Junior Colloquium Fall 2002 Schedule
Tuesdays 4:00 PM in Vincent Hall 570
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9/10 Rick Moeckel Relative equilibria in celestial
(Univ. of Minnesota) mechanics and Smale's 6th problem
ABSTRACT:
The basic problem of celestial mechanics is to study the motion of n
point masses moving according to Newton's laws of motion under the
influence of their mutual gravitational attraction. The simplest known
solutions are the relative equilibrium solutions where the entire
configuration of n bodies just rotates rigidly around the center of
mass. It may seem surprising that there are any solutions at all of
this type, but in fact, there are many different ways to arrange the
masses to obtain such a solution. How many ways is unknown; in fact, it
is an open problem to show that the number is finite. The talk will
provide an introduction to the finiteness problem and other open
problems about relative equilibria.
Suggested undergrad prerequisites: multivariable calculus
9/17 Scot Adams The Banach-Tarski paradox
(Univ. of Minnesota)
ABSTRACT:
The Banach-Tarski Paradox asserts that it is possible to take any ball
(in Euclidean 3-space) of radius 1, separate it into finitely many pieces,
take each piece and move it, using only translations and rotations,
and the moved pieces will reassemble into a ball of radius 2.
We will give a fairly complete proof of this fact, and of various
generalizations of it.
Suggested undergrad prerequisites: none, except for basic notions from
set theory. That is, I'll use words like "intersection" and "union"
without explaining their meaning.
9/24 John Hall Latin squares
(Univ. of Minnesota)
ABSTRACT: A Latin square is an n-by-n array of numbers from the set {1, 2, ..., n}
in which each number appears exactly once in every row and every column. We will discuss
the question of whether a given _partial_ Latin square, one in which only some cells of
the array are filled, can be completed to a Latin square. We will prove that a partial
Latin square with fewer than n cells filled can always be completed. We will also
discuss the existence of _orthogonal_ Latin squares, and will mention some recent
results.
Suggested undergrad prerequisites: familiarity with set notation and basic induction
techniques.
10/1 No Junior Colloquium
due to Faculty Retreat
10/8 Markus Keel Sample problems from
(Univ. of Minnesota) nonlinear wave equations
ABSTRACT: This survey talk is aimed at undergraduates.
We sketch some open problems and basic approaches in the study of certain
nonlinear partial differential equations. What is the shape of a breaking
wave? What are the equations that help describe this shape? What sorts of
mathematical constructions arise from the study of such equations?
How faithful are these constructions to phenomena which actually
arise in nature?
Suggested undergraduate prerequisites: Basic differential calculus.
Familiarity with partial derivatives would be helpful, but
isn't necessary.
10/15 Jeremy Martin Shear transformations, Fibonacci
(Univ. of Minnesota) numbers, and why cats have nine lives
ABSTRACT: I'm going to talk about a remarkable little matrix which is
known in computer graphics as a "shear transformation." I'll explain what
the shear transformation does geometrically and discuss its order as a
group element. In particular, I'll prove that shearing a cat doesn't
really hurt it. As a bonus, we'll see how to use the shear transformation
to get at a formula for Fibonacci numbers.
Suggested undergraduate prerequisites:
I intend the talk to be accessible to undergraduates. Some knowledge of
linear algebra and group theory may be helpful, but is by no means required.
10/22 Jens Rademacher Dynamics of symmetric patterns
(Univ. of Minnesota) in certain oscillator arrays
ABSTRACT:
Coupled oscillators sometimes serve as simple models for networks of
essentially oscillating cells, e.g. heart muscle cells, yeast cells. A
further simplification is that the cells are essentially equal and the
coupling between them is weak.
We consider finite arrays of equal oscillators in the plane with
translation symmetric nearest neighbour coupling, constituting a
homogeneous medium. Formally, copies of equal ordinary differential
equations are used, each having a limit cycle and this system is perturbed
by a coupling function. Some additional properties of the coupling
permit the existence of spiral wave patterns, phase waves and target
patterns, which may be superimposed to create a variety of beautiful
stable patterns in a rather simple setting.
Suggested undergraduate prerequisites:
Basic knowlegde of linear algebra and differential equations in R^n; it
would also be helpful to know what a torus in R^n is.
10/29 Peter Olver How to recognize objects
(Univ. of Minnesota)
ABSTRACT:
In this talk, I will describe a new approach to the problem of object
recognition in images via invariant signatures. The method relies
on the general theory of moving frames and invariants of group
actions. Additional applications to geometry, algebra, differential
equations and numerical analysis may be mentioned.
Suggested undergraduate prerequisites:
Almost none. Curvature and arc length. Elementary group theory.
11/5 Alexander Voronov Morse theory
(Univ. of Minnesota)
ABSTRACT:
Morse theory is one of the oldest ways of studying the topology
of a geometric object, such as a surface or more generally, a manifold. We
will talk about Morse theory at three levels. The first one is elementary
and deals with what Morse theory may tell us about a surface, or a
landscape. The second level consists in dividing a manifold into
unstable manifolds of the gradient flow of a Morse function, which gives a
Morse complex. The third level is due to Witten, a theoretical physicist,
who got a Fields medal partially for his work on Morse theory. Witten
defined a flow on the differential forms, which pushes the forms to the
delta-functions of the unstable manifolds.
Suggested undergraduate prerequisites:
The first level discussed in the talk requires no prerequisites and is
accessible to a high-school student. The second two levels require a
certain knowledge of manifolds, but will be only briefly touched upon at
the end of the talk.
11/12 Ryan Berndt Behavior of Fourier Transforms
(Univ. of Minnesota)
ABSTRACT:
Calculating the Fourier transform of certain functions can be difficult
and time consuming, and sometimes it is enough to only know its general
behavior. For example, in the case of non-integrable functions, we do not
always know that its "Fourier transform" is bounded. In this talk we will
prove an inequality that demonstrates the general behavior of the Fourier
transform of a certain class of functions. The proof uses basic facts
from calculus.
Suggested undergraduate prerequisites:
Calculus II (Mean Value Theorem, Integration by Parts)
11/19 Donald Kahn Self-equivalences of spaces
(Univ. of Minnesota) versus automorphisms of groups
ABSTRACT:
Homotopy self-equivalences and automorphisms of groups are
natural notions with similarity to the idea of a one-to-one
correspondence for sets. For special spaces (Eilenberg-Maclane spaces),
they are essentially the same notion, but we can show that there is only
one non-trival group all of whose automorphisms are trivial.
Surprisingly, there are many spaces with only trivial homotopy
self-equivalences. This leads to other differences and many unsolved
problems.
Suggested undergraduate prerequisites:
very basic facts about groups, metric spaces
11/26 Naresh Jain From coin tossing to Brownian motion
(Univ. of Minnesota) to Ito integral
ABSTRACT: My attempt will be to discuss the topics in the title of
the talk in an intuitive setting with some of the main results along with
some applications highlighted. The need for abstract measure theory
will enter in a very natural manner, and even though I have no intention
of discussing measure theory, I believe the undergraduates will understand
the idea. The goal will be that they appreciate some of the basic ideas
of modern probability theory and its applications.
Suggested undergraduate prerequisites: The talk will be basically for
undergraduates, so a knowledge of calculus and elementary probability will be useful.
12/3 William Messing The work for which Lafforgue
(Univ. of Minnesota) was awarded the Fields Medal
(Beijing 2002)
ABSTRACT: Lafforgue established the validity of the Langlands
Program for the group GL_n in the function field case. I will
explain the origins of this conjectural program in the work of Gauss,
Hilbert, Artin, Hecke, .... I will indicate the various inputs
needed to enunciate the Langlands Program, the various concepts and
techniques used in attacking this cluster of problems and what went
into Lafforgue's amazing work.
(minimal) Backround: modular arithmetic, knowledge of what a group,
ring, matrix is.
12/10 James Swenson Introduction to the Penrose tiles
(Univ. of Minnesota)
ABSTRACT:
We will become familiar with the Penrose tiles, which tile the plane
without any translational symmetry. The tiles have a hierarchical
structure, which we can exploit to give two essentially different proofs
of the tiling's aperiodicity. We shall consider the origins of these
tiles, subsequent modifications, and a practical application.
Suggested undergrad prerequisites: The discussion should
be accessible to a general audience, with no prerequisites beyond precalculus.
12/17 No Junior Colloq. - Finals week
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