Junior Colloquium Spring 2003 Schedule
Tuesdays 3:30 PM in Vincent Hall 570
(with refreshments in 1st floor Tea Room at 3:10 PM)
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1/21 No Jr. Colloquium- 1st day of class
1/28 Paul Garrett Factoring x^n-1: cyclotomic
(Univ. of Minnesota) and Aurifeuillian polynomials
ABSTRACT:
Polynomials of the form x^2-1, x^3-1, x^4-1 have at least
one systematic factorization
x^n - 1 = (x-1)(x^{n-1} + x^{n-2} + ... + x^3 + x^2 + x + 1).
For composite exponent n one can be a little clever and obtain
several different factors
x^6-1 = (x^3)^2-1 = (x^3-1)(x^3+1)
x^6-1 = (x^2)^3-1 = (x^2-1)((x^2)^2 + (x^2) + 1)
Such _algebraic_ factorizations yield _numerical_
partial factorizations of some special large numbers, such as
2^{33} - 1 = (2^{11})^3 - 1 = (2^{11}-1)(2^{22}+2^{11}+1).
There is a pleasant description of the _complete_
factorization of x^n-1 into _irreducible_ polynomials (which
cannot be factored further without going outside the rational
numbers). The irreducible factors are _cyclotomic polynomials_.
For example,
x^{15}-1 = (x - 1)
(x^2 + x + 1)
(x^4 + x^3 + x^2 + x + 1)
(x^8 - x^7 + x^5 - x^4 + x^3 - x +1).
Less well known are {\it Aurifeullian-LeLasseur} factorizations such as
x^4+4 = (x^4 + 4x^2 + 4) - (2x)^2 = (x^2 + 2x + 2)(x^2 - 2x + 2).
More exotic are
x^6 + 27
-------- = (x^2 + 3x + 3)(x^2 - 3x + 3)
x^2+3
x^{10}-5^5
----------- = (x^4 + 5x^3 + 15x^2 + 25x + 25)(x^4 - 5x^3 + 15x^2 - 25x + 25)
x^2-5
x^{12}+6^6
---------- = (x^4 + 6x^3 + 18x^2 + 36x + 36)(x^4 - 6x^3 + 18x^2 - 36x + 36)
x^4+36
x^{14}+7^7
---------- = (x^6 + 7x^5 + 21x^4 + 49x^3 + 147x^2 + 343x + 343) ...
x^2+7 (x^6 - 7x^5 + 21x^4 - 49x^3 + 147x^2 - 343x + 343)
These Aurifeuillian factorizations yield further factorizations of special large
numbers, such as
2^{22}+1 = 4 ... (2^5)^4 + 1
= (2(2^5)^2+2(2^5)+1)(2(2^5)^2-2(2^5)+1)
= 2113 ... 1985 = 2113 ... 3 ... 397
3^{33}+1 27 ... (3^5)^6+1
-------- = -----------------
3^{11}+1 3 ... (3^5)^2 + 1
= (3(3^5)^2 + 3(3^5) + 1)(3(3^5)^2 + 3(3^5) + 1)
= 7 ... 25411 ... 176419
Both the cyclotomic and Aurifeuillian polynomials can be explained
well in terms of roots of unity.
Suggested undergrad prerequisites:
High-school algebra of polynomials, complex numbers
2/4 and 2/11
John Dennis
(Noah Harding Professor Emeritus and Two lectures on
Research Professor at Rice Univ./ nonlinear optimization
IMA Visitor)
ABSTRACT:
For more than 10 years, I have been working with ExxonMobil, United
Technologies, and Boeing on methods for optimal design using virtual
prototyping. We have had some fine successes, but I feel we are just
getting to the real problem. In these two lectures, I will try to give
you a feel for what these problems are like and how mathematicians can help.
Lecture 1 (2/4): What is nonlinear optimization, and why is it hard?
In this lecture, I will try to use pictures to get across the difficulty and
applicability of nonlinear optimization. Then I will show how engineering
design can be modeled by nonlinear optimization and what the features of such
problems are.
Lecture 2 (2/11): Formulations of multidisciplinary design optimization.
In this lecture, I will use aeroelastic design of an airfoil to motivate a
difficult class of multidisciplinary design optimization problems. In such
problems, it is often crucial which of several mathematically equivalent
formulations one tries to optimize.
2/18 Leonard Blackburn Transfinite ordinal numbers
(Univ. of Minnesota) and their applications
ABSTRACT:
We will give a self-contained, informal description
of transfinite ordinals, their properties, and their arithmetic.
We will then briefly discuss their uses in mathematics outside
of logic and set theory. Then we will use transfinite
ordinals to prove Goodstein's Theorem. This counter-intuitive
theorem asserts that certain sequences of natural numbers
(called Goodstein sequences) all eventually stabilize at zero.
It is an important result because it is a straight-forward
number theoretic statement which is not provable in formal
number theory. So, something other than ordinary finite
arithmetic must be employed in its proof.
Suggested undergrad prerequisites:
Conversion of natural numbers to and from bases other than 10.
2/25 Scot Adams Groebner bases and
(Univ. of Minnesota) integer programming
ABSTRACT:
Work in the early 1990s of Conti and Traverso
established a link between computational algebraic geometry
and integer programming. We expose some of the basic ideas
leading to this connection.
3/4 Wojciech Chacholski Approximating topological spaces
(Univ. of Minnesota) and the fundamental groupoid
3/11 Andy McLennan Nash Equilibria of Auctions
(Univ. of Minnesota-
Economics Dept.)
ABSTRACT
The concept of Nash equilibrium, originated by John Nash of "A
Beautiful Mind" fame, will be explained and applied to the some basic
models of competition in auctions.
3/18 No Jr. Colloquium- spring break
3/25 Ryan Gantner What is the best way to
(Univ. of Minnesota) bet on the roulette wheel?
ABSTRACT:
We will address the question of "What is the best way to bet on the
roulette wheel?". Although this will not be completely answered, we get
some results beginning with the standard "gambler's ruin" problem, then
looking at selection systems, wagering systems, and gambling policies.
These terms will be defined and their importance noted. From here, we
apply (and prove) a special case of what has become known as the
Fundamental Theorem of Gambling to a particular kind of roulette game.
Extensions from this will be discussed, and examples will be given.
Suggested undergrad prerequisites:
This talk will make use of some elementary notions of independence of
events (high school probability), but will not presume any
undergraduate work in probability.
4/1 Jon Rogness The Mathematics of
(Univ. of Minnesota) Satellite Images
ABSTRACT:
Everybody is aware of spy satellites in space, but there are a number of
civilian satellites being used to monitor different conditions on earth.
We'll start with basic explanations of what a satellite image is, and what
you can and can't see. We'll talk about how satellites are used to
measure vegetation and the potential for catastrophic fires. We'll also
talk about land clover classification, which has been done for the entire
Earth on a 1km by 1km scale.
There are many problems one has to deal with when working on satellite
images. (For example, cloud cover is the most obvious, but also one of
the easiest to deal with.) The second part of the talk with be devoted to
(a) demonstrating these problems, and (b) describing the mathematics used
to solve them.
Suggested undergard prerequisites:
I plan on showing lots of pictures, and very few details of the
mathematics behind them. Basic arithmetic is the only requirement for
most of the talk; a few definitions towards the end will use multiple
integrals and other common terms from undergraduate mathematics.
4/8 Jay Goldman Modern knot theory
(Univ. of Minnesota)
ABSTRACT:
In 1984, Vaughan Jones discovered a new "knot polynomial" which
moved the study of knots from the backwater of topology to center stage of
exciting mathematics research
I plan to explain the background knot ( and braid) theory, with
some history, and discuss an elementary approach to Jones' work and how it
solved some classic conjectures. If time permits, I will survey some
later work including connections with physics. There will be more
pictures than equations.
Upper division undergraduates should be able to understand most of
the talk.
4/15 Dick McGehee Total Information Awareness:
(Univ. of Minnesota) Should Donald Rumsfeld
be Aware of Thomas Bayes?
ABSTRACT: As part of the War on Terrorism, the Defense Advanced
Research Projects Administration has started a project called
"Total Information Awareness." The project involves building an
enormous database of transactions (airline reservations, credit
card transactions, etc.) with the goal of identifying terrorists
before they act by searching the database for patterns. Putting
aside legal and ethical considerations, a question arises: is the
goal theoretically possible? The Bayes Theorem from classical
probability theory has something to say about the answer.
4/22 Radu Popescu On the topology of
(Univ. of Minnesota) the knot complement
4/29 Roger Howe What algebras can we understand?
(Yale Univ.)
ABSTRACT: Many situations in algebra and algebraic geometry call for understanding
the structure of a finitely generated commutative algebra (over, say, the complex
numbers). For example, invariant theory asks for a description of the algebra of
polynomials which are left invariant by a group of linear transformations.
One standard means of describing an algebra is via generators and relations:
one specifies a collection of elements which generate the algebra, in the sense that
the monomials in these elements span the algebra as vector space. Then one describes
the relations - the collection of polynomials in the generators which give the zero
element in the algebra. Hilbert's Basis Theorem guarantees that one can find a finite
number of relations which imply all relations. Given a system of generators and
relations, Buchberger's algorithm describes how to determine a system of monomials
in the generators which form a basis for the algebra (as a complex vector space).
This generators-and-relationss approach may be thought of as comparing the algebra
to a polynomial ring. However, if the number of generators and relations is large,
it may be hard to develop much intuition from a generators-and-relations description,
and Buchberger's algorithm may take a very long time to run. Thus, rather than always
comparing algebras to polynomial rings, it may be desirable to have a family of
algebras that one considers "understandable" in some sense, and to compare other
algebras to members of this family. This talk will discuss a proposal along these
lines, using the example of the coordinate ring of the flag variety of chains of
subspaces in C^n as an illustration.
5/6 Christopher Sogge Eigenfunctions of the Laplacian
(Johns Hopkins Univ.)
ABSTRACT: We are interested in how global features of a
manifold reflect themselves in properties of eigenfunctions
for very high eigenvalues. A general well-accepted
principle is that the global dynamics of the geodesic flow
(e.g., whether it is "chaotic" or not) should determine
whether L^2-normalized eigenfunctions can "scar" (have very
large sup-norms) or not. I shall present some joint work
with S. Zelditch that, in the real analytic case,
characterizes Riemannian manifolds without boundary that can
have "scarring". I shall also discuss similar problems for
Riemannian manifolds with boundary. Here the boundary
introduces a difficult but interesting feature since convex
boundary points are known to lead to Rayleigh whispering
gallery modes. Making this rigorous and obtaining sharp L^p
bounds for eigenfunctions is quite difficult, though.
Suggested undergraduate prerequisites: Some knowledge of Fourier
analysis and Riemannian geometry. In particular, it would
be helpful to know what geodesics and L^p spaces are.
5/13 No Jr. Colloquium- finals week
Back to the Junior Colloquium Homepage.
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