## Research Topics

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 poly-time 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 auto-correlation, exploit time parameter (vs. Frechet distance as in our paper); experimental validation of the Dolphinn approach, extensions to LSH-able 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), IQ-Means 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 multi-homogeneous 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.