# Analysis Seminar by Lihang (Leon) Liu (UNM)

### Event Description:

**Title: **Cantor set and Space Filling Curve.

**Abstract: ** The ternary Cantor set *C*, constructed by George Cantor in 1883, probably is the best-known example of a perfect nowhere-dense set in the real line. In this talk, I am going to give the basic properties of *C* and study in detail the ternary expansion characterization of *C*. We then consider the Cantor-Lebesgue function F defined on *C*, prove its basic properties and study its continuous extension to [0,1]. We also consider the geometric construction of F as the uniform limit of polygonal functions. Next, we consider the Lebesgue's function defined from *C* onto [0,1]^2 and onto [0,1]^3, as well as their continuous extension to [0,1], i.e. obtained the Lebesgue's space filling curves and Netto's theorem. Last but not least, we discuss, Hausdorff's theorem, which is a natural generalization of the definition of Lebesgue's functions, it states that any compact metric space is a continuous image of *C*.

**About the Speaker: **Lihang (Leon) Liu is a second-year PhD student majoring in Pure Math at UNM. He finished his bachelor's degree in China at Xi'dian University, majoring Telecommunications Engineering, and then he switched to Mathematics and finished his master's degree at Roosevelt University in Chicago under the direction of Professor Wilfredo Urbina.