Research Summary

List of Publications

• Lattices in topological groups. Arithmetic lattices. Number theory and algebraic structures (cyclic algebras with structure).

• Spectral geometry and generalizations. Heat invariants. Spectral optimization.

• Riemannian geometry and dynamics. Metrics on surfaces. Geodesic flow. Ergodic actions. Isometry groups.

• Geometric groups theory. Hyperbolic groups. Group actions. Interesting examples of groups. Limit groups.

• Number theory on groups. Quantitative problems on groups. Probabilistic methods. Profinite groups. Harmonic analysis on groups.

• Elementary problems. Convex geometry and graph theory. Optimization problems. Algorithms.

• Teichmuller theory, mapping class groups, and moduli spaces. Abelian varieties.

• Representation and character varieties. Group actions on varieties. Representations induced from geometric constructions.

• Hyperbolic geometry and generalizations. Associated groups. Covers of manifolds. Symmetric spaces and locally symmetric manifolds. Homogenous manifolds.