# Mathematics Colloquium Winter 2010

Jack Baskin Engineering Room 301A

Refreshments served at 3:40

For further information please contact the Mathematics Department at 459-3158

**January 12, 2010**

**Approximate Groups**

**Ben Green**

University of Cambridge

University of Cambridge

The notion of an approximate group is a central one in additive combinatorics. My talk will be in three parts, in which I aim to answer the following questions: 1. What is an approximate group? 2. What is known about the structure of approximate groups? 3. What applications do they have? I hope to mention some very recent progress (with E. Breuillard, T. Sanders and T. Tao) concerning point 2. I will make strenuous efforts to keep the talk accessible to a general mathematical audience.

**February 9, 2010**

**Early applications of a calculus of differentials**

**Early applications of a calculus of differentials**

**Michael Nauenberg, Emeritus**

UCSC Physics Department

UCSC Physics Department

**February 16, 2010**

**The Geometry of the Normal Euler Number for Smooth and Polyhedral Surfaces in Three-Space (and Four-Space)**

**The Geometry of the Normal Euler Number for Smooth and Polyhedral Surfaces in Three-Space (and Four-Space)**

**Thomas Banchoff**

Brown University

Brown University

The normal Euler number for a curve on an orientable or non-orientable smooth surface immersed in Euclidean three-space is the mod 2 number of intersections of the surface and the curve deformed along a generic normal vector field. We compare two definitions of the normal Euler number that are geometric enough to work as well for polyhedral surfaces, involving inflection faces of polyhedral strips. We then discuss generalizations of these ideas for normal Euler numbers of smooth and polyhedral surfaces in four-space. This presentation will be illustrated by interactive computer graphics.