## Blyth 2003

### Department of Mathematics

University of Toronto

### Tenth Annual R.A. Blyth Lectures in Mathematics

### Professor Juergen Jost

Max-Planck Institute for Mathematics in the Sciences, Leipzig

and the Santa Fe Institute for the Sciences of Complexity

### Will give three lectures on Optimal Shapes and Structures

**Lecture 1**

Monday, March 31, 2003 at 4:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 2102

**Optimal Shapes: From the Beauty of Soap Films to the Construction of Light Roofs.**

Nature finds optimal forms and structures from the laws of physics or through evolutionary processes, whereas man constructs them by design. A particularly beautiful and structurally and mathematical rich class of examples of optimal geometric forms is presented by minimal surfaces. These surfaces exhibit shapes of minimal area under given constraints by balancing internal forces. Soap films assume the shape of a minimal surface, as do certain biological forms, and they inspire novel architectural constructions of light weight roofs. Minimal surfaces play also a fundamental role in string theory. We shall explain basic mathematical concepts for understanding the rich variety of shapes of minimal surfaces observed in nature or in computer graphics, and illustrate these concepts in turn through pictures and graphics.

**Lecture 2**

Tuesday, April 1, 2003, 4:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 5017A

**The Existence of Minimal Surfaces.**

We shall present a general mathematical framework for the problem of finding a minimal surface with given boundary (Plateau's problem). Combining ideas and tools from the calculus of variations, Morse theory, and differential and algebraic geometry, we can understand the variety of shapes, topological types, and bifurcation patterns exhibited by minimal surfaces for varying boundary conditions.

**Lecture 3**

Wednesday, April 2, 2003, 1:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 5017A

**Harmonic Maps between Metric Spaces: Analytic, Geometric, Algebraic, and Stochastic Aspects.**

We formulate a general problem of geometric equilibrium states, as maps minimizing a geometrically defined energy functional. Nonpositivity of the (generalized) curvature of the target space implies a fundamental convexity property of that functional. This leads to a fascinating interplay between analytic and geometric concepts, with implications for algebraic geometry and a new perspective on Dirichlet forms.

The Blyth Lecture Reception will follow the First Lecture on Monday, March 31, 2003, at the Faculty Club, 41 Willcocks Street.

All are invited to attend.