I will discuss three different recent results in fixed point and Banach space theory. Aspects of each were strongly influenced by Brailey's life and career as an extremely fine person, mathematician, teacher and mentor.

Parts (1), (2) and (3) were all partially inspired by Brailey's wonderful seminar series at Kent State University in 1986, entitled: ``The Existence Question for Fixed Points of Nonexpansive Maps." As a new Ph.D. student at KSU, meeting and interacting with Brailey during my first year was one of the highlights of my Ph.D. studies - and my life, both generally and mathematically.

Further, my interest in characterizing the fixed point property in $c_0$, which led in part to Item (3), was initiated by a paper of Enrique Llorens-Fuster and Brailey: ``The fixed point property in $c_0$", Canadian Mathematical Bulletin 41(4) (1998), 413-422.

1. C. Lennard and V. Nezir have proven that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings. (Nonlinear Analysis 95 (2014), 414-420.)

   More recently, C. Lennard, V. Nezir and L. Piasecki (in preparation) have proven that an arbitrary Banach space is reflexive if and only if it has an equivalent norm that has the fixed point property for mappings of cascading nonexpansive type. This second new class of mappings strictly includes both nonexpansive mappings and cascading nonexpansive mappings.

2. J. Burns, C. Lennard and J. Sivek have proven the existence of a contractive mapping on a weakly compact convex set in a Banach space that is fixed point free. This answers a long-standing open question. (Studia Mathematica 223(3) (2014), 275-283.)

3. T. Gallagher, C. Lennard and R. Popescu show that there exists a non-weakly compact, closed, bounded, convex subset $W$ of the Banach space of convergent sequences $(c, \|\cdot\|_{\infty})$, such that every nonexpansive mapping $T : W \longrightarrow W$ has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett.

   This is also the first example of a non-weakly compact, closed, bounded, convex subset $W$ of a Banach space $X$ isomorphic to $c_0$, for which $W$ has the fixed point property for nonexpansive mappings. (J. Mathematical Analysis and Applications 431 (2015), 471-481.)

A semitopological group (topological group) is a group endowed with a topology for which multiplication is separately continuous (multiplication is jointly continuous and inversion is continuous). In this talk it will be shown that certain topological properties of a semitopological group automatically imply that it is, in fact, a topological group. In particular, it will be shown that every Cech-complete semitopological group is a topological group. This proof relies upon the use of topological games. A technique that was first introduced to the speaker by Brailey Sims back in May of 1990.

Having a top-notch research mathematician as a father is an interesting and, obviously, formative experience. For example, the mathematical puns don't get appreciably better; but your sense of humour gets steadily worse until, eventually, you think they are hilarious. And that's symptomatic, because I think that what makes an excellent mathematician, far beyond mathematical ability, is a combination of a deep thirst for knowledge, a wonderfully active imagination, and the sort of ability to squint at a concept from a hundred different angles that leads inevitably to those awful puns. I'm going to talk a bit about growing up around Dad; about some of the ways in which he passed on that wonderful imagination and curiosity and squinty-angled way of peering deep into the insides of everything; and about some of the ways I keep seeing his deep influence I myself — and not just expressed as bad puns.