# AISENSTADT CHAIR 2016

**Selim Esedoglu (University of Michigan)**

## SLIDESHOW OF THE LECTURE

## VENUE: Centre de recherches mathématiques

Pavillon André-Aisenstadt, Université de Montréal

Room 6254

## Monday, April 4, 2016 – 4:00 pm

*Algorithms for motion by mean curvature of networks of surfaces,
and applications*

Many applications in science and engineering call for simulating the evolution of interfaces (curves in the plane, or surfaces in space), including networks of them, under motion by mean curvature and related geometric flows. These dynamics arise as gradient descent for energies that contain the sum of (often weighted and anisotropic) surface areas of the interfaces in the network. Such applications include image processing, computer vision, machine learning, and materials science. I will focus on the central role motion by mean curvature plays in models of microstructural evolution in polycrystalline materials. These materials, which include most metals and ceramics, are composed of tiny single crystal pieces, called grains, stuck together. When a polycrystalline material is heated, the boundaries of the individual grains, which crisscross the material and constitute a network of surfaces, evolve to decrease its energy. The shapes and sizes of the grains resulting from this evolution is known to have substantial implications for the physical properties of the material, such as its conductivity and strength. I will describe a particularly efficient and almost miraculously simple class of algorithms, known as threshold dynamics or diffusion generated motion, that can be used to simulate motion by mean curvature for networks of surfaces. These algorithms allow simulating microstructural evolution in polycrystalline materials at an unprecedented scale. I will present simulation results and compare them to available experimental data.

**Lecture suitable for a general scientific audience**

*Coffee will be served at 3:30 pm and a reception will follow the lecture at the Salon Maurice-L'Abbé, Pavillon André-Aisenstadt (room 6245).
*

## Thursday, April 7, 2016 – 2:30 pm

## LIEU: Centre de recherches mathématiques

Pavillon André-Aisenstadt, Universite de Montreal

Room 5340

*Threshold dynamics for networks with arbitrary surface tensions*

Threshold dynamics, also known as diffusion generated motion, is a miraculously simple and extremely efficient algorithm for evolving surfaces, including networks of them, via mean curvature motion and related flows. It was proposed by Merriman, Bence, and Osher in 1992. Its efficiency stems from how it generates the entire dynamics, including the relevant conditions at free boundaries known as junctions (along which three or more surfaces meet) by merely alternating two simple and fast operations: Convolution with a kernel, and thresholding. The original algorithm applies only to networks in which the surface area of each interface is isotropic and is weighted equally. Extending the algorithm to the more general setting in which the surface area of each interface is weighted by a possibly different constant (known as surface tension) remained elusive. I will describe how to extend threshold dynamics to this level of generality, which is required by materials scientists, while maintaining its extreme simplicity and efficiency. The key is a new, variational formulation of the original algorithm. This is joint work with Felix Otto.

## Coffee Break 3:30 pm

## 4:00 pm

##
*Threshold dynamics for anisotropic surface energies*

Extending threshold dynamics to anisotropic surface energies -- where the surface tension depends on the direction of the normal to the interface -- has been extensively studied, starting with the original paper of Merriman, Bence, and Osher in 1992. Yet many questions remain, such as the class of anisotropies that can be handled by this approach. A new, variational formulation of the basic threshold dynamics algorithm sheds new light on these questions.