Georgia Institute of Technology College of Sciences

Rank reduction as an effective technique for dimension reduction is widely used in statistical modeling and machine learning. Modern statistical applications entail high dimensional data analysis where there may exist a large number of nuisance variables. But the plain rank reduction cannot discern relevant or important variables. The talk discusses joint variable and rank selection for predictive learning. We propose to apply sparsity and reduced rank techniques to attain simultaneous feature selection and feature extraction in a vector regression setup. A class of estimators is introduced based on novel penalties that impose both row and rank restrictions on the coefficient matrix. Selectable principle component analysis is proposed and studied from a self-regression standpoint which gives an extension to the sparse principle component analysis. We show that these estimators adapt to the unknown matrix sparsity and have fast rates of convergence in comparison with LASSO and reduced rank regression. Efficient computational algorithms are developed and applied to real world applications.

Mechanical forces play a key role in the shaping of versatile morphologies of thin structures in natural and synthetic systems. The morphology and deformation of thin ribbons, plates and rods and their instabilities are systematically investigated, through both theoretical modeling and table-top experiments. An elasticity theory combining differential geometry and stationarity principles is developed for the spontaneous bending and twisting of ribbons with tunable geometries in presence of mechanical anisotropy. Closed-form predictions are obtained from this theory with no adjustable parameters, and validated with simple, table-top experiments that are in excellent agreement with the theoretical predictions. For large deformation of ribbons and plates, a more general theory is developed to account for mechanical instability (slap-bracelet type) induced by geometric nonlinearity, due to the competition between inhomogeneous bending and mid-plane stretching energy. This comprehensive, reduced parameter model leads to unique predictions about multistability that are validated with a series of table-top experiments. Furthermore, this study has been extended to interpret a different type of snap-through instability that the Venus flytrap has been actively employing to capture insects for millions of years, and the learnt principle is used to guide the design of bio-mimetic flytrap robot.

The talk focuses on positive equilibrium (i.e. time-independent)solutionsto mathematical models for the dynamics of populations structured by ageand spatial position. This leads to the study of quasilinear parabolicequations with nonlocal and possibly nonlinear initial conditions. Weshallsee in an abstract functional analytic framework how bifurcationtechniquesmay be combined with optimal parabolic regularity theory to establishtheexistence of positive solutions. As an application of these results wegivea description of the geometry of coexistence states in a two-parameterpredator-prey model.

In this talk we introduce the notions of the contact angle and of the holomorphic angle for immersed surfaces in $S^{2n+1}$. We deduce formulas for the Laplacian and for the Gaussian curvature, and we will classify minimal surfaces in $S^5$ with the two angles constant. This classification gives a 2-parameter family of minimal flat tori of $S^5$. Also, we will give an alternative proof of the classification of minimal Legendrian surfaces in $S^5$ with constant Gaussian curvature. Finally, we will show some remarks and generalizations of this classification.