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In this talk, I will discuss past research using concepts from algebraic topology and algebraic geometry in the form of invariants to define and classify the type of singularities of maps associated with inverse kinematics for spacecraft attitude control and robotic manipulation uses. The work presented here is related to the work on topological robotics by Michael Farber. Also to be discussed is the numerical issues associated with the calculation of pseudoinverses used for steering algorithms associated with singularity type of a given arrangement of a system of actuators.

In this talk I will discuss how a class of solutions of the mass transport problem are related to two key interests of Smale -- linear programming and dynamical systems. I will discuss the problem both in the finite-dimensional setting of adjoint orbits and in the infinite-dimensional setting of the semigroup of measure preserving transformations of the annulus. In both cases links will be made with the geometry of convex polytopes. This is joint work with Tudor Ratiu.

My talk reveals the potassium resistance R and sodium resistance RKNa shown in Figure 1 of the monumental 1952 paper which Hodgkin and Huxley had mis- identified as time-varying resistances , are erroneous , and must be reclassified as memristors , the fourth basic electrical circuit element . It has since been identified as synapses in neuromorphic system models, and as future workhorse for model parameter learning and massive AI computing. From a foundational perspective , I will use the memristor to resolve 4 heretofore unsolved classic problems in neuroscience , namely, the Galvani's irritability , dating back 243 years, the classic "all-or-none" mystery , the Turing Instability , and the Smale Paradox. My resolution of these unresolved problems was made possible by exploiting the Principle of local activity , which , within a certain relatively small parameter region, could harbour a physical state , dubbed edge of chaos. In my talk , I will provide the explicit formula for calculating, via matrix algebra , the precise parameter range where a nonlinear system is locally active , orenturing deeper into an often miniscule domain of edge of chaos . Unlike numerous unsuccessful attempts , such as Boltzmann's assay for decreasing entropy, Schrodonger's futile search for negative entropy, Prigogine's quest for the "instability of the
homogeneous", and Gell-Mann's musing on "amplification of fluctuations" , the principle of local activity provides an explicit formula to identify the parameter nook where edge of chaos reigns supreme.

Principal foliations of a surface embedded in three space are aligned with directions of maximal and minimal normal curvature. They are analogous to vector fields but are non-orientable. The singular points are umbilics where normal curvatures are constant in all directions. A century ago, Caratheodory stated the conjecture that compact convex surfaces have at least two umbilics. Several purported proofs of this conjecture have been presented, but there is still doubt about their veracity. This first part of this talk describes new results about this problem. Sotomayor and Gutierrez studied the analogy between principal foliations and two dimensional vector fields from a dynamical systems perspective. The second part of this talk describes results that extend their work, emphasizing the relationship of foliations with dense leaves to quasiperiodic flows on the two dimensional torus.

Oscillations are ubiquitous in biology. The science inspired by such phenomena has a wide range of motivations. Here I focus on the distinction between work that highlights the mathematical and/or “universal” structure of some phenomena and work intended to highlight the functional importance of biological detail.

No abstract available.