Modern algebra has its roots in the mathematics of the ancient world, arising out of the basic problem of solving equations. Following an explosive development in the twentieth century, it is now a vibrant, multi-faceted and wide-ranging branch of mathematics, having ties with almost every field of mathematics and computer science.
The interests of the algebra group at Northeastern include algebraic geometry, commutative algebra, representation theory, homological algebra, and quantum groups, with connections to combinatorics, singularities, Lie groups, topology, and physics.
- Anthony Iarrobino studies Artinian algebras, and the punctual Hilbert schemes, which are related to singularities of maps, and to combinatorics of partitions. He also studies pairs of commuting nilpotent matrices.
- Donald King studies nilpotent orbits associated with semisimple Lie groups as symplectic manifolds and algebraic varieties. He investigates the interplay of these properties with issues in representation theory.
- The research interests of V. Lakshmibai are the geometric & representation-theoretic aspects of flag varieties and related varieties. Flag varieties constitute an important class of homogeneous spaces. Being at the cross-roads of algebraic geometry, algebraic groups, commutative algebra, combinatorics & tepresentation-theory, the study of flag varieties (and related varieties) has important implications to these areas in mathematics.
- Ivan Losev works in Representation theory including that of reductive algebraic groups and Lie algebras, W-algebras, Symplectic reflection algebras, quantum groups and quantum Hamiltonian reductions. He is also interested in related subjects such as Kac-Moody categorifications, Hamiltonian group actions, Deformation quantization, combinatorics of crystals and geometry of symplectic resolutions of singularities.
- Alex Martsinkovsky works in homological algebra and representation theory, but is interested in other things as well: singularity theory, local algebra, abstract homotopy theory, deformation theory, noncommutative geometry, algebraic number theory, differential equations, integrable systems, and special functions.
- David Massey studies singular spaces, especially complex analytic singular spaces. Many of his results revolve around finding effectively calculable algebraic data that describe or control the singularities in the space. This work requires a great deal of commutative algebra and intersection theory. In addition, his frequent use of the derived category of constructible complexes of sheaves requires a large amount of homological algebra.
- Egon Schulte studies automorphism groups of geometric or combinatorial structures such as polytopes, complexes, tessellations, or graphs, as well as reflection groups.
- Alexander Suciu works in topology, and how it relates to algebra, geometry, and combinatorics: in particular he studies algebraic and topological invariants of hyperplane arrangements.
- Gordana Todorov works in representation theory of Artin algebras, Non-commutative algebra, Representations of quivers, cluster categories, cluster algebras and semi-invariants. Her joint paper Tilting theory and Cluster Combinatorics was a recent fast breaking aricle.
- Valerio Toledano Laredo works in representation theory, particularly loop groups and quantum groups. One of the themes of his research has been to explore how quantum groups describe the branching behaviour, or monodromy of solutions of certain systems of differential equations in the complex domain. More recently, he has explored the semiclassical aspect of quantum groups and uncovered, in collaboration with Tom Bridgeland of Sheffield University, a novel [and fascinating] dictionary between wall-crossing in Algebraic Geometry and Stokes phenomena for differential equations with irregular singularities in the complex plane.
- Jonathan Weitsman works in quantum field theory, combinatorics, symplectic geometry, and the role of mathematical physics in geometry and topology.