Topological

In this talk I will introduce some techniques for topological reasoning within the purview of graph search-based motion planning.

Our work with x-ray micro-CT images of complex porous materials has required the development of topologically valid and efficient algorithms for studying and quantifying their intricate structure. As an example, simulations of two-phase fluid displacements in a porous rock depend on network models that accurately reflect the connectivity and geometry of the pore space. These network models are usually derived from curve skeletons and watershed basins. Existing algorithms compute these separately and may give inconsistent results.

Pieces of string or curves in three dimensional space may be knotted or
braided. This topological tool can be used to study certain types of nonlinear
differential equations. In particular, such an approach leads to forcing
theorems in the spirit of the famous period three implies chaos for interval

In this talk we introduce the basic construction of topological RNA structures.
We introduce shapes and the associated shape polynomial and its connection to
RNA folding. We then establish the connection to unicellular maps and outline
the combinatorial constructions that facilitate genus induction. We furthermore
show applications of this framework to the uniform generation of RNA structures
of fixed topological genus and how to deal with RNA-RNA interaction structures.

The hippocampus plays an important role in representing space (for spatial navigation) and time (for episodic memory). Spatial representation of the environment is pivotal for navigation in rodents and primates. Two types of maps, topographical and topological, may be used for spatial representation. Rodent hippocampal place cells exhibit spatially-selective firing patterns in an environment that can be decoded to determine the animal’s location, heading, and past and future trajectory.

Interest in manifold learning for representing the topology of large, high dimensional nonlinear data sets in lower, but still meaningful dimensions for visualization and analysis has grown rapidly over the past decade, including analysis of hyperspectral remote sensing data.

I will discuss some stylized topological inference problems and give some information about Kolmogorov, sample and computational complexity. I will then discuss some tentative first steps towards a theory of feasibly computable invariants.

Persistent homology and persistent diagrams have been developed as tools of topological data analysis. They provide a robust topological characterization of a given (discrete) geometrical data. In this talk, I will present our recent researches on applying persistent diagrams to protein structural analysis. Topological characterizations of protein compressibilities and H/D exchanges will be explained in detail.

Over the last ten years, a number of methodologies have been developed which leverage topological techniques and ways of thinking to provide understanding of point cloud data. These include ways of measuring shape via homological signatures, topological mapping techniques, and applications of certain kinds of diagram constructions to collections of samples. I will survey these methods, and propose some direct connections of these ideas with mainstream machine learning.

The theory of smectic liquid crystals is notoriously difficult to study. Thermal fluctuations render them disordered through
the Landau-Peierls instability, lead to anomalous momentum dependent elasticity, and make the nematic to smectic-A transition
enigmatic, at best. I will discuss recent progress in studying large deformations of smectics which necessitate the use of nonlinear
elasticity in order to preserve the underlying rotational symmetry. By recasting the problem of smectic configurations geometrically