Research Topics
GRAPES supports 2 PhD positions in our Lab on Scientific / geometric computing and on Machine learning of 3D shapes. Applications are now open until they are filled and no later than Fall 2020! We also have an opening on geometric optimization, and imminent positions in scientific / geometric computing for microchip design.
PhD, MSc, and BSc theses can be undertaken in all of our Lab's topics (details below in English/Greek), or in the framework of existing projects.
Here is some administrative info for PhD students and instructions on applying for BSc/MSc theses (in Greek).
Geometric computing / Geometric modeling
 Sampling and volume estimation of (non)linear (non)convex regions (volesti software) by polytime methods; applications to scoring financial portfolios (our paper), predicting their composition, and to forecasting stockmarket crises via copulas.

There are various useful representations of geometric objects, most notably implicit, parametric, and point clouds. We develop efficient ways for switching between representations while exploiting properties such as sparseness. Specific operations of current interest include swept volume, surfaces of revolution and the development of robust predicates on them. Also searching and mining among such objects.
 Voronoi diagrams of complex objects and the corresponding algebraic operations for constructing them. Large datasets, nonlinear objects and applications (paper for ellipses).
Machine Learning
 Approximate Nearest Neighbors (ANN) and randomized projections (following our first paper): Queries lying in a subspace (bio data); ANN on complex 1D objects, e.g. time series, compare autocorrelation, exploit time parameter (vs. Frechet distance as in our paper); experimental validation of the Dolphinn approach, extensions to LSHable metrics beyond L1 and L2; farthest point in query direction; recovery for sparse coding and basis pursuit.
 Clustering of complex objects (possible representation by Neural Networks), IQMeans algorithm.
 Deep geometric learning for 3d shapes (representation, mining, classification, segmentation).
Algebraic algorithms / polynomials
 Discrete geometry for capturing and exploiting algebraic properties. Mixed volume and root bounds via the Permanent.
 Approximate Mixed volume via the volume approximation methods (telescoping method on mixed cells or fraction of "mixed" sample); sample of mixed cells for MV estimation and sample of root homotopies.
 Polynomial system solving by linear algeba. Sparse resultants and Macaulay formulae. Structured systems such as multihomogeneous systems, and syzygies.
Bioinformatics
 Molecular 3d structure: folding and docking. Geometric, statistical (sampling), and Machine Learning methods (geometric learning).
 Rigidity theory (counting rigid graphs), and Distance geometry on NMR data for ab initio structure prediction (in Greek).
Robotics
 Parallel robots, kinematics, calibration, collision avoidance. Applications to molecular kinematics
 Distance geometry and pose counting. Overconstrained algebraic systems and applications in calibration.