Given a set $K$ in a metric space $M$ one may ask when is $K$ contained in
$\Gamma$, a connected set of finite 1-dimensional Hausdorff length, and
for estimates on the minimal length of such a $\Gamma$ (up to
multiplicative constants). This was first answered for $M$ = the
Euclidean plane by P. Jones and extended to $M=R^d$ by K. Okikiolu.
Recently, there have been several new relevant results (by several people)
for $M$ being: a Hilbert space, the Heisenberg group, and a general metric
space. for some of these one restricts the discussion to $K$ and $\Gamma$
in specific categories. In some of these categories which we will discuss
the anwswer is in IFF form, whereas in others it is not. The answer to
this question usually comes together with a multiresolutional analysis of
the set $K$ and a construciton of a $\Gamma$ containing $K$ which is not
'too long'. Essentially no prior knowledge in analysis is needed to
understand this talk.

SCIENCES VIEW OF BIOLOGYS FRONTIERS: The Editorial Board of Science celebrated Sciences 125th Anniversary by posing 125 critical questions on What Dont We Know?. Those 125 described crucial gaps in knowledge that we have a chance of filling, or knowing how, in the next 25 years. Complex Systems Models for Cell Cycles of Life and Existence of Uncertainty beyond our knowledge provide tools for advancing 3 of the 4 biology questions: What Determines Species Diversity? Why Do Humans Have So Few Genes? How Will Big Pictures Emerge from a Sea of Biological Data?

OUR SYSTEMS MODELS: Our Systems Models for Protein Cycle of Life and (Draft) Natures Dance of Life are revolutionary in form and comprehensive yet comprehendible, and probably the first such models. General form of Protein Model has five Stages, with each having inputs, outputs and interactions: Cellular Needs Specification + Cellular Needs Definition, Cellular Needs Transmission + Chromatin Remodeling, Transcription + Splicing & Processing, Translation + Secondary Structuring, and Tertiary Structuring + Feedback Regulation. Protein Cycle, Cell Cycle, Organism Cycle and Evolution in the Environment are partners in Natures Dance of Life, while also playing Uncertain Game against Nature to either repeat the Determined Part or spread more Uncertain Diversity.

TO BE AND NOT TO BE: A Cycles Complexity increases as its number of Process Stages or Repetitions do, and creates more Diversity about Determinism. Determinisms Certainty needs a bridge to Cell Behavior, and Uncertainty is the keystone of that bridge:
Cellular Behavior = Determined Part + Determinable Part + An Uncertain Diversity.
As the number of Stages and Repetitions increases, Determined becomes less likely (as a product of Stage and Repetition probabilities), while Diversity becomes more likely (as a sum of Stage and Repetition probabilities) and has increasing Variance. If our question to Nature is To be or not to be, then Natures answer to us is To be and not to be.

UNCERTAINTY BEYOND KNOWLEDGE: Uncertainty is due to complexity of all Cell Cycles, and is generated by cascading effects of intricate processes compounded with interactions. This produces order so complex that it lies beyond our abilities to grasp, which is closely related to chaos theory and its mathematics. An awareness of Uncertainty is increasing in the world, as there was an Uncertainty Session at 2005 World Academy of Arts and Sciences Congress in Zagreb. Is it not as plausible to believe in an Uncertain world with growing areas of known Certainty, as it is to believe in a Certain world with shrinking areas of unknowable and inherent Uncertainty? Democritus said it first and best: Everything existing in the universe is the fruit of chance and necessity.

In this talk, I will discuss the relative $K$-stability and modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds. I will show a sufficient condition in the sense of polyhedrons associated to toric manifolds for both relative $K$-stability and modified $K$-energy. In particular, our result holds for toric Fano manifolds with vanishing Futaki invariant. We also verify our result on toric Fano surfaces.