Title: Convexity, star-shapedness, and multiplicity of numerical range and its generalizations

Abstract:  Given an $n\times n$ complex matrix $A$, the classical numerical range (field of values) of $A$ is the following set associated with the quadratic form:
$$W(A) = \{x^*Ax: x*x=1, x\,\text{ is a complex }\, n\text{-tuple}\}$$We will start with the celebrated Toeplitz-Hausdorff (1918, 1919) convexity theorem for the classical numerical range. Then we will move on to introduce various generalizations and we will focus on those in the framework of semisimple Lie algebras and compact Lie groups. In our discussions, results on convexity, star-shapedness, and multiplicity will be reviewed, for example, the results of Embry (1970), Westwick (1975), Au-Yeung-Tsing (1983, 84), Cheung-Tsing (1996), Li-Tam (2000), Tam (2002), Dokovic-Tam (2003), Cheung-Tam (2008, 2011), Carden (2009), Cheung-Liu-Tam (2011) and Markus-Tam (2011). We will mention some unsolved problems.