Exciting Math-Physics Interfaces

Newsletter 2003

EXCITING MATH-PHYSICS INTERFACES

STRING THEORY

In recent years, there has been much interaction between mathematics and physics in the area of string theory and M-theory. These are theories that unify all forces in the universe. The development of these theories uses many different branches of modern mathematics, including algebraic geometry, differential geometry, symplectic geometry, combinatorics, representation theory and so on. On the other hand, these physical theories give mathematicians new insights, which lead to many exciting and surprising conjectures and problems in mathematics. Among them are the structure of Gromov-Witten invariants, the mirror symmetry conjecture for Calabi-Yau manifolds and the geometry of G2 manifolds.

Our School of Mathematics has a very strong group of mathematicians working on the interface between string theory and mathematics. This includes Ionut Ciocan-Fontanine on Gromov-Witten theory, Naichung Conan Leung on Differential Geometry, Tian-Jun Li on Symplectic Geometry and Alexander Voronov on Representation Theory. Dunham Jackson Assistant Professor Junho Lee and several graduate students also work on such topics. There are many fruitful interactions among people in our group. There are also many seminar talks on this interface in our department. Furthermore, the theme of the next Yamabe Symposium in the Fall of 2004 is 'Geometry and Physics'. Speakers for the symposium include I.M. Singer, C. Vafa, S.T. Yau and other mathematicians and physicists.

Naichung Leung, Professor of Mathematics

INFINITESIMAL MACHINES DESIGNED BY THE MATHEMATICS OF SHAPE

Can the "infinitesimal machines" envisioned by Feynman be constructed from transformations between cubic and tetragonal lattice structures by the magnetostrictive crystal Ni₂MnGa? What kind of shape change can we construct if we coherently mix cubic and tetragonal cells, or cubic and orthorhombic cells? Professor Mitchell Luskin in the School of Mathematics is working with Professors Richard James (Department of Aerospace Engineering and Mechanics) and Chris Palmstrøm (Department of Chemical Engineering and Materials Science) to develop theory, computational methods, and experimental techniques to design and grow single crystal films as thin as 90 nanometers to build infinitesimal machines that utilize lattice transformations. The mathematics of shape change guides the search for the most effective atomic composition and crystallographic orientation of the film.

Mathematical and computational challenges are presented by the disparate space and time scales, from atomistic to continuum, needed to model small materials. Luskin, James, and Palmstrøm are utilizing ideas based on weak convergence and geometry to develop multiscale mathematical and computational methods. More details can be found at http://www.math.umn.edu/~luskin/.

Major contributions to this research program on the mathematics of shape change and related computational methods have been made by graduate students and postdocs in Luskin’s group, and a new program is being developed to provide an interdisciplinary experience in mathematics, science, and engineering for high school students in the University of Minnesota Talented Youth Math Program.

(a)Theory, (b) design, (c)computation, and (d)experiment for a small scale actuator. (a) The idealized two-dimensional crystal has gray square cells at high temperature and symmetry-related yellow and red rectangular cells at low temperature. (b) The film is flat and in the gray phase at high temperature. At low temperature, the film is tent-shaped as the upper and lower triangular regions of the film reversibly transform to red rectangular cells, and the left and right triangular regions reversibly transform to yellow rectangular cells (Bhattacharya & James). (c) Numerical model and simulation of a “melting” tent (Belik & Luskin). (d) Experiment with a single crystal CuAlNi film (Cui & James).

Mitchell Luskin, Professor of Mathematics