What are Geometry and Topology?

Knowledge of geometry has always played a central role in science.

Above the gate to Plato's Academy in Athens, the world's first major university, was the inscription:

Let no man ignorant of geometry enter here.

and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker of apprehension than one who has not.

In the work of Newton and Einstein, new forms of geometry (Cartesian and Riemannian) enabled the formulation of new physical theories. Differential and algebraic geometry form the basis of current research in string theory and quantum field theory. Geometry also plays a key role in all areas of applied mathematics, from computer graphics and visualization to robotics and the properties of proteins.

Topology studies qualitative features of geometric objects, such as how many holes they contain and whether they are connected or knotted. These properties do not change when an object is stretched or squeezed, as long as it is not torn. Applications of topology are numerous, ranging from analysis of large data sets to determining the shape of the universe and the actions of enzymes on DNA.

Current interests of the members of the geometry-topology group include:

• Low-dimensional topology, knot theory and its applications to the mathematical biology.
• Riemannian and symplectic geometry.
• Geometry of convex polytopes and related problems in optimization.
• Algebraic geometry.
• Application of geometric ideas to mathematical physics.
• Geometric algorithms and their computational complexity.
• Applications of topology and geometry in biology and engineering

Benham has investigated the topological properties of protein structures and their functional and evolutionary correlates, and enumerated possible topological types that can arise.

Hass, in collaboration with Hutchings and Schlafly, solved the double-bubble conjecture in three dimensions. The conjecture, first noted by the ancient Greeks, asserts that among all closed containers in three dimensions that have two chambers with equal volume, a pair of round bubbles that meet at a flat face has the least total surface area.

Kapovich, in collaboration with Leeb and Millson, applied ideas and methods of geometry of nonpositively curved spaces to representation theory and theory of algebraic groups. For instance, these methods allow one to answer the following question: Suppose that A and B are symmetric n-by-n matrices with the given sets of eigenvalues. What are possible eigenvalues of A+B?

Kuperberg used topological methods to give a solution of an old problem about the number of n-by-n alternating-sign matrices.

De Loera, in collaboration with Below and Richter-Gebert, showed that finding the most economical triangulation of a 3-dimensional polytope is NP hard.

Mulase found a solution to the Riemann-Schottky problem by relating it to a system of partial differential equations called the KP hierarchy. Although the KP equation was originally proposed as a model for shallow water waves, several mathematicians independently discovered the equation and the hierarchy have a deep algebraic structure related to isospectral deformations of linear ordinary differential operators.

Schultens established a structure theorem for Heegaard decompositions of 3-dimensional graph-manifolds.

Schwarz in collaboration with Connes, Douglas, Nekrasov and others, have shown how to apply methods of noncommutative geometry to the problems of quantum field theory.

Thompson used the notion of "thin position" to establish that an algorithm put forward by Rubinstein decides if a given 3-manifold is homeomorphic to the 3-dimensional sphere.