September 9, room NAC 4113: Peter Neumann (Oxford University),
Did Galois deserve to be shot?
Abstract:
When Evariste Galois was shot in a duel and died in May 1832,
he was only twenty years old. Yet he had already published one
important paper on his mathematical researches, and had completed
an even more important work which, after some vicissitudes, was
published posthumously in 1846. His ideas led to a fundamental
change in the way mathematicians approached the study of equations.
They led directly to group theory and to what later became known as
`abstract' or `modern' algebra. In this lecture I will try
to paint a word-picture of Galois' turbulent life and his astonishing
achievements.
October 7: Ethan Akin (CCNY), About chaos
Abstract:
Among the many definitions of chaos is the concept of
"sensitive dependence upon initial conditions". We
discuss the history and some of the subtle meaning of this
idea.
October 14: George Havas (The University of Queensland),
How hard is computing with matrices?
Abstract:
The calculation of standard forms for matrices plays an important role
in many computations. In 1851 Hermite described a standard form for
integer matrices which is triangular. Conversion of a matrix representing a
set of linear equations to a triangular form provides a way of solving the
equations by back-substitution. This form is important in
computational linear algebra and number theory. In 1861 HJS Smith
described a standard form for integer matrices which is diagonal.
This standard form is important in control theory, digital signal
processing and in group theory.
Our aim is to develop effective algorithms for finding standard forms
of matrices. In principle it is easy, but it can be difficult if the
matrix is large.
October 28: Chuck Miller (University of Melbourne),
Computability in algebra and geometry
Abstract: This is
an overview of the theory and practice of doing
computations about algebraic and geometric objects. I will
discuss some of the history and meaning of the theoretical
limitations on abstract mathematical computation. But I will
also indicate why doing computations with groups and geometric
objects is still useful even though it is impossible.
My intention is to assume very little specific knowledge on
the part of the audience. Hopefully a wide range of
mathematicians and computer scientists will be able to
follow the discussion.
November 11: Igor Rivin (Temple University),
How to measure an egg
Abstract:
We use probabilistic methods to get formulas for
computing the surface area and other "integral mean curvatures" of an
ellipsoid. The formulas involve hypergeometric functions, but we show
that there are very simple approximations which become more and more
accurate as the dimension of the ellipsoid becomes bigger.
December 2, room NAC 4113: Olympia Hadjiliadis (Columbia University),
Change-point detection in the Brownian motion model with two-sided
alternatives and its connection to the gambler's ruin problem with
relative wealth perception.
Abstract:
In this presentation we address two problems; the problem of
change-point detection in the Brownian motion model with multiple
alternatives and the gambler's ruin problem with relative wealth perception.
Although the two
problems appear very different, it turns out that there exists a
connection that is carefully examined and discussed.
In the former problem, the objective is to detect a change in the
constant drift by means of a stopping rule when there are multiple
possibilities for such a change. Particular attention is drawn to two-sided
alternatives, whereby the two-sided CUSUM stopping rule (2-CUSUM) is
employed. In particular, it is shown that within two specific classes
of 2-CUSUM rules (the harmonic mean rules and the equalizer rules), a
particular choice of 2-CUSUM rules enjoys a very strong asymptotic
optimality property.
In the latter problem, we review the traditional gambler's ruin problem
but assume that the gamblers make their decisions based on the relative
change of their wealth as opposed to the absolute change. In other
words, gamblers are assumed to quit after their wealth makes a significant
upward rally or a significant downward fall. In this setting we compute the
probabilities of winning (i.e. quitting on the upward fall) or losing
(i.e. quitting on the downward fall).
Using the above probabilities one can compute the first moment of a
broader class of 2-CUSUM stopping rules than the specific class of
harmonic mean rules and thus set the first stepping stone on examining
the optimality properties of 2-CUSUM stopping rule beyond the traditional
setting of equal thresholds in its respective one-sided CUSUM branches.