Abstract: The numbers that do not pass thru
the sieve of Eratosthenes naturally fall into useful sets. A study of
these sets shows that certain constellations, such as the so called twin primes,
occur in regular patterns. It is shown that, in the case of the twin
primes, there is an infinite number of these that are prime.
Abstract: Like prime numbers in the set of integers, prime ideals play an analogous role in ring theory. Anderson and Smith generalized the notion of a prime ideal by defining weakly prime ideals for commutative rings. We extend their definition to any ring, and investigate the structure of rings in which every ideal is weakly prime.
Date: Friday, April 9 , 2010
Speaker:
Jeremy Leach,
Yale University
Title:
An Introduction to Hyperbolic Knot and Weave Complements
Subject:
Hyperbolic Geometry
Abstract:
Hyperbolic knot complements have provided a rich class of finite volume
hyperbolic 3-manifolds in recent decades. The aim of the talk is to first
present a concise introduction to these manifolds, constructing some simple
examples by gluing ideal polyhedra, and then to describe how ideal polyhedra can
be used to tile the complement of an alternating weave in R^3 to construct
infinite volume hyperbolic 3-manifolds.
Date: Friday, April 16, 2010
Speaker:
Keith Mellinger,
University of Mary Washington
Title:
Conics in the finite projective plane and their applications to coding and
cryptography
Subject:
Finite Geometry
Abstract: In the classical finite projective
plane, PG(2,q), there is only one non-degenerate conic – a set of points
satisfying a quadratic form. These simple structures have a rich theory. For
instance, when q is odd, they can be defined synthetically, without any regard
to their natural algebraic definition. In this talk, I will survey some of the
broad uses of conics in the study of coding theory and cryptography. We look at
optical orthogonal codes, low-density parity-check codes, and a certain
cryptographic protocol. In each case, we use configurations of conics to provide
new examples, and to study properties of the objects in question. Along the way,
I plan to discuss the involvement of undergraduates in this line of research.
Date: Friday, April 23, 2010
Speaker:
Chi-Kwong Li,
The College of William & Mary
Title:
A gentle introduction to Quantum Information Science
Subject:
Matrix Analysis
Abstract: Quantum information science concerns
the study of the use of quantum properties in constructing fast computing
devices and designing secure communication schemes. Advancement is this area
will have significant impact on activities in business, industry, engineering,
and many branches of sciences. For instance, it will lead to great improvement
in security and efficiency issues on financial transaction, data storage and
transmission, network connection, computing and simulation of quantum systems In
this talk, a gentle introduction of the subject will be given. (No quantum
mechanics background is needed to understand the talk.)
Date: Tuesday, September 30, 2014 (AC-104)
Abstract: This is a brief introduction to a software available on-line called Eigenmath, which I have introduced to students in classes such as Calculus III. Eigenmath is a lightweight application that can help students work with mathematical equations, by providing them with the necessary tools for seamlessly writing complex expressions. It can be easily figured out, even by users with no experience in software programs.
Abstract:
An important concept in modern group theory is that of a
free group,
a particular type of group which has no structure beyond that required by the
classical group axioms. Free groups arise early on in the study of groups: any
finitely generated group can be constructed by starting with a finitely
generated free group and imposing additional structure. However, free groups are
also interesting in and of themselves and are thus the subject of a tremendous
amount of contemporary mathematical research. In this talk, we will survey some
of the algebraic, algorithmic, and geometric aspects of free groups that have in
turn led to some of the most important developments in modern group theory.
Date: Friday, November 21, 2014
Abstract:
After briefly discussing the concept of infinity-category, I will present two constructions rooted in higher category theory, both of which have for aim to give a geometrical answer to the problem of having equations of motion that lack a proper mathematical foundation. The first construction is that of infinity-4-manifolds, a generalization of 4-manifolds for which braids are replaced by infinity groupoids, and the second construction is that of a sheaf of symmetric monoidal infinity-categories over a Grothendieck site whose morphisms are roof diagrams over infinity-groupoids. This will pave the way for introducing classes of derived algebraic stacks intermediate between those Toen is currently working on and the infinity-topoi of Lurie, the idea being that for such classes, equations of motion can be replaced by purely algebraic constructs.
Abstract: We begin our discussion with the question of "Which
integers are represented by a sum of two squares?". This question falls under
the category of "representations by quadratic forms", and we will take a
leisurely stroll through this area of number theory. Although we will not be
able to consider all details, we will see many of the "main attractions", and
visit both old and new questions. Our discussion will be self-contained and
should be accessible to all who are interested.
Abstract: The study of growth in groups has a rich
history dating to the 1950s. Growth in algebras is a related study which sheds
light on some questions in group growth. In this talk we briefly introduce the
concepts of groups and algebras. We define growth functions and then narrow our
focus to uniform exponential growth. We discuss the group algebra as a link
between growth in groups and algebras. We give a sufficient condition for an
algebra over a field to be of uniform exponential growth and apply this to
provide a concrete example of an algebra of uniform exponential growth.
Date: Thursday, June 2, 2015