J. Arlo Caine
Visiting Assitant Professor
Ph.D in Mathematics, University of Arizona. 2007
B.Sc. in Applied Mathematics with a minor in Computer Science, California State University Hayward (now CSU East Bay). 2000
Office: 248 Hayes Healy Hall
Phone: (574) 631-8711
Fax: (574) 631-6579
For additional information see Arlo Caine’s Personal Page.
Poisson Geometry, Lie Theory, Mathematical Physics, Computational Topology, Teacher Education.
Poisson manifolds arise as phase spaces of families of classical mechanical systems or their reductions. Symmetries of the Poisson structure can sometimes be used to reduce the complexity of the mechanics and integrate the system. Poisson Lie groups arise as semi-classical limits of quantum groups and their geometry relates the representation theory of the classical and quantum groups. I study Poisson manifolds acted on by Poisson Lie groups such that the action map carries the product Poisson structure on the group cross the manifold to the Poisson structure on the manifold. This class includes Poisson structures which are invariant under the action of the group (when the Poisson structure on the group is zero) and many more general examples such as flag varieties of semi-simple groups and symmetric spaces. Such structures have found applications in representation theory, combinatorics, and integral formulas. I am also interested in infinite dimensional Poisson manifolds such as loop groups and loop spaces of symmetric spaces. In these settings, we use Poisson geometry to decompose the space into strata and determine local coordinate systems for calculations in quantum field theory.
Computational topology is a newly developed field of applied mathematics and computer science which attempts to discretize the continuous mathematics of algebraic topology in an effort to describe the shape of point-cloud data. Scatter plots are examples of two dimensional point clouds and scientists infer correlation between the variables from the shape of the cloud. One would like to do the same with higher dimensional data sets, but beyond dimension 3 one cannot examine it visually without using a projection and projections may lose information. Algebraic topology enables one to describe characteristic features of shapes in any dimension and computational topology attempts to bring these tools of description to bear on higher-dimensional point clouds. I am interested in developing software which implements these ideas and applying them to problems in text recognition, path planning in robotics, and big data analysis.
Teachers of science and mathematics at the elementary, secondary, and collegiate levels face a variety of professional challenges. Elementary teachers who are well-versed in child development and pedagogy for multiple subjects are often poorly trained in mathematics and science. Yet, with the vertical structure of these subjects, their work with students at the foundational level is very important and can have a lasting impact on student learning. Secondary and collegiate teachers of mathematics and science have greater expertise in their subjects but are often less informed with regard to teaching methods and curriculum design. I am Co-PI for the RESPeCT grant, a 5 yr, $7.8M project funded by the NSF which is testing an implementing a video-based analysis-of-practice professional development program for science teaching with the elementary teachers in the Pomona Unified School District (California). We are testing whether sustainable student and teacher learning gains can be achieved through the PD program when led by peer teachers in the district. At the secondary and collegiate level, I have also been involved in a number of curriculum design projects, most notably ConcepTests for Calculus (http://www.cpp.edu/~conceptests) at Cal Poly Pomona, authoring classroom voting questions for use in Calculus instruction.
- Toric Poisson Structures, Moscow Mathematics Journal, 11 (2) 2011, pp. 205-229
- (with D. Pickrell) Homogeneous Poisson Structures on Symmetric Spaces, International Mathematics Research Notices 2009, no. 1, 98-140
- Compact symmetric spaces, triangular factorization, and Poisson geometry, Journal of Lie Theory, 18 (2), pp. 273-294, 2008
- (with D. Pickrell) Noncompact groups of Hermitian symmetric type and factorization, Transformation Groups, to appear
- (with K. Roth, et al) The power of design principles for adapting video-based professional development programs to new contexts, AERA 2016, to appear.