer University**

Tuesday, March 26 4 PM Q207

Refreshments 3:30 PM Building D Second Floor

This talk will begin with a broad introduction to the field of mathematical fiber arts. The talk will then focus on presenting the 14 finite spherical symmetry groups and will show that they can be manufactured via the Japanese craft of temari (embroidered thread balls). Time permitting, the conclusion of the talk will move to a discussion of other interesting temari problems.

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**Give Me a Place To Stand, and I Will Make You PiTh**

**ursday, March 14, 3:14 PM Q 202**

In Celebration of Pi Day

Professor Steven Edwards

In Celebration of Pi Day

Professor Steven Edwards

We explore the life, times and achievements of Archimedes, the greatest engineer and mathematician of antiquity. Archimedes was the first to give a method that could be used to calculate pi to any degree of accuracy. In this talk we will give an introduction to some of Archimedes’ most interesting accomplishments, including his visionary method for calculating pi .

**Mariana Montiel, Georgia State U. Mon. Feb. 25th talk at 4pm in Q206Meet the speaker: (coffee & cookies) 3:30pm 2nd floor DMathematical Music Theory: Some Selected Topics from a Research Field with Roots in
the Origin of Modern Science.**Abstract: In the first part of this talk we will give an overview of some historical
antecedents of modern Mathematical Music Theory, and how the study of musical objects
has contributed to the advancement of scientific knowledge. We will also give a brief
description of how certain notions and structures that this field has produced,
such as maximal evenness and Vuza tiling (rhythmic) canons have led to the solution
of open problems in Mathematics, as well as to applications in other sciences, such
as Physics. In the second part of the talk, we will learn to identify well-formed
scales by the symmetry condition, understand the duality relation (Sturmian involution)
as it applies to the six authentic modes, and get a notion of the mathematical motivation
behind the development of the software Rubato composer. whose Forms and Denotators
provide a means for implementing an important part of Computational Category Theory.
If time permits, we will describe the notions behind Euclidean Rhythms.

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**Fall 2012 Colloquia**

**Can Mathematics Heal All Wounds?**

Richard Schugart, Western Kentucky UniversityTuesday, November 27 3:00 PM

**Q 108**

Refreshments at 2:30 2nd floor of the Mathematics Building (D)

In this talk, I will begin by providing background on the wound-healing process, the importance of oxygen in the wound-healing process, and methods for treatment with oxygen therapy. A mathematical model using differential equations will be presented for the treatment of a bacterial infection in a wound using oxygen therapy. Analysis of a sub-model (i.e., with two equations) and simulation results will be presented, with an emphasis on research results produced by undergraduate students. A modification to the model for the treatment of a chronic wound using optimal control theory will also be presented. Previous work will also be highlighted and briefly discussed.

**How to Tame a Dragon**Scott M. Bailey, Clayton State University

**Monday, October 29 4:15 PM**

**Q 109**

Paul Levy introduced the "Levy Dragon" fractal in 1938. Although it has become a well-known and well-studied example in the field of fractals, the question of "What does the Levy Dragon look like?" has never been answered. In this talk, we will answer this question by journeying deep inside Paul Levy's dragon, and find that it is very tame after all.

**Euclid and Dehn in the Operating Room**

Margaret Symington, Mercer University

Tues. Sept. 25 4pm Q 108

Refreshments at 3:30 2nd floor of the Mathematics Building (D)

In this talk I will describe how a quest for the mathematics behind a

surgical improvement led to some interesting Euclidean geometry and

highlighted a little known kinship between dermatologists and geometric

topologists. Along the way I will share some topological gems and pose a set of geometry
questions that anyone can work on.