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SCU Mathematics Colloquium Series Schedule
Preliminary
Spring 1999
SCU Mathematics Colloquium Series Schedule
Preliminary
Spring 1999
Talks will be at 4:00 Tuesdays in O'Connor 107 (note room change!). There will be refreshments before all talks in O'Connor 31 starting around 3:40pm.
Title: The Jacobi Symbol
The Jacobi symbol is a function which arises frequently in number theory. We will introduce the Jacobi symbol from scratch, explore its basic properties, and describe some recent results.
Title: G.H. Hardy's Years in Oxford
Title: On Interesting 3-Dimensional Puzzles
I will start on a puzzle derived from an inequality, but the major part of the talk will be on 6-piece burr puzzles and their relatives. The 6-piece burr puzzles have been in existence for about two hundred years. However, the new generation of the 6-piece burr puzzle was invented by William Cutler in the late 70's. I used his idea to create many more related puzzles. An example of such a puzzle can be seen here . I will be bringing many puzzles so that people can try to have fun with them.
Title: In the Right Field: Climbing up Towers while Digging for Roots
We will show how to use Groebner bases (defined for several variables) to find roots of f(x). We're able to speed calculuations in places where otherwise we would resort to hand-computed groups or solving for fixed fields. This theory combined with Maple tools yields an eficient method to help understand minimal polynomial towers, automorphisms and invariants. We'll state results we can carry into old approaches which are slower than ours.
Title: Optimal Card Stacking
Given a deck of n cards, each of unit length, how should they be stacked on a table so that the overhang beyond the edge of the table is maximized? Different versions of this problem have appeared in a variety of mathematics and physics books. Most frequently we see the following question: what is the maximum overhang that can be obtained when each card extends beyond the card below? Optimal solutions for this and other variations of the basic problem will be given.
Title: Discriminant Bounds and Infinite Class Field Towers
A complex number which is a root of a polynomial with integer coefficients is called an algebraic number. The set of finite rational linear combinations of powers of an algebraic number form a field called, not surprisingly, an algebraic number field or number field for short. As a rational vector space, this field has finite dimension. This dimension is called the degree of the field. Number fields are fundamental objects of study in number theory.
Just as the field Q of rational numbers has a distinguished subring Z consisting of the integers (where "arithmetic happens"), a number field K has a subring Z(K) consisting of the "algebraic integers" in K. These integers form a lattice in n-dimensional Euclidean space, where n is the degree of K; roughly speaking, this means that the integers are spaced far apart from one another and at regular intervals. How tightly the integers are packed together can be measured by an invariant called the discriminant of the field and dnoted d(K). The root discriminant of a number field is the n-th root of |d(K)|. It weighs the discriminant according to the dimension of the lattice. Lattices with tight packing are very important in practical applications, such as telephone networks.
As an example of a number field, consider the set of numbers a+bi, where a and b are rational numbers. This is a number field of degree 2 whose ring of integers is the set of numbers a+bi with a and b integers, which form the familiar lattice of Gaussian Integers in the plane. The discriminant of this field is -4. Its root discriminant is 2.
In 1891, Minkowski introduced a fundamental method ("geometry of numbers") for estimating root discriminants. Recently (in the last 30 years), new analytic methods for estimating root discriminants have also been introduced, leading to drastic improvements of Minkowski's results. It is thought that these estimates are very sharp. After introducing all the necessary background, I'll discuss briefly a new method for proving sharpness of these bounds which has led to the first progress on the problem since 1978.
Title: Cubical Complexes, Curvature, and Coxeter Groups
A tiling of the plane with n-gons (not necessarily congruent) is called "right-angled" if there are exactly four n-gons meeting at each vertex. It is well-known that such tilings exist if and only if n is greater than 3. Moreover, if such a tiling exists, it is unique (up to combinatorial isomorphism).
More generally, a "right-angled tiling of Rn by a
polytope P" is a decomposition of Rn into regions each
of which is combinatorially isomorphic to P and such that 2n
of these regions meet at each vertex (like the 2n orthants
in Rn). We will consider the two questions:
(1)
for which polytopes P does Rn admit a right-angled tiling
by P?
(2) Can Rn admit more than one right-angled tiling by a
given P?
Anyone wishing further information about any of these talks should contact the coordinator of the Colloquium Series, Rick Scott, rscott@schubert.scu.edu
The list of talks from previous quarters are available via this archive link.
Last Updated: 7 May 1999
Maintainer: dsmolarski@math.scu.edu
