Georgia Institute of Technology College of Sciences

This systematic review of mathematics educational technology literature identified 1356 manuscripts addressing the integration of educational technology into mathematics instruction. The manuscripts were analyzed using three frameworks (research design, teacher knowledge, and TPACK) and four supplementary lenses (Data sources, outcomes, NCTM Principles, and NCTM Standards) to produce a database to support future research syntheses and meta-analyses. Preliminary analyses of student and teacher outcomes (i.e., knowledge, cognition, affect, and performance) suggest that graphing calculator and dynamic geometry technologies have been abundantly studied, but the strength of the evidence measures (i.e., validity and reliability) may be lacking. More specifically, research on mathematics educational technology appears at first glance to be ubiquitous, the usefulness of this research to practitioners and researchers is limited by lack of attention to research design and validity, reliability, and threats to validity (Rakes et al., 2011). Additionally, much of the research appears to be unorganized, with topics such as graphing calculators studied often, while other topics such as virtual manipulatives understudied (Ronau et al., 2010).

Steady fluid solutions can play a special role in characterizing the dynamics of a flow: stable states might be realized in practice, while unstable ones may act as attractors in the unsteady evolution. Unfortunately, determining stability is often a process substantially more laborious than computing steady flows; this is highlighted by the fact that, for several comparatively simple flows, stability properties have been the subject of protracted disagreement (see e.g. Dritschel et al. 2005, and references therein). In this talk, we build on some ideas of Lord Kelvin, who, over a century ago, proposed an energy-based stability argument for steady flows. In essence, Kelvin’s approach involves using the second variation of the energy to establish bounds on the growth of a perturbation. However, for numerically obtained fluid equilibria, computing the second variation of the energy explicitly is often not feasible. Whether Kelvin’s ideas could be implemented for general flows has been debated extensively (Saffman & Szeto, 1980; Dritschel, 1985; Saffman, 1992; Dritschel, 1995). We recently developed a stability approach, for families of steady flows, which constitutes a rigorous implementation of Kelvin’s argument. We build on ideas from bifurcation theory, and link turning points in a velocity-impulse diagram to exchanges of stability. We further introduce concepts from imperfection theory into these problems, enabling us to reveal hidden solution branches. Our approach detects exchanges of stability directly from families of steady flows, without resorting to more involved stability calculations. We consider several examples involving fundamental vortex and wave flows. For all flows studied, we obtain stability results in agreement with linear analysis, while additionally discovering new steady solutions, which exhibit lower symmetry. Paolo is a candidate for J Ford Fellowship at CNS. To view and/or participate in the CNS Webinar from wherever you are: evo.caltech.edu/evoNext/koala.jnlp?meeting=MeMMMu2M2iD2Di9D9nDv9e

Oscillatory integrals arising as Fourier transforms of local and global height functions play an important role in the spectral analysis of height zeta functions. I will explain a general geometric technique which allows to evaluate such integrals. This is joint work with A. Chambert-Loir.

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham;  The Georgia Institute of Technology;  Emory University;  The University of Tennessee Knoxville.  The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. Catherine Williams (Columbia U);  Hugh Bray (Duke U);  Simon Brendle (Stanford U);  Spyros Alexakis (U of Toronto);  Alessio Figalli (U of Texas at Austin).   There will also be an evening public lecture by plenary speaker Hugh Bray (Duke U) entitled From Black Holes and the Big Bang to Dark Energy and Dark Matter: Successes of Einstein's Theory of Relativity.