Crystals, proteins, stability and isoperimetry

Abstract

Abstract. We attempt to formulate several mathematical problems suggested by structural patterns present in biomolecular assemblies. Our description of these patterns, by necessity brief and over-concentrated in some places, is self-contained, albeit on a superficial level. An attentive reader is likely to stumble upon a cryptic line here and there; however, things will become more transparent at a second reading and/or at a later point in the article.

Morse Spectra, Homology Measures and Parametric Packing Problems

“…Also it may be interesting to augment the k-volume by other (integral) invariants of Y = ψ −1 (T 0 ), where the natural candidates in the case of k-dimensional (mildly singular) Y = ψ −1 (0) would be curvature integrals expressing the kvolumes of the tangential lifts of these Y ⊂ X to the Grassmann spaces Gr k (X), k = dimY , of target k-planes in X (compare section 3 in [26]).…”

Algorithmic Aspects of Analysis, Prediction, and Control in Science and Engineering

“…In this section, following the ideas of a renowned mathematician M. Gromov [52], we use symmetries and similarities to show that under reasonable assumptions, the empirically observed shapes of cylindrical spirals and planes are indeed the best families of simple approximating sets. In this section, following the ideas of a renowned mathematician M. Gromov [52], we use symmetries and similarities to show that under reasonable assumptions, the empirically observed shapes of cylindrical spirals and planes are indeed the best families of simple approximating sets.…”

Moduli spaces and macromolecules

“…Gromov asks the question in [50], "Is there mathematics in biology?" and he then goes on to give affirmative examples.…”