Tenured, Tenure Track & Permanent Faculty

Research Interests

Inverse Problems, Uncertainty Quantification, Computational Mathematics, Computational Statistics

Research Interests

Analysis

Research Interests

Geometric topology, especially knot theory and hyperbolic 3-manifolds.

I also used to be a sculptor and printmaker.  Most of my artwork was designed using random numbers and elementary mathematics.

Research Interests

Mathematical & Applied Statistics

Research Interests

Applied Mathematics, Asymptotic Methods

Research Interests

Discrete Mathematics, Optimization, Theoretical Computer Science

Research Interests

Finite dimensional division algebras, the Brauer group, valuation theory and algebraic geometry

Research Interests

Combinatorics, Matroids, Graph Theory

Research Interests

Graph theory and combinatorics

Research Interests

Applied statistics, classification

Research Interests

My research explores the educational implications of treating mathematics as a human activity. I take the perspective that all human activity is inexorably bound with culture: activity is "mediated" by cultural artifacts, and cultural artifacts are produced through human activity. I therefore take a cultural perspective on learning, studying the mutually constitutive nature of mathematical artifacts—including models, tools, strategies, representations, algorithms, and notation systems—and mathematical activity. I explore the ways in which artifacts are reinvented and made-meaningful in activity, and the ways in which these artifacts mediate future activity. 

"Mediation" is a complicated word for a fairly simple idea. Here is the idea: Human activities take place in a cultural milieu and therefore thay always incoprorate pieces of prior human activiites. Thus human actions “involve not a direct action on the world but an indirect action, one that takes a bit of material matter used previously and incorporates it as an aspect of action” (Cole & Wertsch, 1996, p. 252). Here, the “bit of material matter” is a cultural artifact. In recruiting thses artifacts into activity, we distribute the labor across these artifacts. This is surely true for physical actions—consider how the activity of pole vaulting, for example, is distributed across the pole vaulter and the pole—but it is also true for cognitive actions. What this means is that I believe cognition is not something that happens “in the head”, but rather is distributed across resources in the environment. As Edwin Hutchins explains, “humans create their cognitive powers in part by creating the environments in which they exercise those powers.”

Broadly, then, I am interested in the ways that cognition gets distributed in math classrooms; how artifacts—such as models, tools, representations, and symbol systems—mediate learning in mathematics. To study this, I conduct classroom design experiments in high schools. My most-recent study took place in an Algebra II classroom, and explored how students learn quadratic functions. Prior work has explored how students develop a robust understanding of slope, and how students come to understand fractions-as-division.

Artifacts do more than meditate activity. As humans act with artifacts, the artifacts “act back” on the human actors, such that humans are constituted by culture. Thus, I am also interested in how students become particular kinds of people, and the ways that mathematics mediates students' educational and biographical trajectories. To study this, I conduct ethnographic work in educational and community settings.

Finally, I am interested in educational measurement and psychometrics. In this line of research, I interrogate the design principles behind educational assessments, and to explore ways of bringing learning theories to bear on the design of assessments. For example, if students "create their cognitive powers in part by creating the environments in which they exercise those power," then what does this mean for "traditional" tests which—by design—occur in cognitively impoverished environments?

Research Interests

Mathematics Education

Research Interests

My current research interests are focused on the development of statistical methods for high dimensional  data analysis.  High dimensional data problems arise in various fields such as functional neuroimaging, DNA and metagenomic microbial communities sequencing.  I worked on fMRI voxel-wise connectivity analysis problems using wavelet shrinkage denoising methods and large covariance matrix estimation for dependent data.  My current research projects involve algorithmic and eigen-decomposition based dimension reduction techniques for next generation microbiome sequencing data visualization and health outcomes prediction. I also work on evaluating the effect of  data pre-processing bias on statistical analysis.

I collaborate with microbiologists, statisticians and mathematicians to achieve both biologically relevant and mathematically justified  solutions to these interdisciplinary problems.  As a part of our work, we provide software solution and data analysis methods manuals  to our collaborators working in clinical and translational research.

Research Interests

Differential Geometry

Research Interests

Creativity; Learning Sciences; Interdisciplinarity (Math-Art-Science); International Mathematics Education; Epistemology; Encyclopaedism

Research Interests

Functional Analysis

Research Interests

Predictive analytics

Research Interests

Applied Math and Dynamical Systems

Research Interests

Complex & Functional Analysis, Operator Theory

Research Interests

Algebra, in particular non-commutative ring and invariant theory.

Research Interests

Mathematics Education