March 12 - 14, 2013

Wednesday, March 13 at 3:00pm

Location - Engineering 100 (Hammond Auditorium)

**Title:** Bifurcation and symmetry, a mathematical view on pattern formation in nature

**Abstract**: Patterns in Nature are not of so many types. The coat of a zebra is stripped while the coat of a leopard is spotted (and a cougar has a uniformly colored fur). Honey bees build incredibly regular hexagonal cells. Many plants or sea organisms present a high degree of symmetry, like the icosahedral shell of certain radiolarians. These quite simple patterns are extremely common, not only with living creatures but also in inanimate matter, think of the regular patterns in crystals like the cubic symmetry of salt for example, or the spiral patterns which can form on the heart muscle and provoke a heart attack. The common denominator of these examples is the underlying mathematics, which model the formation of regular patterns. Although more complex patterns have recently been observed, like quasi-crystals, the mathematical theory of pattern formation, which was initiated by the celebrated mathematician Alan Turing, is an example of the "unreasonable effectiveness of mathematics in natural science" as Nobel Prize winner Eugene Wigner used to say.

*Reception following lecture in 117 Weber Building from 4:00-5:30pm.*

Tuesday, March 12 at 4:00pm

Weber 237

Title: *Pattern formation and the bifurcation of heteroclinic cycles*

Abstract: Robust heteroclinic cycles (RHC) are flow-invariant bounded sets that naturally occurs in certain types of dynamical systems (typically in systems with symmetry). The presence of a RHC can explain intermittent switching between steady-states or periodic orbits, which are sometimes observed in physical experiments. Heteroclinic cycles in pattern formation systems can exist but are usually associated with codimension 2 (or higher) bifurcations. I shall show an example of a generic, codimension 1 bifurcation of robust heteroclinic cycles when the domain is the hyperbolic plane.

*Reception prior to talk in 117 Weber Building from 3:30-4:00pm.*

Thursday, March 14 at 11:00am

Weber 237

**Title:*** Pattern formation on compact Riemann surfaces and applications*

Abstract: Pattern formation on the sphere and torus has been widely studied in relation to the occurrence of periodic patterns in classical hydrodynamical systems and in biochemical models of reaction-diffusion equations. Recently a model for images texture perception by the visual cortex was introduced, which involves neural field equations posed on the hyperbolic plane. Looking for pattern formation in this non euclidean geometric context comes back to analyzing the bifurcation of patterns on compact Riemann surfaces of genus > 1. This leads to new and sometimes unexpected results, which open the door to a classification of patterns on Riemann surfaces.

*Tea Time reception prior to talk in 117 Weber Building 10:30-11:00am*

The Arne Magnus Lectures are given annually in the Department of Mathematics at Colorado State University in honor of Dr. Arne Magnus, our friend and colleague for 25 years.

The lectures are supported by the Arne Magnus Lecture Fund and the Albert C. Yates Endowment in Mathematics.

Contributions to the Magnus Fund are greatly appreciated and may be made through the Department of Mathematics. Please contact Sheri Hofeling (hofeling@math.colostate.edu) at at (970)-491-7047 for specific information.

All lectures are free and open to the public.