Seminar

The study of straightedge-and-compass geometric constructions goes back to ancient geometry. Many centuries later, the mathematicians became interested in straightedge-only constructions. Recall that a straightedge (a.k.a. a ruler without marks) is a device that can be used to connect any two points by a line but not to measure distances or draw parallel lines. Surprisingly, many things can be achieved with only a straightedge: for example, one can construct tangent lines to a given circle.

In this talk, we will discuss a ''new'' way to define an integral. Instead of using the standard definition of $\displaystyle \sum_{i=1}^{n} f(x_{i}) \hspace{3pt} \Delta x_{i}$, can we use an infinite product? How will this change the definition of an integral? Will this also change the definition of a derivative? This talk will examine the new mysteries of the so-called ''star-integral'' and ''star-derivative''.

 We focus on the interaction of an incompressible fluid and an elastic structure. Two cases are considered: 1. the elastic structure covers part of the fluid boundary and undergoes small displacement and 2. the elastic structure is immersed in the fluid and features large displacement. For the first case, we propose an Arbitrary Lagrangian-Eulerian (ALE) method based on Lie’s operator splitting.