Whitney
Berard -- Program in Mathematics, University
of Arizona
Braid Groups
Imagine you are studying the way that enzymes form
knots in a strand of
DNA. The first question you may ask the
mathematicians is: How do I distinguish between
two different knots? This is a surprisingly
difficult
question to answer! The way we attack the
problem is by constructing link
invariants -- a number, or polynomial, etc. --
such that if you calculate the invariant for the
projections of two knots and you get different
results, then the knots must be distinct.
Now, we can form a braid from a knot by cutting the
knot in strategic locations and straightening it
out. We can then put a very nice algebraic
structure on these braids. In this talk, I will
define the braid group and show how we can use
this algebraic structure to define some nice
polynomial link invariants.
To understand this talk, you should know what a
group is and be able to work with equivalence
classes.
Feb
19
Daniel
Schultheis -- Department of Mathematics,
University of Arizona
Moduli Spaces
and Curve Counting
Moduli spaces give us a way of parametrizing
collections of geometric objects, like the
collection of all lines through the origin, or all
curves in space. More than just collections
of objects, moduli spaces can allow us to
intersect geometric shapes and get accurate counts
of the intersections without getting bogged down
by coordinates. In this talk, I'll give a
several examples of moduli spaces, discuss the
importance of compactness, and mention at least a
couple intersection problems which can be solved
using moduli theory.
To understand this talk, you should have taken linear
algebra and you should also understand the concept
of compactness. Exposure to ring theory would be
helpful, but is not necessary.
Mar
5
Sarah
Mann -- Program in Applied Mathematics,
University of Arizona
Topics in
Information Theory
Information theory is the study of how information
(numbers, words, sounds, images, movies) are
represented, stored, and transmitted
electronically. These techniques rely
heavily on a wide variety of mathematical
ideas. In this talk, I will introduce a few
areas in information theory and discuss their
foundations in mathematics. I will discuss
how sound is represented digitally and how it can
be compressed using the discrete cosine transform
thus allowing you to fit even more music on your
ipod. I will also introduce error correcting
codes which allow information to be transmitted in
the presence of errors without information
loss. This is the technology that supports
communication with deep space probes, allows you
to talk on a wireless telephone, and protects CDs
against skipping.
This talk will be accessible to students who have
completed calculus.
Some knowledge of Fourier series, Lagrange
interpolation, and finite fields could prove
useful, but not necessary.
Mar
19
Ivan
Ventura -- Department of Mathematics,
University of Arizona
Can You "Hear"
the Shape of a Radial Schrödinger Operator?
Inverse problems are a large class of both
theoretical and applied problems that have
captivated the mathematical community for over
half a century. During this time numerous
applications have arisen in a variety of fields,
such as medical imaging and cloaking. In the
first half of this talk I will discuss, by
example, general inverse problems and how they
arise in the real world. In the second half
I will focus specifically on spectral inverse
problems, starting with the classic "Can you hear
the shape of a drum?" problem in the case of the
sphere. Finally I will present an analogous
result for semiclassical Schrödinger operators.
Mar 26*
Sarah Post --
Department of Mathematics, University of Hawaii
at Manoa
Quantum Computing, Perfect
State Transfer and Orthogonal Polynomials
In this
talk I will give a brief introduction to the
basics of quantum computing and quantum
information. I will then discuss the
phenomena of perfect state transfer which allows
for the transport of a quantum state to another
with probability one. Hamiltonians with
this property can be constructed from spin
chains and remarkably their existence can be
completely characterized by families of
orthogonal polynomials. The situation is
much more complicated in systems on a lattice
though I will discuss on such model which
corresponds to the discrete analog of the
harmonic oscillator.
I will finish
the discussion with a neat trick to use such
perfect transfer to "teleport" an unknown
quantum system across long distances.
Apr
2
Michelle
Hine -- Program in Applied Mathematics,
University of Arizona
Elasticity
Theory and Finite Elements Using a Porous Media
Formulation
The theory of poroelasticity is used in
applications as varied as soil mechanics and soft
tissue deformation. In this talk, I will first
introduce elasticity theory and then a model for
porous media using the effective stress principle
to decouple stress into solid and fluid
components. I will then introduce the finite
element method, a well-known computational tool
used to solve boundary value problems, and discuss
how to apply it to the model.
This talk will be accessible to students who have
completed calculus. Differential equations and
some numerical background would be helpful, but
not necessary.
Apr
16
Rohit
Thomas -- Department of Mathematics,
University of Arizona
Calculus and Differential
Equations in Geometric Topology: Morse Theory and
Handle Decompositions
Morse theory is a tool for connecting
analytic information (the critical points of a
manifold) to topological information (a handle
decomposition of the manifold). This talk will
begin with a brief introduction to the topology of
surfaces, followed by an introduction to handle
decompositions of
n-manifolds. We will then see how a
simple application of calculus and differential
equations (critical points and gradient flows)
yields a powerful topological tool (Morse theory).
This talk should be accessible to anyone who has
completed or is currently taking vector calculus
and is willing to exercise their visualization
skills. Depending on audience background and
interest, we may also see connections to
bifurcation theory or singular homology.
Apr 23*
Brie
Finegold
-- Department of Mathematics, University of
Arizona
Groups Acting on TreesCANCELED!
We'll look at a classic example of a group acting
on a tree: SL(2,Z)
(the set of 2x2 integer matrices with determinant
1) acting on a trivalent tree that is the dual to
the Farey graph. By examining this action,
we will prove that any 2x2 matrix can be written
as the product of powers of just two
matrices. In fact, we'll show that SL(2,Z) is
isomorphic to an amalgamated free product of Z4
and Z6.
This example illustrates the interplay between
group theory and topology. I will then give
one of my results which has to do with SL(3,Z).
To understand this talk, you should know what a
group is, how to multiply
matrices, and it would be helpful to know what a
group action is although I
will talk about this briefly. If you know
what a presentation is that will
also be helpful, but not necessary.
Apr
30
Bill
McCallum -- Department of Mathematics,
University of Arizona