We are very proud to present the Mathematics Research Seminar at UHH!  This will be a frequent seminar given by mathematicians and other interested parties in topics of their choosing.  The talks are to be aimed at advanced undergraduates or curious researchers in the area.

All current talks are scheduled for Mondays from 3:00 - 4:15 in K 109 at the UH-Hilo campus.

All are Welcome!






Monday, November 16
Title: Buffon and His Random Needle
Speakers: John Robertson, Calvin Luscombe, and Chai Young Kim
Time: 3:00 - 4:15 PM
Room: K 109

Abstract:  Suppose we have a floor made of parallel strips of wood,
each the same width, and we drop a needle onto the floor. What is
the probability that the needle will fall on a line between two strips?

Buffon’s needle problem asks exactly this:  What is the probability
that a needle of a given length l will land on a set of parallel lines
set distance l apart from each other. This problem was first posed
by the French naturalist Buffon in the 18th century.

In this talk, we will discuss how to understand this geometric
probability question by constructing a probability measure on a
configuration space and computing the probability of a needle falling
on a line.  We will also demonstrate how this computation and a
little experimentation will lead to physical way of approximating a
very important mathematical constant. 

This talk will be very accessible to undergraduate Mathematics students.


Monday, November 23
Title: Noncooperative Collusion Under Imperfect Price Information
Speaker: Calvin Luscombe
Time: 3:00 - 4:15 PM
Room: K 109

Abstract: We will be reviewing a paper by Edward J. Green and Robert
H. Porter entitled "Noncooperative Collusion Under Imperfect Price
Information".  We will see how imperfect price information can cause
episodic industry price drops eluding to competitive forces that may
actually be oligopolistic behavior. We look at equilibrium's robust
enough to handle deviation from the collusive norm without complete
break down. We will use our model to draw clear-cut conclusions about
the presence of absence of collusion in specific industries on the basis
of market data. In this presentation we will use techniques from game
theory, dynamic programing and calculus.


Monday, November 30
Speakers: Calvin Luscombe, John Robertson, Joshua Loving
Time: 3:00 - 4:15 PM
Room: K 109

~~~~~~~ARCHIVED TALKS~~~~~~~~


Tuesday, October 28
Speaker: Bob Pelayo
Title:  Braids Groups and Braid Representations in Knot Theory
Time: 3:15 - 4:15 PM
Room: MSB 101

Abstract:  In recent years, braid groups have received much attention in areas as widespread as knot theory, topological quantum field theory, and operator algebras.  The intrigue of these objects lies in the topological interpretation of a braid as a collection of vertical strands intertwined or 'braided' together in a natural way.  Via concatenation, these braids inherit a group structure with a very interesting group presentation.  We will discuss the relationship between braid groups and knot theory as well as investigate some uses of braid groups in finding quantum invariants.
This talk will be motivated by geometric arguments and should be accessible to advanced undergraduates.



Thursday, November 13
Speaker: Efren Ruiz
Title: Shift Spaces, Graphs, and their Invariants
Time: 3:15 - 4:15 PM
Room: MSB 101

Abstract:  Dynamical systems arose in the study of systems of differential equations used to model physical phenomena like motions of the planets or of molecules in a gas.  One way to study these systems is to relate them to a discrete space consisting of infinite sequences of symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator.  This method is known as symbolic dynamics.   Symbolic dynamics has also been used in data storage and transmission.

In this talk, I will introduce the notion of shift spaces and discuss their relationship to directed graphs.  I will also discuss various notions of shift equivalences and their invariants.



Tuesday, November 25
Speaker: Brian Wissman
Title: Numerically Solving Differential Equations
Time: 3:15 - 4:15 PM
Room: MSB 101

Monday, January 26, 2009
Speaker: Shuguang Li
Title: Survey of Research in Primitive Root - Part I
Time:  3:00-4:15 PM
Room:  UCB 111

Abstract:  Primitive roots were traditionally defined for a special
type of integers, such as primes. Although the concept
was extended to composite integers by American Mathematician,
R. Carmichael, research of it stayed at very fundamental
level until about the end of the 20th century when study of
distribution of primitive roots began. Soon it was discovered
that the distribution of primitive roots for composite integers
is significantly different from those for primes, which is
known as Artin's conjecture. The first talk on the primitive
roots will survey the results achieved so far on the primitive
roots and its distribution.


Monday, February 2, 2009
Speaker: Shuguang Li
Title: Survey of Research in Primitive Root - Part II
Time:  3:00-4:15 PM
Room:  UCB 111

Abstract:  Primitive roots were traditionally defined for a special
type of integers, such as primes. Although the concept
was extended to composite integers by American Mathematician,
R. Carmichael, research of it stayed at very fundamental
level until about the end of the 20th century when study of
distribution of primitive roots began. Soon it was discovered
that the distribution of primitive roots for composite integers
is significantly different from those for primes, which is
known as Artin's conjecture. The first talk on the primitive
roots will survey the results achieved so far on the primitive
roots and its distribution.


Monday, February 9
Speaker: Ramon Figueroa-Centeno
Title: Imbedding PG(m,q) with m+1 a prime
Time: 3:00 - 4:15
Room: UCB 111

Abstract: The standard non-Euclidean geometries, hyperbolic geometry and elliptical
geometry, both arise by negating the parallel postulate of Euclid. Both these geometries
share with Euclidean geometries an infinitude of points and lines. But also possible are
many finite geometries. Among these are the class of projective geometries PG(m,q)
of projective dimension m (m ≥ 2). These mathematical objects, although primarily
geometric in nature, provide related structures of combinatorial interest: block designs.
Now, A. T. White added a topological flavor to the study of the geometries PG(m,q). In particular, when m = 2 he found models, as imbeddings of Cayley graphs
on surfaces and pseudosurfaces, using voltage graphs (obtained from the classical field
construction of PG(m,q)). In this talk, we discuss an extension of his work for the case when
m + 1 is a prime number, in a similar manner.


Monday, February 23
Speaker: Bill Wright
Title: Developments in Modeling Concurrency
Time:  3:00 - 4:15
Room: UCB 111

Abstract:  Multithreaded programming and concurrent computing is becoming more important as we rely increasingly upon web services and multithreaded applications.  Aside from the issue of how to design correct concurrent programs and/or scheduling techniques without wasting resources, one might ask, how are we to verify the correctness of a concurrent program?

We illustrate concurrency generally, issues of program correctness arising from the typical non-determinacy of possible interleaved execution paths such as deadlock and race conditions, a model of discrete states and arcs representing the execution paths, and state space explosion, in which increasing the number of computing threads renders deadlock identification with accepted heuristics a computationally infeasible problem.  A review of established analysis techniques motivates a look at current efforts to reduce the state space, by geometric methods, to categories about which we might reason.



Monday, March 16
Speaker: Philippe Binder
Title: Dynamics and Forecasting of Two Chaotic Stars
Time: 3:00 - 4:15
Room: UCB 111

Abstract:  The analysis of real-life data that may be chaotic is
illustrated with a study of the light curves of two variable stars.
They turn out indeed to be chaotic, and the analysis goes as far
as inferring the geometry of the trajectories in phase space and
to make (so far, unconfirmed) predictions of future behavior over
a time horizon of ten months.


Monday, March 30
Speaker: Brian Wissman
Title:  Using Matrices to find Eigenvalues of Boundary Value Problems
Time:  3:00 - 4:15
Room:  UCB 111

Abstract:  When one studies the motion of a vibrating string or the cooling of
a metal plate, its analysis can often be boiled down to solving an ordinary
differential equation.  Unfortunately, these differential equations must satisfy
conditions not at one point (like an initial value problem), but at two different
points.  To make matters worse, solutions only exist for certain values of an
unknown parameter.

For the most basic boundary value problems, these certain unknown values,
called eigenvalues, and their corresponding solutions, called eigenfunctions, can
be explicitly found.  When studying harder problems, this process is more
complex as solutions to the boundary value problems are no longer elementary. 
We will show that using a finite difference approximation to the differential
equation, the eigenvalues of the boundary value problem are approximated by
the eigenvalues of a certain NxN matrix. Using this we will construct an
approximate solution to a vibrating string with spatially varying wave speed.
 


Monday, April 6
Speaker: Keisuke Nakao
Title: How Can Minority Representation Be Ensured by Racial
Redistricting?: A Theoretical Approach
Time: 3:00 - 4:15
Room: UCB 111


Abstract:  Some political scientists have expressed the concern that
concentration of minority voters into a few districts in order to
promote minority representation induces a partisan "perverse effect"
of reducing the number of seats won by Democrats. Our theory explains
how the perverse effect can occur and demonstrates that racial
redistricting generates a trade-off between the number of minority
legislators and the number of Democratic legislators. In light of the
trade-off, we subsequently explore two alternative approaches proposed
in recent debates. One is to create "coalition districts," and the
other is to maintain "second-order diversity." Our theory suggests
that proportional representation of minorities without partisan
electoral consequence may require the diversification among districts
as well as the concentration of minorities.





Monday, April 27
Speaker: Bob Pelayo
Title: Heegaard Decompositions and the Algebra of 3-manifolds
Time: 3:00 - 4:15
Room: UCB 111

Abstract:  If we take a cirlce and cut it in two places, we are
left with two line segments.  Similarly, if we take a (2-dimensional)
sphere and cut it in half, we are left with two disks.  In the same
spirit, we seek to understand how to decompose 3-dimensional
spaces (called 3-manifolds) by cutting along a 2-dimensional
surface into two simpler pieces.  Such decompositions are known
as Heegaard splittings and have greatly advanced topologists'
understanding of 3-manifolds.  These decompositions are relatively
easy to describe and provide a quick method for computing
algebraic invariants of these spaces.  This talk will be largely
visual and intuitive and requires only an open-minded approach
to Mathematics.




Monday, May 4
Speaker: Efren Ruiz
Title: An Introduction to C*-Algebras
Time: 3:00 - 4:15
Room: UCB 111

Abstract: 
This talk will be an introduction to C*-algebras and the K-theory
of a C*-algebra.  The definition of K-theory is different from
topological K-theory and algebraic K-theory.  I will show that in
the case where the C*-algebra is a commutative C*-algebra,
then all three definitions agree.


Monday, October 12
Speaker: Brian Wissman
Title:  Attractor Reconstruction with a Metric
Time: 3:00 - 4:15 PM
Room: K 109

Abstract: Often when scientists study a nonlinear process one
cannot observe all the variables in the system. It would then
seem impossible to study the overall dynamics of the system
without knowing all the variables or the underlying differential
equations. In reality, one can typically use the information from
just one variable to reconstruct the “attractor” of the system,
the geometric object, often with fractal like properties, on which
the dynamics of the system unfold. In this talk we will give a
short introduction to the attractors of some nonlinear systems
and a method to reconstruct them only knowing a limited amount
of information. We will also discuss recent joint work with Philippe
Binder on how one can obtain metric quantities “for free” during
this process.

Monday, October 19
Speaker: Efren Ruiz
Title:  An Introduction to the classification of C*-algberas
Time: 3:00 - 4:15 PM
Room: K 109

Abstract:  We give an introduction to the classification of
C^*-algebras which is now known as the Elliott Classification
Program.  The classification program for the most part
progressed independently for the classes of infinite and finite
C^*-algebras. In the finite case, we will cover the pioneering
work of George Elliott in the classification of AF-algebras. 
An AF-algebra is a C^*-algebra that can be approximated
by finite dimensional C^*-algebras.  For the infinite case,
we will cover the results of Kirchberg and Phillips in the
classification of purely infiinite simple C^*-algebras.



Monday, October 26
Speaker: Efren Ruiz
Title:  The classification of Graph C*-algebras
Time: 3:00 - 4:15 PM
Room: K 109

Abstract:  A graph C^*-algebras are a family of C^*-algebras
 which are associated to directed graphs.  It turns out that
every simple graph C^*-algebra is either an AF-algebra or
purely infinite.  Hence, every simple graph C^*-algebra can
be classified using the results of Elliott and Kirchberg-Phillips.
We give a classification result of the class of graph
C^*-algebras with exactly one non-trivial ideal.

This is joint work with Soren Eilers and Gunnar Restorff.

Questions or Comments: Send to robertop AT hawaii DOT edu