2002 - 2003
October 11, 2002
Title: "Topics in Complex Analysis"
Valleri Bond
October 18, 2002
Title: "Topics in Real Analysis"
Deanne Newborg
October 18, 2002
Title: "Topics in Algebra"
Andrei Ghenciu
October 25, 2002
Title: "Topics in Topology"
Ross Bryant
November 1, 2002
Title: "On Kneser's spiral theorem"
Dr. Anghel
Abstract: A beautiful result in plane differential
geometry, due to Kneser, states that spirals can be equivalently
characterized as --- curves with an increasing radius of
curvature function, or --- curves which pass from the inside
to the outside of any disk of curvature, or --- curves admitting
a nested family of disks of curvature. In this talk I will
give a different proof to this result, based on the Cauchy-Schwarz
inequality
November 8, 2002
Dr. Gao: Title and abstract TBA.
2001-2002
October 5, 2001
Title: "Differential-Algebraic Equations: Applications
and Numerical
Solutions"
Ahmed Rashed, University of North Texas
Abstract: Differential-Algebraic Equations
(DAEs) describe dynamical processes that are restricted
to some constraints. They arise in many applications, e.g.,
in the electric circuit analysis (network modeling), chemical
reaction kinetics, motion of mechanical systems (constrained
variational problems), control theory, singular perturbation
problems, and discretization of partial differential equations
(PDEs).
Numerical solution of DAEs is relatively new field within
mathematics introduced in the early 1970's). Only a few
existing numerical solvers are capable of handling such
systems. In this talk we will:
· Introduce the different forms of DAEs
· Give a brief description of the relevant problems
· Survey some applications of DAEs
· Give some basic theoretical aspects of numerical
methods and
solutions.
September 21, 2001
Title: "The Complexity of Classification Problems
in Mathematics"
Su Gao, University of North Texas
Abstract: Your advisor gave you a classification
problem to work on as your thesis subject. You worked hard
but still couldn't solve it. Is there a way out of graduate
school (with a Ph.D.)? Well, you probably can write a thesis
saying that the problem is too hard -- and you'll pass if
you know how to say it properly. In this talk I will discuss
a (descriptive set theoretic) way to analyze the complexity
of classification problems in mathematics.
November 2
Title: "A very short introduction to Galois
theory"
Charles Conley, University of North Texas
Abstract: We will describe the Galois
group of a field extension with a minimum of technical detail.
The emphasis will be on specific examples illustrating things
such as the following: a systematic way to find the solutions
of the cubic and the quartic, the insolvability of the quintic,
and the fact that the Galois group of a finite field extension
is a cyclic group generated by a power of the Frobenius
map. We will assume no more than a little familiarity with
the polynomial ring Q[x] over the rationals and the fields
of integers modulo primes.
November 9
Title: "Polya's Theorem on Simple Random Walks"
Barret Thompson, University of North Texas
Abstract: "Suppose you stood on a
line and randomly took steps forwards or backwards. For
instance, you could flip an unbiased coin and take a step
forward if the coin turns up heads or backwards if the coin
turns up tails. This is a simple random walk on the line.
Similarly, you could stand on a plane and take random steps
forwards, backwards, to the left, or to the right (roll
a four ided die to determine which direction to go). This
would be a simple random walk in the plane. The notion of
a simple random walk in any dimension can be easily generalized.
If you take a simple random walk on the line or in the plane,
you will always end up back where you started. However,
this is not true in 3 or more dimensions. In other words,
if you leave home in 3 or more dimensions, there is a non-zero
probability that you will never return home (hence you could
be lost in space forever). This curious result was proven
by Polya in 1921. I intend to give a very brief introduction
to the theory of random walks, then I will give a sketch
of the proof of Polya's theorem. Time permitting, I will
calculate the exact probability of returning to the origin
on a 3 dimensional random walk."
March 8, 2002
Title: "Forest Landscape Dynamics: Modeling
and Management"
Michael Monticino, University of North Texas
Abstract: There is an increasing realization
that forest ecosystem management decisions are best made
by considering the consequences on a landscape scale. Mathematical
models involving complex spatial processes have been developed
to simulate forest landscape dynamics under various management
scenarios. These models generally either concentrate on
representing the details of forest time dynamics --- growth
and succession --- or focus upon simulating the interactions
between neighboring sections of the landscape. This talk
discusses current research on developing models that include
both spatial interactions and succession dynamics. Besides
the mathematics behind the models, simulation examples derived
from the H.J. Andrews Forest in Oregon will be presented.
April 5, 2002
Title: "The 65,537-gon"
Charles Conley, University of North Texas
Abstract: The ancient Greek geometers
posed the following problems regarding constructions with
a compass and straightedge:
1. Can the cube be doubled?
2. Can angles be trisected?
3. Can the circle be squared?
4. Which regular polygons can be constructed?
They remained open for over 2000 years. We will give their
modern solutions, focusing on the last, which was solved
by Gauss at the age of 19. The usual solution does not yield
explicit constructions of the constructible regular polygons.
We will outline explicit constructions, related to finite
Fourier transforms. There are no "prerequisites:"
we will not use any Galois theory, and we will develop what
little field theory we need as we go. "
April 12, 2002
Title: "The Banach-Tarski Paradox"
Barrett Thompson, University of North Texas
Abstract: The Banach-Tarski theorem asserts
that you can cut a ball up into a finite number of pieces
and rearrange the pieces (via isometries) to produce two
balls, each the same volume as the original ball. This striking
result is so counter-intuitive that it is usually referred
to as the Banach-Tarski paradox. It is actually one of the
first theorems encountered in the broader theory of paradoxical
decompositions.
After a brief introduction to some standard terminology
(that will make the statement of the problem precise), I
will give a careful proof of the Banach-Tarski paradox.
The implications of this theorem and the role of the axiom
of choice will also be briefly discussed. Only some basic
knowledge of linear algebra and group theory will be assumed.
I will supply a full set of notes to everybody, so the difficult
parts of the proof can be reviewed.
April 19, 2002
Title: "Transfinite Induction, Recursion, and
Ordinal Numbers"
Ross Bryant, University of North Texas
Abstract: What does "transfinite
induction on the ordinals" mean? How does it differ
from "transfinite recursion on the ordinals"?
What are the ordinal numbers and how do they correspond
to other number systems? To answer these, we reexamine the
familiar finite induction and recursion theorems which use
the natural numbers in the same way that their transfinite
counterparts use the ordinals. Once done, we develop the
ordinal numbers, ordinal arithmetic, and their relationship
to cardinal numbers.
This talk is intended for the beginner; we will give examples
rather than proofs in most instances. Our goals are ambitious,
and as such, this talk favors intuition over rigor. All
are welcome to attend.
2000-2001
April 20, 2001
Title: "Approximations of continuous functions
by Lipschitz
functions"
Radu Miculescu, University of North Texas
Abstract: We shall present two techiques(one
based on the existence of LIP partition of unity and the
other on Zorn lemma and some extension results) for obtaining
results about approximation of (uniformly) continuous functions
by (LIP) Lipschitz functions.
February 16, 2001
Title: "Making Your Retirement Fund Last Forever"
Dr. John Quintanilla, University of North Texas
Abstract: Suppose that regular disbursements
are to be made from a portfolio consisting of one or more
stocks. Since the return on securities is uncertain, financial
planners must choose a disbursement level and/or portfolio
mix which ensures a certain probability that the portfolio
survives forever. We consider a discrete-time model for
the value of a portfolio with random multipicative returns
and constant withdrawls. This model allows the multiplicative
returns of the stock to follow any specified distribution,
including the commonly used lognormal distribution. An integral
equation for the survival probability function is developed,
allowing for its numerical evaluation. We also study the
sensitivity of the survival probability upon the parameters
of market return and volatility.
February 9, 2001
Title: "Some problems in classical analysis
and planar geometry"
Dr. R. Daniel Mauldin, University of North Texas
Abstract: I will discuss some simply stated
long standing problems in real analysis and geometry due
to Erdos, Schmidt, Chung, Steinhaus, etc. and their status.
February 2, 2001
Title: "On the classification of self-similar
sets determined by two contractions on the plane"
Kiko Kawamura
Abstract: In the long history of mathematics,
we have seen some discoveries of strange functions, which
gave us a strongly impact; for examples, the Takagi function,
constructed as a simple example of a nowhere differentiable
but continuous function, the Von Koch curve, a continuous
Jordan curve, admitting no tangent line anywhere, and the
L\'evy curve, a continuous curve but its area is positive,
and so on. Each of these curves was discovered independently
and initially, no relationships between them were known
for a long time. However, in 1984, Hata and Yamaguti showed
the close relationship between the Takagi function and Lebesgue's
singular function, a monotone increasing continuous function
whose derivative is zero almost everywhere.
A generalization of this relation was considered by Sekiguchi-Shiota
in 1991, and showed that it has a nice application to an
open problem about digital sums. Also, Tasaki, Antoniou
and Suchanecki pointed out Hata-Yamaguti's result has some
valuable applications in physics. The purpose of our work
is to extend Hata-Yamaguti's results by finding close relationships
among other strange functions from the viewpoint of the
theory of fractal geometry. We define binary self-similar
sets as self-similar sets defined by two contractions on
the plane, and classify them into four classes determined
by the form of their functional equation. Also, we show
that our main theorems have nice applications to several
open problems in other fields.
December 1, 2000
Title: "Things you can do with 3 by 3 matrices"
Dr. Charles Conley
Abstract: We will describe the representations
of the SU(2) and SU(3), the groups of 2 by 2 and 3 by 3
matrices whose inverses are equal to their conjugate transposes.
These are important symmetry groups in physics, and their
representations are in some sense the various disguises
in which they can appear.
The physical importance of SU(2) is explained by the fact
that it is the double cover of the group SO(3) of rotations
of 3-space, but the appearance of SU(3) as the symmetry
group of the strong nuclear interaction is quite surprising
and mysterious. If there is time, we will conclude with
the story of Gell-Mann's stunning prediction of the Omega
particle in 1964.
Or strategy will be algebraic: the problem comes down to
studying the representations of the Lie algebras of traceless
2 by 2 and 3 by 3 complex matrices. We will describe the
results without assuming any prior knowledge of Lie groups,
Lie algebras, or their representations.
November, 10, 2000
Title: "Unique Range Sets for Polynomials"
Dr. William Cherry
Abstract: A set is called a "unique
range set" for polynomials if the inverse image of
the set uniquely determines the polynomials. Ostrovskii,
Pakovitch, and Zaidenberg have given a nice geometric characterization
of unique range sets for polynomials of fixed degree. I
will discuss their proof and mention some related open problems.
November 3, 2000
Title: "Introduction to Quantum Computing"
Dr. John Neuberger, University of North Texas,
Abstract: A framework which models some
aspects of quantum computation is given. The model is in
terms of linear algebra. An attempt is made to describe
the potential power of quantum computation and why it might
be revolutionary. Two questions are discussed:
1. What are the prospects for constructing a useful
quantum
computer?
2. What are some algorithms that might be implemented
if such a
computer were made?
October 27, 2000
Title: "Things to do with 2 by 2 matrices"
Dr. Robert Donley, University of North Texas
Abstract: This talk is about the connection between the
Lie group SL(2,R), the two by two matrices of determinant
1, and the Lie algebra of traceless two by two real matrices.
The key ingredient is the exponential map. We also introduce
the notion of a representation and give an application.
October 13, 2000
Title: "Sigma-finite invariant measures for
the exponential family"
Dr. Mariusz Urbanski
Abstract: The family ofmaps $z\mapsto
\lambda \exp(z)$, $\lambda \ne 0$ of the complex plane,
called the exponential family will be explored. The concept
of Julia sets will be introduced. The distortion property
in the spherical metric of branches of logarithms will be
established. This property along with Marco Martens criterion
for the existence of $\sigma$-finite invariant measures
will be applied for the maps from the exponential family.
Sept. 17, 2000
Title: "The Minimum Distance Theorem"
Rhonda Huettenmueller, UNT Teaching Fellow
Abstract: Given two smooth nonintersecting
curves such that the minimum distance between them exists,
how can the points at which the minimum distance occurs
be found? There is an elementary formula which can find
them. A couple of consequences of the formula will also
be discussed.
1999-2000
Oct. 8, 1999
Title: "Applications of Numerical Methods for
Stochastic Processes"
Frederi Viens, UNT Math Faculty
Abstract: Stochastic partial differential equations (SPDEs)
are some of the most useful tools in mathematical modeling,
in situations in which quantities depend on time and other
parameters (e.g. space), and have a highly irregular random
behavior. We will briefly introduce these equations, focusing
on the "parabolic" type. We will discuss the possible
way of writing explicit representation formulas for parabolic
SPDEs, including the particle method, and how such formulas
can be used to numberically simulate SPDEs.
Examples of possible practical applications of these techniques
to well-known open problems in applied mathematics will
be discussed, including:
* estimation of stochastic volatility (finance),
* the calculation of Lyapunov exponents, as connected
to
magneto-hydrodynamics (fluid dynamics),
* the study of stability properties for stochastic filtering
(signal processing).
Subject to departmental approval, a beginning graduate
level course (MATH 5820) covering the above topics may be
offered as soon as next Spring.
Oct. 15, 1999
Title: "Dynamical Systems and their Minimal
Sets"
Alex Clark, UNT Math Faculty
Abstract: We will discuss vector fields
and how they give rise to a dynamical system. We will discuss
limit sets of dynamical systems and how the topology of
minimal sets is related to properties of the system being
modelled.
Oct. 22, 1999
Title: "Transcendence of numbers with low block
complexity expansions"
Luca Zamboni, UNT Math Faculty
Abstract: It is conjectured that the digit
expansion (in any integer base) of an algebraic irrational
(e.g., root 2) is random in every sense of the word. For
example, somewhere in the decimal expansion of root 2 one
should find (in fact infinitely often) a block consisting
of a billion and one consecutive 7's. An alternative formulation
of this idea is that if the digit expansion of an irrational
number is constructed by a simple algorithm, then the number
must be transcendental. For instance the Champernowne number
0.1234567891011121314..., obtained by concatenating the
decimal expansions of consecutive integers, is a transcendental
number. Analogously, it is believed that the sequence of
partial quotients in the continued fraction expansion of
an algebraic irrational of degree greater than two is also
extremely random. In particular it is conjectured that if
the sequence of partial quotients of an irrational x is
bounded, then x is either quadratic or
transcendental. These conjectures are extremely difficult,
and appear completely out of reach at the present time.
We will discuss some recent partial results in case the
expansion (digit or continued fraction) has linear block
growth.
Oct. 29, 1999
Title: "Applications of Locally Lipschitz Partition
of Unity"
Radu Miculescu, UNT Math Teaching Fellow
Abstract: J. Luukainen and J. Vaisala have
shown that there exists a LIP (locally Lipschitz)-partition
of unity for metric spaces and that we have a LIP version
of Dowker Theorem. G. Georganopoulos has sown that a continuous
function f: X \rightarrow B, where X is a compact metric
space and B a convex subset of a real normed space Y, is
a uniform limit of Lipschitz maps from X to B. This result
is obtained using a Lipschitz partition of unity. We show,
using the LIP version of partition of unity, due to Luukainnen
and Vaisala, a generalization of this result. Namely, our
first application of LIP-partition of unity states that
a bounded continuous function f:X \rightarrow B where X
is a metric space and B is a convex subset of the real normed
space Y, can be approximated, in the uniform norm, by LIP
function. Obviously, when X is compact, we get the Georganopoulos's
result. The next application is based on the LIP-Dowker
Theorem. Using this result, we give a characterization of
the distance between a Lipschits function f: X \rightarrow
R and a certain subset of the set of Lipschitz functions
from X to R, X being a compact metric space. Here, f and
the set of Lipschitz functions are viewed as elements of
C(X, R) endowed with the uniform norm. As a third application
of LIP-partition of unity we prove that giving an open convex
neighborhood V of 0_Y in Y and a function \phi : X \rightarrow
P(Y) having certain properties, where X is a metric space
and Y is a real normed space, we can pick elements f(x)
in \phi(x) + V, so that f:X \rightarrow Y is LIP.
Nov. 5, 1999
Title: "Experiences of Doctoral Students in
Mathematics in New Zealand"
Margaret Morton, UNT Visiting Professor (University of Auckland,
New Zealand) Abstract: In New Zealand (NZ)
there has been a rapid growth in the number of mathematics
doctoral students over recent years. Moreover, mathematics
education is now an area of doctoral study, and these students
are often associated with mathematics departments. Whilst
the literature contains a number of students about the experiences
of graduate students in other countries little has been
written about the situation in NZ.
In order to obtain information about factors affecting
doctoral study in mathematics and mathematics education
in NZ universities, a questionnaire was developed and mailed
individually to all appropriate registered doctoral students.
The type of information sought from the students was their
previous education, the financial support they're receiving,
what motivated them to pursue a doctorate, how they negotiated
their choice of topic and supervisor, the level of satisfaction
they are experiencing from their studies, and their career
expectations. Also of interest was the gender composition
of the group and whether gender was a critical factor in
their recruitment and subsequent experiences as doctoral
students.
The findings from this survey will be presented. Comments
will be most welcome.
Nov. 12, 1999
Title: "Continuous Newton's Method for Polynomials"
John W. Neuberger, UNT Math Faculty
Abstract: Continuous Newton's method for
finding roots of polynomials is defined. This process is
seen to be much more orderly than conventional Newton's
method. Results give rise to a multivalued dynamical system.
Will give a five line routine written in Mathematica for
visualizing the position of roots of a polynomial. This
talk might be considered as introductory to the study of
continuous descent methods for finding solutions to differential
equations.
Nov. 19, 1999
Title: "An Introduction to Expectation Pricing
and Arbitrage Pricing and the Straightforward Probability
and Statistics Needed to Set up a Forwards Stock Model-
A Book Report"
LeRoy Valdes, UNT Math Teaching Fellow
Abstract: A stock is being sold forward
when two parties enter a contract where one agrees to give
the other the stock at some agreed point in the future in
exchange for an amount agreed now. What amount should be
written into the contract now to pay for the stock one year
in the future? Using Kolmogorov's Strong Law Of Large Numbers
and the Law of the Unconscious Statistician, one can answer
the above pricing question. But is this the correct value?
A model for forwards that does not require expectation or
even a distribution on the stock price can be developed
to set an arbitrage price. It is possible that the strong
law sets up a price that could lead to unlimited riskless
profits for one of the parties involved. If this is the
case; Mortgage the House, Sell the Dog, and Take Advantage.
Feb. 4, 2000
Title: "Where Did Functional Analysis Come
From?"
John W. Neuberger, UNT faculty
Abstract: Functional analysis may be thought
of as linear algebra generalized to infinite dimensional
linear spaces. Problems in differential equations related
to some of the main scientific developments of the past
century raised deep questions as to the possibility of such
generalization. I will try to trace some of this history.
Feb. 18, 2000
Title: "Real-time statistical data analysis:
Internet marketing, the hype, hope and horror"
Michael Monticino, UNT faculty
Abstract: Using cookies, registration,
platform and access information, ebsites can learn a great
deal about visitors by tracking their item-by-item click
histories -- or clickstreams. This information can be employed
to customize sites dynamically and target ads and content
to individual visitors. At least that is the hope and the
hype of Internet marketing companies, like DoubleClick.
The reality is that there are significant data management
and statistical challenges in analyzing and interpreting
clickstream data. This talk will discuss these challenges
and some potential solutions. The talk is based on Dr. Monticino's
work with IBM and his current work with a Silicon Valley
start-up company.
March 24, 2000
Title: "Getting an Academic Job at a Primarily
Undergraduate Institution"
(Part I) "An Exploration of Two Dimensional
Manifolds" (Part II)
Cami Sawyer, Ph.D UNT 1999, TWU Visiting Asst. Professor
Abstract: Since I just went through the
academic job search process, my hope is to pass on some
hints and advice that will make the process smoother for
you. I have suggestions for you if you are just starting
on your masters (what to do now to build up your resume)
or if you are going to be applying for jobs next year (helpful
web sites).
At each interview I had to give a talk. The talk I gave
was titled "What Is This Surface?", subtitled
"An Exploration of Two Dimensional Manifolds."
I give the classification of all two dimensional manifolds
and show some of the motivations behind it.
April 14, 2000
Title: "Extensions of Lipschitz and Locally
Lipschitz Functions"
Radu Miculescu, UNT graduate student
Abstract: Tietze's theorem concerning
extensions of a continuous map is a central theorem in mathematical
analysis. It is natural to try to find similar results for
other classes of functions. The class of Lipschitz functions
is of particular interest since the problem of extension
of such functions comes up in geometry also. The basic results
in this direction are due to McShane and Kirszbraun. Namely
a Lipschitz function from a subset of a metric space to
R, or from a subset of R^n to R^m can be extended to the
whole space conserving the Lipschitz constant. There are
a lot of generalizations of these results due to Valentine,
Schonbeck,Flett, Mustata, Czipszer & Geher, Vaiasala
& Luukkainen. The speaker will also present one original
result about extensions of LIP functions between a closed
subset of a separable metric space and a "Lipschitz
manifold modelled into m."
April 21, 2000
Title: "Financial Derivatives: A story on Straddles
and Strangles"
Ruth Michler, UNT faculty
Abstract: We will give a brief introduction
to Financial Derivatives and give examples on their use.
April 28, 2000
Title: "On the Classification of Self-Similar
Sets"
Kiko Kawamura, post-doctoral visitor, Nara Women's University,
Japan Abstract: We classify self-similar
sets constructed by two similar constructions into four
classes from the viewpoint of functional equations. In this
classification, we can not only show the close relationship
among functions with self-similarity but also give solutions
to some open problems in several fields.
