**Algebra and Number Theory:** Commutative and Homological Algebra (Blowup algebras, Hilbert functions, local cohomology, projective modules, complete intersections, Gorenstein rings). Algebras with involution, Galois cohomology of classical groups. Representations of algebraic groups. Reductive groups and related structures. Automorphic forms, Lfunctions, Diophantine equations.

**Analysis:** Functional Analysis (Operator Theory, unbounded subnormals, Hilbert modules, Operator algebras), Numerical Functional Analysis (Approximate solutions of operator equations and eigenvalue problems, spline theory), Real Analysis (Mean periodic functions, generalized integrals)

**Combinatorics:** Polyhedral combinatorics (approximation algorithms, combinatorial optimization) Enumeration (posets, generating functions, qanalogues), Graphs (colouring problems, eigenvalues, trees), Codes (linear codes associated to projective algebraic varieties), Design theory (finite geometries, symmetric designs and related structures), Extremal combinatorics. Theoretical computer science (computational complexity, analysis of algorithms).

**Geometry & Topology: **Algebraic Topology (stable homotopy theory), Differential Topology (harmonic manifolds, matrix varieties). Algebraic Geometry (Schubert varieties, determinantal varieties, varieties over finite fields), Lie groups (discrete subgroups, congruence subgroups, homogeneous spaces, probability measures), Ergodic theory.

**Partial Differential equations and Numerical Analysis:** Partial Differential Equations (Hyperbolic systems of quasilinear partial differential equations, nonlinear waves, partial integrodifferential equations, homogenization), viscoelastic fluid flow problems, Shock waves in hyperbolic systems of conservation laws. Numerical Analysis (Finite element methods, finite volume methods).

**Statistics and Probability:** Computational Biology (Biostatistics, Bioinformatics), Statistical data mining in proteomics (probabilistic optimization problems in Molecular Biology), Reliability Theory, Industrial Statistics, Construction of reliability test plans, Statistical Inference (Geostatistics, modeling bivariate distributions), Stochastic Differential Game Theory (Stochastic Control Theory, Mathematical Finance), Applied Probability, Statistical Inference. Linear and generalized linear models, design of experiments