SCU Mathematics Colloquium Series Schedule

*Final*

Spring 2000

Talks will be at 4:00 **Wednesdays**, room TBA.
There will be refreshments before all talks in O'Connor 31 starting
around 3:45pm.

**Apr. 12**David Sul, Santa Clara UniversityTitle: Calculus 100 - Your Most Direct Route to Calculus!

In June, 1999, the Office of College Special Programs (OCSP) at Santa Clara University sponsored the Calculus 100 program in an attempt to positively impact the AP Calculus enrollment of African American and Latino students in the East Side Union High School District. In a one-year time frame (1998-1999 to 1999-2000), overall AP Calculus enrollment in the district increased by 0.6% yet increased by a remarkable 22% for the targeted groups after just one summer of the program. The ramifications of the program on the district, Santa Clara University, the community of east San Jose, and the mathematics education community will be presented. Discussion will also include the research and evaluation design for the project, the "traditional versus reform mathematics" debate, and issues of educational equity.

**Apr. 19**Steve Givant, Mills CollegeTitle: The algebraic theory of binary relations

The calculus of binary relations --- created by Augustus De Morgan, Charles Saunders Peirce, and Ernst Schroeder in the second half of the nineteenth century --- was an important forerunner of first-order logic. In 1941, the calculus was recast as an abstract algebraic theory and extensively developed by Alfred Tarski and his collaborators. The theory has found important applications in logic, algebra, computer science, and several other domains.

The first part of this talk will trace the original motivations of the theory and some of the important results that were achieved by Tarski and others. The second part will present some recently discovered connections with the theory of groups.

**Apr. 26**Brian Sittinger, Santa Clara UniversityTitle: A View to Higher Reciprocity

In virtually all introductory number theory courses, one encounters the beautiful Law of Quadratic Reciprocity, which answers the question of the solvability of the quadratic congruence x

^{2}= a (mod p). Do similar laws exist for higher-powered congruences? In fact, they do exist, and we will discuss, in particular Cubic Reciprocity (and time permitting, Quartic Reciprocity) after having developed the proper amount of background to state them clearly. Anyone who understands the notion of a congruence will be able to understand this talk.

**May 3**Michael Aissen, Naval PostGraduate School, Monterey.Title: A Family of 0,1 Matrices

Two disjoint, discrete sets of real numbers separate each other iff every interval containing at least two elements of either of the sets, contains at least one element of the other. Non-trivial examples occur widely, notably in the the theory of orthogonal polynomials.

Given a family of m sets of real numbers, we say that the family is cyclically ordered (C-O), iff every pair of distinct sets in the family separate each other. We use the phrase `cyclically ordered sets' as an abbreviaton for `(a) cyclically ordered family of sets'.

For m > 2, the attention to C-O sets in the literature seems to be extremely sparse. and for m > 3, I am aware of only one non-trivial example.

The main result in the talk refers to a two parameter family of 0,1 matrices, A(b,k). The eigenvalues of A(b,k) are all non-negative, and the positive eigenvalues are all simple. If F(b,k) is the set of positive eigenvalues of A(b,k) then the sets F(b,j),F(b,j+1),...,F(b,j+k) are C-O for each j.

In the talk, the characteristic polynomials of the A(b,k) will be given explicitly, and asymptotic information about the eigenvalues will be discussed.

The interest in the matrices A(b,k) arises in a counting problem involving k-ary trees. How many k-ary trees on n vertices, with bias b, are there ? The answer is the (1,1) entry in the nth power of A(b,k). The concept of bias will be described in the talk. Bias is interesting, because it leads to an effective numerical coding of k-ary trees.

**May 10**Dan Ostrov, Santa Clara UniversityTitle: Determining 3-D Nonsmooth Surface Using a (2-D) Photograph

We will explore the Shape-From-Shading equation, which is a first order nonlinear partial differential equation (PDE) that relates surface height to photographic information. By using methods from control theory, we will be able to solve this PDE and extend this solution to photographs of nonsmooth surfaces where the brightness in the photographic image is discontinuous. If time permits, we will extend these ideas to a broad class of first order nonlinear PDEs that depend upon discontinuous data.

**May 17**Ed Schaefer, Santa Clara University.Title: The Birch and Swinnerton-Dyer conjectures

We will describe the two Birch and Swinnerton-Dyer conjectures for elliptic curves. There is a lot of empirical evidence for these amazing conjectures. Kolyvagin and Logachev have given partial proofs and the recent extension of Wiles' proof of Fermat's last theorem has made the conjectures more reasonable.

**May 24**Joe Pulford, Santa Clara University.Title: Where O Where Did Our Conformal Maps Go?

Conformal mappings, mappings which preserve the size of angles, are abundant in 2 dimensions but "virtually" disappear in higher dimensions. We will prove the standard result in the plane that a mapping

*f(z)*is conformal if and only if*f(z)*is analytic with nonzero derivative using a nonstandard approach. We will then sketch a recent proof of Liouville's theorem which states that the only conformal mappings in higher dimensions are Mobius transformations. The ideas involve the uniqueness of a system of partial differential equations. Time permitting, we will consider quasiconformal mappings, a broader class of mappings which generalize many of the salient features of conformal mappings in higher dimensions.

Title: The Dynamics of Dipolar Domains: Patterns Formed by Current Ribbons.

The maze-like labyrinthine patterns formed in a number of diverse physical systems are strikingly similar. These systems are explored both experimentally and theoretically in an attempt to understand the essential mechanism responsible for how these patterns form. A particularly simple experimental system that exhibits this behavior is a ferrofluid trapped in the quasi two-dimensional geometry of a Hele-Shaw cell. In an applied magnetic field, these fluid droplets undergo a fingering instability that leads to a complex, branched structure. This talk will focus on the details of this dynamical process and how the interactions between multiple droplets can affect the final patterns that are observed.

If you have a disability and require a reasonable accommodation, please call Rick Scott, rscott@schubert.scu.edu, 1-408-554-4460 or 1-800-735-2929 (TTY - California Relay).

The list of talks from previous quarters are available via this archive link.

*Last Updated: 14 May 2000*
Maintainer: webmaster@math.scu.edu