The algebra group has interests in commutative and non-commutative algebra, algebraic geometry, algebraic groups, Lie algebras and representation theory, and algebraic number theory.

The analysts in the Mathematics department reflect interests in harmonic analysis, complex analysis, potential theory, geometric measure theory, functional analysis, and summability theory. Several members of this group engage in research that is closely related to other groups in the department, such as Complex Analysis, Geometry, PDE and Applied Mathematics.

The complex variables group is relatively small, but very active with many visitors.

A diverse and energetic group of internationally recognized faculty at Indiana University pursues research in dynamical systems and ergodic theory. Many of these faculty have overlapping interests in geometry, complex analysis and/or probability. Areas of interest include complex dynamics and its generalizations, connections between hyperbolic geometry and conformal iteration, ergodic theory, hyperbolic and partially hyperbolic dynamics, geodesic flows including billiards and the Teichmuller flow, and dynamics and rigidity of large group actions.

Research interests include the interaction between curvature and topology, geometry of Lie group actions, analysis of geometric variational problems from PDE and geometric measure theory viewpoints (e.g. Einstein manifolds, minimal submanifolds, and the Laplacian), and smooth dynamics.

The logicians in the Mathematics department are part of the Indiana University Program in Pure and Applied Logic. Matching the interdisciplinary nature of logic, the program involves faculty members in Cognitive Science, Computer Science, History and Philosophy of Science, Informatics, Mathematics, and Philosophy. Taken together, IU has an impressive collection of logicians, mainly working on applied areas of the subject.

The mathematical physics group at Indiana University has a long and strong tradition. The interests of current mathematical physics group include 1) field theory for the four fundamental interactions, 2) theoretical cosmology and astrophysics, 3) statistical physics, and 4) classical and geophysical fluid dynamics. The main objectives of the study of the group is to derive experimentally verifiable laws of Nature based on a few fundamental mathematical principles, and to provide new insights to a number of challenging problems in theoretical physics. We focus on symbiotic interplay between modern physics and advanced mathematics.

The group in PDE, applied mathematics and computation is one of the strongest such groups in the country. Research covers a wide array of nonlinear phenomena and involves work ranging from pure analysis to scientific computation.

Faculty in Probability and Combinatorics are interested in random walks, probability on graphs, statistical mechanics, relations to geometric group theory, ergodic theory, and analysis, applications of probability to problems in physics and biology, and algebraic, extremal, probabilistic and topological combinatorics, with an emphasis on connections to other areas of mathematics. Their research is well known internationally, has produced many published papers, and has led to a highly acclaimed graduate textbook.

There is a very active group with interests including Geometric Topology (eg. Classical Knot Theory, 4-Manifold Theory, Higher Dimensional Knot and Link Theory, and Surgery Theory) and Algebraic Topology (eg. K-theory and Transformation Groups).