Project AA: Factorization of fewnomials over finite fields
This project proposes to investigate factorization patterns
of monic polynomials having a small number of terms. The
coefficients are assumed to be in a finite field of
p elements.
The focus is on carrying out computations with the view of
formulating new conjectures, discovering concrete families
of fewnomials having interesting factorization patterns such
as no factors (irreducibility) or no factors of small degrees
etc. and possibly proving some related results.
Project AB: Bounds on the roots of cubic units
This project aims to explore the following problem: If
u
is a root of an irreducible monic cubic polynomial with integer
coefficients and constant term ±1, then which fractional
powers of
u are integral linear combinations of 1,
u,
u2? The focus of the investigation is on
finding explicit bounds either in the form of a conjecture
that is well supported by the computational evidence or by
proving related theorems.
Project GT: Topology of finite posets
The term poset stands for "partially ordered set" and refers
to any set that has an ordering in the usual sense, except
that some pairs of elements may not be comparable. For
example, the set of all nonempty proper subsets of
the set {1,2,3,4} is partially ordered by inclusion.
For example {1}≤{1,2} but {1,4} and {2,3} are not
comparable because neither is a subset of the other.
Posets can be used to construct topological spaces. One
starts with a vertex for every element of the poset. Then one
adds an edge for every pair of comparable elements, a
triangle for every triplet of comparable elements, etc. In
the above example, there are 14 vertices, one for each nonempty
proper subset. Since {1}≤{1,2} there is an edge
between those vertices. Since {2}≤{1,2}≤{1,2,3}
there is a triangular face with those vertices as corners.
It is very difficult to draw, but the resulting space is
topologically a 2-dimensional sphere.
Posets abound in mathematics, and in this project students
will analyze the topology of various posets. One poset of
particular interest, where the topology is unknown, is
the poset of subsets of the vertices of an n-dimensional
cube which avoid a k-dimensional face. This poset arises
in the study of Boolean formulae, and is related to the
Millenium Problem that asks whether P=NP, a solution for
which the Clay Mathematics Institute is offering a
million dollar prize.
updated: 01/9/08
