Algebra and Geometry

The research interests of our group cover a wide range of topics on the interface between algebra and geometry.

Geometric objects can be conveniently encoded using algebra. For example, a circle or ellipse in the plane can be described by a quadratic polynomial equation. Higher degree polynomials give more interesting spaces called "algebraic varieties", which we can study using tools from ring theory, or using more advanced tools like derived categories. The symmetries of a geometric object form a group, which can be studied using pure algebra and representation theory.

Conversely, geometric techniques can often be applied to algebraic objects. For example, you can often deform algebraic objects in families called moduli spaces, which have natural notions of geometry and topology. Or you can deduce something about a group by letting it act on a well-understood metric space.

In our department, the algebra and geometry group conducts research mainly in:

• representation theory of groups and algebras,
• group theory and geometric group theory,
• Lie algebras and algebraic groups,
• homotopical and homological algebra,
• algebraic topology,
• deformation theory,
• noncommutative geometry,
• symplectic topology.