The faculty in this group work collaboratively on many different projects, often involving vertically integrated teams of undergraduates, graduates, and post-docs. Here are some current examples.

• Gene Expression
  Harer and Mukherjee study the yeast cell cycle with Steve Haase in the Duke Biology Department. They are working to discover the transcriptional network that underlies the yeast cell cycle. Gene expression data collected at different times gives a high dimensional data set, and periodic genes give points distributed near to a circle. This data set is inherently noisy, includes non-periodic genes and has missing data. These issues are dealt with by a mixture of topological, statistical and biological tools. Software already exists to analyze the data and collaboration with biologists at Duke make this an effective and interesting project.

• Mathematics Meets Art History
  At Princeton, Daubechies mentored Josephine Wolff, a mathematics major whose senior thesis explores how mathematical analysis can capture the distinctive signature style of different artists. She started from high quality images of paintings by Gossen van der Weyden (GvdW), mastered wavelet analysis, and learned firsthand the advantages of dual-tree complex wavelet transforms for image analysis. Josephine looked at patterns of wavelet coefficients across different scales to capture signature features, and learned how techniques from machine learning can be used to generate strong classifiers. GvdW made extensive use, as was customary in his time, of preliminary drawings on the panel to be painted, typically in charcoal of blank ink; these can now be uncovered with infrared reflectometry. The style in these underdrawings has been hypothesized to be related to whether the mast or an apprentice did the underpainting, and art historians have developed a classification of these styles. The goal of the project was to see whether the features of the visible painting layer could be used to recover the classification by style, and using the machine learning Bloit Boost algorithm she achieved a remarkably good 90% rate. Further validation is under way and plans are being made by Prof. Martens from Ghent, who provided the original images, to incorporate this into his research. This will provide further opportunities for interesting projects for budding researchers.

• Social Networks
  Mukherjee is involved in a collaborative project in modeling social networks with Prof Banks in Statistical Science, Prof. Moody in Sociology and Prof. Alberts in Biology. Geometric approaches are being used to infer higher-order interactions, hypergraphs, and the dynamics of these interactions in two data sets: wikipedia links, and a group/social dynamics in a wild population that Prof Alberts has studied. Preliminary software for his problem exists. Prof. Willett is also studying social networks, especially their temporal evolution based on meeting or communication patterns. Preliminary studies of the email database for the three years surrounding the downfall of the notorious Enron Corporation (conducted by an undergraduate using methods developed by more senior lab members) shows that simply knowing who communicates with whom is a valuable predictor of what is being communicated. However, social networks can easily contain hundreds of thousands of people or more, resulting in very large data sets and requiring sophisticated yet scalable analysis methods.

• Forest Change
  Gelfand is involved in a project modeling forest dynamics on large spatial scales, e.g., the eastern United States. Forest change is best understood at the level of the individual tree. Individual-level survival, growth, fecundity, flowering, root development, etc., are much studied. Immediately, all of the foregoing issues arise in proceeding from individual level performance to forest scale response. Such a project provides an excellent example of collaboration between stochastic modelers (statisticians), geometric computing algorithm experts (computer scientists) and tree physiologists (ecologists) and an excellent opportunity for young researchers to gain knowledge and expertise across multiple fields.

• Volcanic Flows
  An example of a project with Wolpert that can involve both undergraduate and graduate students is the computer simulation of the frequency and magnitude of pyroclastic flows and the resulting flow levels at hundreds or thousands of locations, and the comparison of these to historical records. Students with a first undergraduate course in probability and a bit of experience in programming already have enough back- ground to do this under simplifying assumptions about the underlying physics, topography, geological pro- cesses, and their probability distributions; several Ph. D. students are now in the process of extending known theory and methodology to make more reliable risk predictions without those simplifying assumptions. Both the RHIC study and the volcanology project entail computer modeling, quantifying the uncertainty about the deterministic outcome of large slow computer models at thousands or millions of possible input values, most of them as yet untried; this too is an area where undergraduate and graduate students can work together on simple and (almost!) hopelessly complex versions of the same models, respectively.

• Voting Data
  An interesting recent problem being studied by Carin involves integration of document data with other data that may come in the form of an (incomplete) matrix. Specifically, he has looked at voting data in the US Senate and House, from the first congress to the present and learned the latent structure associated with legislatures and of legislations. Since the model is time evolving, one can examine how the latent characteristics of a senator/congressman evolve with time. Additionally, he has a database of the legislation (document) itself, and therefore can integrate nonparametric Bayesian topic models with matrix analysis, to examine how the legislation wording impacts its latent structure and hence the voting. He plans to also integrate speeches. Similar analyses may be performed for Supreme Court votes and cases (documents). The analysis allows one to infer the lowdimensional latent space (manifold) on which both legislatures and legislation reside.
  

• Tree-based Processes
  An example of a project with Schmidler concerns statistical inference and stochastic sampling of tree topolo- gies. Methods for efficient sampling of the space of trees are being developed for applications in statistical classification/regression trees, phylogenetic tree construction using 3D protein structures, and segmentation of blood vessels in retinal images (in collaboration with Carlo Tomasi of Duke CS). Currently one under- graduate and two graduate students are involved, one in each of the three applications; opportunities range from the highly applied to the computationally intensive to the highly theoretical (e.g. mixing times of non- random-walk Markov chains on complex tree distributions, or methods for a-stable processes (with Prof Wolpert) to compute marginal likelihoods for model selection).
  

• Expander Graphs and Compressed Sensing
  Mentored jointly by Calderbank and Willett, CS Graduate Student Sina Jafarpour derived performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. Jafarpour developed a new sensing paradigm based on expander graphs and proposed a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on signal reconstruction error. In fact it is the geometry of the expander graphs that makes them provably superior to random dense sensing matrices, such as Gaussian or partial Fourier ensembles, for the Poisson noise model. There is also some very recent work that uses expander graphs for sparse monitoring of data streams.