It has been nearly 80 years since Frank Plumpton Ramsey proved his eponymous theorem. So what has been done since then? We will present some history, the big theorems in this area of combinatorics, as well as present some recent results in the subarea of Ramsey theory on the integers. We will ¯nish by presenting some open questions that are appropriate for undergraduate research.
The MAA has recently launched a national initiative to change the focus in college algebra and related courses, courses that are taken by a million students each year. Who are these students? Why do they take these courses? Where do they go after taking these courses? What mathematics do they really need? What should be the focus in these courses to better serve the needs of the students and other disciplines?
Movies have great moments and sports have great moments, but mathematics has the greatest moments of all. In this talk, we will review some basic properties of the Riemann zeta function and we will explore its moments (also known as mean values), revisiting the momentous discoveries of Hardy, Littlewood, and Ingham in the 1920s. We will also discuss memorable moments that today's number theorists have encountered as they work on open problems related to the Riemann Hypothesis.
The talk is focused on special material formations termed the dynamic materials (DM). DM are defined as structures assembled from conventional materials distributed in spacetime. If such assemblages occur on a microscale, they become spatio-temporal, or dynamic, composites (DC). When a low frequency disturbance propagates through DC, it may perceive this one as a medium with some effective properties detected through homogenization. A discussion of such properties along with some special effects they produce in material design is the central objective of the talk. Such effects include material screening, elimination of a cutoff frequency in waveguides, amplification and generation of waves, compression of impulses, frequency multiplication, and so on. A DM is a linear system with coefficients (material parameters) variable in space and time, and their variability may produce said effects, some of them being typical for the non-linear systems. This is accompanied by an exchange of energy and momentum between DM and the environment. The energy/momentum transformation in DC is examined in the context of electrodynamics of moving dielectrics. Waves of negative energy may particularly emerge through the material mixing in space-time, and such waves may, in special circumstances, demonstrate instabilities and open the way to power generation. The effective properties of DC are needed for the purpose of optimal layout in dynamics. The bounds for such properties related to the mixtures of two or more original dielectrics will be discussed in the talk for one-dimensional wave propagation without shocks. Such bounds appear to be sharp, i.e. attainable by laminates of multiple rank. Other microstructures may demonstrate quite a different performance, not necessarily characterized by the effective properties. For example, a spatio-temporal checkerboard may create “synchronized waves” in one spatial dimension, i.e. the waves with profiles initially occupying some space in one dimension and eventually being compressed into much smaller intervals that shrink to a point as t ! 1. Energy supplied from without into the waves traveling through a checkerboard is accumulated within the impulses as they contract, and this creates very high energy concentration in space.
The oldest competition for an optimal shape (area-maximizing) was won by the circle. But if the fixed perimeter is measured by the line integral of |dx| + |dy|, a square would win. Or if the boundary integral of max(|dx|,|dy|) is given, a diamond has maximum area. For any norm in R^2, we show that when the integral of ||(dx,dy)|| around the boundary is prescribed, the area inside is maximized by a ball in the dual norm. When || || is the l^2 norm, that ball is a circle (!). Our proof comes directly from the calculus of variations, where Busemann's original proof (and most of the 999 isoperimetric proofs)used inequalities from convex geometry. This problem has applications to computing minimum cuts and maximum flows in a plane domain.
Cooperative learning is a method of active learning in which stable groups of students produce a significant amount of work in a course. Their work is assessed and counts in the course grade. We will discuss formation of student groups, initial activities for groups, groups in the classroom and computer lab, assignments outside class, di±culties with groups, monitoring the groups, modes of operation within groups, assessment in the courses, and group testing. Courses considered include Mathematics for the Liberal Arts, Calculus with Review, Calculus I-III, Discrete Mathematics, Abstract Algebra, and Topology.
Mathematics and music have been associated throughout history, and mathematicians in all ages have felt the a±nities, even if they have not been able to pin down exactly what it is about the subjects that ties them together. In this talk, we will present evidence for the assertion that people from many cultures have created musical compositions using processes that are closely to mathematics. Our examples will focus on various types of symmetry, and will include pieces from both Western classical music and traditional African music. Musical thought and mathematical thought have much more in common than might be apparent at first glance, and mathematics can be used to understand the structure of music in very deep ways.
This workshop will address the question of how to get beyond teaching math skills in an online environment. We will explore ways to develop higher order thinking in math using features such as the discussion tool available in course management systems. The workshop is not only for those who teach courses online, but also for those contemplating teaching online and those who wish to provide web-enhancements to their on-ground courses.