Basilis Gidas Professor Division of Applied Mathematics Brown University email:  gidas@dam.brown.edu

RESEARCH INTERESTS

Past Research Interests

Probability Theory/Mathematical Physics: 
Probability theory on spaces of generalized functions. Gibbs distributions on spaces of tempered distributions. Construction of 2-D and 3-D quantum field theories. Renormalization of quantum field theory Hamiltonians. Spectral properties of Quantum Hamiltonians. Borel summability of ground states asymptotic expansions. 

Elliptic Partial Differential Equations/Differential Geometry:
Singular Solutions of the Yang-Mills equations.   Free boundary problems and Quark confinement. Symmetry properties, uniqueness, and a priori bounds of solutions of elliptic partial differential equations.   Classification of  singularities of conformal deformations of Riemannian metrics and other nonlinear elliptic equations.


Present Research Interests

Bayesian Statistics/Computer Vision/Speech Recognition:

Metropolis-type Monte Carlo simulation algorithms and simulated annealing. Simulation and optimization via the Langevin equation. Markov Random Field (MRF) estimation and consistency of pseudo-likelihood estimators, and of maximum likelihood estimators from complete or incomplete data. A variational method for estimating MRFs. Nonparametric estimation for continuous-time stochastic processes arising in speech recognition. Object identification via classification trees and stochastic grammars. Renormalization group methods for multiscale/multilevel image processing.  Texture representation via MRFs with polynomial interactions. Tracking of moving objects via particle filters. Speech signal representation via nonlinear transformations and wavelets. Classification and clustering of stop consonants via nonlinear transformation and nonlinear discriminant analysis.


Computational Molecular Biology: 
Probabilistic hierarchical/ syntactic models (analogous to  Chomsky grammars) for identifying, representing, and analyzing transcription regulatory networks and signal transduction pathways.  Identification of genes regulated directly and indirectly by combining microarray expression data, ChIp-chip data, and cross-species comparison information; identification of downstream pathways through which Myc functions in cell growth, cell-cycle proliferation, and apoptosis. Identifying phosphorylation sites motifs on the basis of tandem mass spectrometry data, protein-protein interactions, and structural information about kinases and substrates. Protein representation and ab initio folding via hierarchical/syntactic (also known as compositional) models.