# Geometry

Geometry is concerned with the shape, size, and orientation of objects in space, and indeed such properties of space itself. The particular objects studied and the tools used in investigating their properties create subfields of geometry, such as *algebraic geometry* (which generally uses tools from algebra to study objects called algebraic varieties that are solution sets to algebraic equations) and *differential geometry* (which generally uses tools from analysis to study objects called manifolds that generalize Euclidean space). Another example is analytic geometry (which generalizes algebraic geometry by considering spaces and maps defined locally by analytic functions). Other subfields of geometry represented in our Department include *discrete geometry* (which studies combinatorial properties of finite or discrete objects) and *symplectic geometry* (which studies objects with structure generalizing that of the phase space of certain dynamical systems).

**Algebraic Geometry **

Anthony Iarrobino works on secant bundles and the punctual Hilbert scheme. V. Lakshmibai studies the geometric aspects of flag varieties and related varieties. Alina Marian works on moduli theory in algebraic geometry. Alex Suciu studies the topology of algebraic varieties. Ana-Maria Castravet works on algebraic geometry, with focus on birational geometry and moduli spaces, arithmetic geometry, combinatorics, and computational algebraic geometry. Emanuele Macri works on algebraic geometry, homological algebra and derived category theory, with applications to representation theory, enumerative geometry and string theory.

**Differential Geometry **

Chris Beasley works on gauge theory, as well as problems concerning manifolds with special holonomy. Maxim Braverman works on various problems in differential geometry including analytic torsion. Robert McOwen has applied nonlinear PDEs to the study of conformal metrics and scalar curvature on noncompact Riemannian manifolds. Peter Topalov applies various analytic techniques to problems in Riemannian geometry.

**Singularities in Analytic Geometry **

Terence Gaffney studies the topology and geometry of singular spaces and maps, in the smooth, real analytic, and complex analytic settings, with the equisingularity of sets and maps being a particular interest. David Massey works on stratified spaces and the local topology and geometry of singular analytic spaces, making heavy use of the derived category and the Abelian category of perverse sheaves.

**Discrete/Combinatorial Geometry **

Egon Schulte works on discrete geometry, with an emphasis on combinatorial aspects and symmetry.

**Symplectic Geometry **

Jonathan Weitsman works on symplectic geometry, and the role of mathematical physics in geometry and topology.